CN111341104B - Speed time-lag feedback control method of traffic flow following model - Google Patents

Speed time-lag feedback control method of traffic flow following model Download PDF

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CN111341104B
CN111341104B CN202010142358.3A CN202010142358A CN111341104B CN 111341104 B CN111341104 B CN 111341104B CN 202010142358 A CN202010142358 A CN 202010142358A CN 111341104 B CN111341104 B CN 111341104B
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靳艳飞
孟经伟
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Beijing Institute of Technology BIT
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    • G—PHYSICS
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    • G08G—TRAFFIC CONTROL SYSTEMS
    • G08G1/00—Traffic control systems for road vehicles
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Abstract

The invention discloses a speed time-lag feedback control method of a traffic flow following model, belonging to the field of traffic flow control. According to the method, a classical optimization speed following model is established under a traffic flow system, and the stability of the optimization speed following model is determined; a classical optimization speed following model is improved through a speed time-lag feedback control method, and the stability condition of the improved following model containing a time-lag feedback control item is determined by combining the control theory; determining the value range of the feedback gain coefficient; solving a corresponding stable time-lag interval according to a stability switching principle of a time-lag differential equation and a constant integral stability analysis method; the controller is designed according to the feedback gain coefficient and the stable time-lag interval, so that the unstable parameter combination of the optimized speed following model becomes stable, the stable parameter combination interval of the original optimized speed following model is expanded, the stability of the following model is improved, traffic jam is inhibited, the traffic trip efficiency and the traffic resource utilization rate are improved, and the stability of a traffic flow system is effectively improved.

Description

Speed time-lag feedback control method of traffic flow following model
Technical Field
The invention relates to a traffic flow control method, relates to a speed time-lag feedback control method of a traffic flow following model, and belongs to the field of traffic flow control.
Background
The vehicle following model is a dynamic system composed of a large number of vehicles, has dynamic characteristics such as multi-parameter, coupling, nonlinearity and uncertainty, and is very complex in modeling and automatic control problems. The existing effective control method is to add a time-lag feedback control item at the right end of a following model to improve the stability of uniform traffic flow and achieve the purpose of inhibiting traffic jam.
Asymptotic stability of traffic flow is associated with the movement fluctuations of any vehicle and the propagation of a series of vehicle behaviour fluctuations. The adaptive cruise control system may collect distance and speed differences from the vehicle in front and the corresponding controller may perform acceleration or deceleration. Controllers are now installed in some autonomous, intelligent vehicles to help avoid traffic congestion. There are many control methods in the control device. To ensure linear stability of the traffic flow, different control strategies have been proposed to alleviate traffic congestion and improve the efficiency of the traffic flow. The most classical method is a displacement time-lag feedback control method, and traffic jam is restrained in an optimized speed following model. Based on the displacement time-lag feedback control method, later people also develop some improved control methods to inhibit traffic jam and stabilize traffic flow.
However, most of the previous controller parameter designs design a feedback gain coefficient based on a control theory, and then adopt a method of bringing each time-lag point into a stability condition to verify the stability of a model, so as to estimate a time-lag interval, so that an accurate stable time-lag interval cannot be simply and conveniently obtained, and the calculation result precision of the time-lag interval is low.
Disclosure of Invention
The invention discloses a speed time-lag feedback control method of a traffic flow following model, which aims to solve the technical problems that: a classical optimization speed following model in traffic flow dynamics is improved through a traffic flow speed time-lag feedback control method, speed time-lag feedback control of the traffic flow following model is achieved, stability of a traffic flow system is effectively improved, and the purposes of traffic jam suppression, traffic trip efficiency improvement and traffic resource utilization rate improvement are achieved. In addition, the feedback gain coefficient and the time-lag interval are respectively solved through the control principle and the dynamic time-lag system stability analysis, the combination of the feedback gain coefficient and the time-lag interval is combined, a control item is introduced to improve the original classical optimization speed following model and design a corresponding controller, the speed time-lag feedback control precision is improved, and further the traffic flow system control precision is improved. The invention can also reduce the time cost of controller design and has the advantage of high control efficiency. The invention is particularly suitable for the field of traffic flow control and unmanned traffic flow test.
The purpose of the invention is realized by the following technical scheme.
The invention discloses a speed time-lag feedback control method of a traffic flow following model, which comprises the steps of establishing a kinetic equation of a classical optimization speed following model under a traffic flow system, selecting an optimization speed function, and determining the stability of the optimization speed following model; a classical optimization speed following model is improved through a speed time-lag feedback control method, and the stability condition of the improved following model containing a time-lag feedback control item is determined by combining the control theory; determining the value range of a feedback gain coefficient according to the stability condition of a following model containing a time-lag feedback control item; after the feedback gain coefficient is determined, solving a corresponding stable time-lag interval according to a stability switching principle of a time-lag differential equation and a fixed integral stability analysis method; according to the obtained feedback gain coefficient and the stable time-lag interval, a controller is designed, so that the unstable parameter combination of the classical optimized speed car-following model becomes stable, the stable parameter combination interval of the original optimized speed car-following model is expanded, namely the original unstable optimized speed car-following model becomes the stable car-following model containing the time-lag feedback control item, the stability of the car-following model is effectively improved, the purposes of restraining traffic jam, improving traffic trip efficiency and traffic resource utilization rate are achieved, and the stability of a traffic flow system is effectively improved.
The invention discloses a time-lag feedback control method of a traffic flow optimized speed following model, which comprises the following steps:
the method comprises the following steps: and establishing a dynamic equation of the optimized speed following model under a traffic flow system, selecting an optimized speed function, and determining the stability of the optimized speed following model.
The first implementation method comprises the following steps:
step 1.1: and establishing a kinetic equation of the traffic flow optimization speed following model, and determining the uniform stable flow of the optimization speed following model by selecting an optimization speed function.
The optimized speed following model has the following working conditions: the total length of the single lane is L, the number of vehicles is N, and the overtaking condition does not exist. The dynamic model of the optimized speed following model is established as follows:
Figure BDA0002399527510000021
wherein v isj(t) represents the speed of the jth vehicle at time t; y isj(t)＝xj+1(t)-xj(t) represents the inter-vehicle distance between the jth vehicle and the preceding vehicle; alpha is the driver reaction coefficient, related to the characteristics of the driver, and has the unit of s-1；V(yj(t)) is an optimum speed function with respect to the inter-vehicle distance yj(t) is the optimal speed of the vehicle determined by the driver according to the distance between the vehicle and the preceding vehicle. Optimal speed function V (y)j(t)) may be selected in a variety of forms including formulas that are fitted to or derived empirically from measured road data. Optimal speed function V (y)j(t)) includes an exponential function form, a hyperbolic function form, and preferably, the optimal velocity function V (y) in step 1.1j(t)) takes the form:
V(yj(t))＝16.8[tanh0.0860(yj(t)-25)+0.913] (2)
the optimized speed following model is used for describing the behavior of the traffic flow, including the transition from free flow to congested flow, the density-flux relationship and the stop-and-go traffic wave behavior. When all vehicles run at the same uniform speed, the optimized speed following model is in a uniform stable flow solution state in which the inter-vehicle distances of all vehicles are the same, i.e., the equilibrium solution of the equation is expressed as:
Figure BDA0002399527510000031
h*indicating the initial inter-vehicle distance.
Step 1.2: and (3) determining the stability of the optimized speed following model in the step 1.1 by utilizing a small disturbance and a Taylor expansion method.
Perturbing the tiny displacement to xijAnd velocity disturbance ηjTo the homogeneous stream: y isj(t)＝h*+ξj(t)，vj(t)＝V(h*)+ηj(t), the linearized equation around the equilibrium point is described as:
Figure BDA0002399527510000032
V′(yj) Is the equation V (y)j) To variable vehicle spacing yjDerivative function of, order
Figure BDA0002399527510000033
Setting etaj＝z·eikj+ztThen, the expression is obtained: xij＝eikj+zt·(eik-1)，
Figure BDA0002399527510000036
Substituting the above-mentioned perturbation expression into equation (4) to obtain:
z2-αΛ(eik-1)+αz＝0 (5)
combining with Taylor expansion method, expanding the form of z into z ═ z1(ik)+z2(ik)2+ …, taken into equation (4) and taking only the first and second expansions of (ik), the following equation is obtained:
Figure BDA0002399527510000034
and respectively calculating the real part and the imaginary part of the separation equation to obtain:
Figure BDA0002399527510000035
z1and z2Corresponding to the imaginary and real parts of z, respectively. According to the stability criterion, when z2When the speed is more than 0, the optimized speed following model described in the equation (1) is stable, and the stability condition of the finally obtained optimized speed following model is as follows:
α＞2Λ(8)
the reaction coefficients alpha and lambda are two important parameters of the optimized speed following model, and the stability of the optimized speed following model is determined by value taking. When the value meets the stability condition alpha is larger than 2 lambda, the optimized speed following model is stable; when the parameter does not satisfy the stability condition, namely alpha is less than or equal to 2 lambda, the optimization speed following model is unstable. After the traffic flow is subjected to micro disturbance, the micro disturbance is amplified along with time, the speed amplitude of each vehicle is continuously increased, the oscillation waves are simultaneously propagated to the upstream and downstream of the traffic flow, and finally the oscillation waves are evolved into a stop-and-go traffic flow form to cause traffic jam.
Step two: and (3) improving the classical optimization speed following model in the step one by a speed time-lag feedback control method, and determining the stability condition of the improved following model containing a time-lag feedback control item by combining a control theory.
The second step is realized as follows:
step 2.1: introducing a time-lag speed feedback control signal Fj(t) improving the original optimized speed following model and improving the stability of the model, FjThe specific form of (t) is expressed as follows:
Fj(t)＝κ(vj(t)-vj(t-τ)) (9)
where κ is the feedback gain coefficient and the unit s-1(ii) a τ is the feedback lag, in units of s. The feedback gain coefficient kappa and the feedback time lag tau are two important parameters in a feedback control signal, and the stable range interval of alpha and Lambda in the model is enlarged by selecting the numerical values of the two important parameters, so that the original unstable interval alpha is less than or equal to 2 Lambda, and one part of the unstable interval alpha is a stable parameter interval, thereby improving the stability of the original classical optimization speed following model.
Substituting a feedback control signal equation (8) into the optimal speed following model (1), and establishing a microcosmic traffic flow following model containing a time-lag control item, wherein the specific expression is as follows:
Figure BDA0002399527510000041
step 2.2: and (3) determining the stability condition of the 2.1 following model with a time-lag control term by applying a dynamic control principle.
The linearized equation around the equilibrium point is described in matrix form as:
Figure BDA0002399527510000042
wherein the content of the first and second substances,
Figure BDA0002399527510000043
is a constant. Fj(t)＝κ(vj(t)-vj(t- τ)) is the feedback control signal mentioned in step 2.1. By performing a laplace transform on equation (11):
Figure BDA0002399527510000044
description of the drawings: l (-) represents the laplace transform of the corresponding variable. V, Y, Γ correspond to the V, Y, F transformed variables, respectively. After laplace transformation, equation (11) is expressed in the form:
Figure BDA0002399527510000051
wherein each coefficient expression is as follows:
Figure BDA0002399527510000052
Vj(s) and Vj+1(s) the following relationship exists:
Vj(s)＝G(s)Vj+1(s) (15)
wherein G(s) is a transfer function. Derivation of Vj(s) and Vj+1(s) relationship between:
Vj(s)＝G11(s)(1-G12(s)H(s))-1Vj+1(s) (16)
combining equation (13) and equation (14), the form of G(s) is derived as follows:
Figure BDA0002399527510000053
definition d(s) as characteristicTerm, d(s) ═ s2+αs+αΛ-sκ(1-e-sτ). According to the principles of control, the stability condition of a follow-up model (9) containing a time-lag control term is given by satisfying two property conditions of a transfer function g(s):
the first condition is as follows: the characteristic polynomial d(s) in g(s) as denominator is asymptotically stable;
and a second condition: h of G(s)∞Norm | g(s) | non-woven phosphor∞≤1。
d(s) is progressively stable and | | | G(s) | non-woven phosphor∞And the stability condition of the follow-up model with the time-lag control item is not more than 1.
Step three: and determining the value range of the feedback gain coefficient according to the first stability condition of the following model containing the time-lag feedback control item in the step two.
The third step is realized as follows:
in conjunction with the small gain theorem, the sufficient requirement that the characteristic polynomial d(s) is asymptotically stable is:
||G12(s)||∞·||H(s)||∞＜1 (18)
according to G12(s) and expressions of H(s), deriving G12H of(s) and H(s)∞Norm form is as follows:
Figure BDA0002399527510000054
Figure BDA0002399527510000061
will | G12(s)||∞And | | H(s) | non-woven phosphor∞The expressions (19) and (20) are substituted into the expression (18), and the value range of the feedback gain coefficient k is deduced:
Figure BDA0002399527510000062
step four: and step two, expressing the microcosmic traffic flow car-following model containing the time-lag control item into a time-lag differential equation, and solving a corresponding stable time-lag interval after determining a feedback gain coefficient according to a stability switching principle of the time-lag differential equation and a fixed integral stability analysis method.
The implementation method of the fourth step is as follows:
step 4.1: according to the stability analysis of a time-lag kinetic system, setting lambda as a characteristic value of an equation (9), and solving a characteristic value equation f (lambda) of a follow-up model containing a time-lag feedback control term and a derivative function f' (lambda) of the characteristic value equation to the characteristic value lambda by combining a stability analysis method of a time-lag differential equation.
Step 4.2: on the basis of selecting a feedback gain coefficient kappa, a constant integral stability analysis method is applied to set a time lag interval tau epsilon [0, T ∈ ]]For calculating the interval, the number of unstable characteristic roots contained in the characteristic value equation f (lambda) under the selected time lag value is calculated
Figure BDA0002399527510000068
And further judging the stability of the kinetic system. The specific method comprises the following steps:
(a) taking the eigenvalue equation expression F (λ) in step 4.1, and calculating F (ω) ═ i-nf (i ω), where n is the power of the highest order term of the eigenvalue equation. The real part of F (omega) is taken and is recorded as R (omega).
(b) Selecting a feedback gain coefficient kappa based on the feedback gain coefficient range obtained in the step three0In the interval [0, T]In selecting the time lag value tau0Solving equation R (omega) to 0, and taking the maximum positive real root as omegamaxRandom T0＞ωmaxAs the upper limit of integration for the next step.
(c) Calculation of F (0, T) according to the following equation0) The numerical value of (A):
Figure BDA0002399527510000063
(d) calculating at time lag τ0Number of lower unstable characteristic roots
Figure BDA0002399527510000064
Figure BDA0002399527510000065
Step 4.3: imaging the calculation result of each time lag point in the step 4.2 to obtain the number of unstable characteristic roots
Figure BDA0002399527510000066
Discrete point diagram, in the figure, as a function of time lag τ
Figure BDA0002399527510000067
The corresponding time lag interval is the feedback gain coefficient kappa of the power system0The lower corresponding settling time lag interval.
Step five: and designing a time-lag speed feedback controller by introducing a control item feedback gain coefficient and a stable time-lag value according to the feedback gain coefficient and the stable time-lag interval obtained in the third step and the fourth step, and bringing the time-lag speed feedback controller into the original classical speed optimization following model to form an optimized speed following model with a time-lag feedback item. Substituting each parameter into the stability condition II in the step II to enable the model to meet the stability condition of the model; through the feedback control of the controller, the unstable parameter combination of the classical optimized speed car-following model is changed into stable, the stable parameter combination interval of the original optimized speed car-following model is expanded, namely the original unstable optimized speed car-following model is changed into the stable car-following model containing the time-lag feedback control item, the stability, the control precision and the control efficiency of the car-following model are effectively improved, the purposes of restraining traffic jam, improving traffic trip efficiency and traffic resource utilization rate are achieved, and the stability of a traffic flow system is effectively improved.
The method comprises the following steps: obtaining the feedback gain coefficient kappa according to the third step and the fourth step0And a steady time lag interval [ tau ]1,τ2]By introducing a control term to feed back the gain factor k0And the steady time lag value tau0Designing a time-lag velocity feedback controller and bringing it into the original channelIn the typical speed optimization following model, an optimization speed following model with a time-lag feedback term is formed. The parameters alpha, lambda and kappa are respectively measured0,τ0And substituting a stability condition II in the step II to meet the stability condition of the model:
Figure BDA0002399527510000071
namely, the following conditions are satisfied:
Figure BDA0002399527510000072
through the feedback control of the controller, the unstable parameter combination of the classical optimized speed car-following model is changed into stable, the stable parameter combination interval of the original optimized speed car-following model is expanded, namely the original unstable optimized speed car-following model is changed into the stable car-following model containing the time-lag feedback control item, the stability, the control precision and the control efficiency of the car-following model are effectively improved, the purposes of restraining traffic jam, improving traffic trip efficiency and traffic resource utilization rate are achieved, and the stability of a traffic flow system is effectively improved.
Further comprises the following steps: and carrying out numerical simulation on the original classical optimization speed following model and a new following model obtained after introducing a controller. The change conditions of the speed, the position and the phase track of the vehicle on the annular road section along with the time are simulated, the numerical simulation results of the speed, the position and the phase track are compared, and the speed time-lag feedback control method is verified to be effective, namely the traffic jam can be effectively inhibited after the time-lag feedback is introduced.
Has the advantages that:
1. the invention discloses a speed time-lag feedback control method of a traffic flow following model, which adopts a time-lag feedback control method, adds a speed time-lag feedback control item on the basis of a classical optimized speed following model, inhibits the traffic jam phenomenon and enhances the stability of the traffic flow.
2. The invention discloses a speed time-lag feedback control method of a traffic flow following model, which adopts a control theory and a time-lag dynamic system stability analysis method to respectively calculate a feedback gain coefficient range and a stable time-lag interval when designing parameters of a controller, so that the obtained result is more accurate, the defect of estimating the time-lag interval in the past is overcome, and the calculation precision is higher.
3. When a stable time lag interval is obtained, a verification method is generally adopted in the prior art, and the value of each time lag point needs to be brought into the stability condition of the model to verify whether the time lag value meets the stability condition of the model, so as to estimate the stable time lag interval. The invention discloses a speed time-lag feedback control method of a traffic flow following model, which can directly obtain a stable time-lag interval by calculation only by adding a feedback control item on the basis of the original classical optimized speed following model without respectively verifying each point, thereby reducing the complexity of a verification method in the prior art and improving the efficiency of parameter calculation of a controller.
Drawings
FIG. 1 is a flow chart of a speed time lag feedback control method of a traffic flow following model;
FIG. 2 is a schematic view of a circular road used in the present invention to create a traffic flow dynamics model;
FIG. 3 is a graph of speed of 1 st, 50 th, 100 th vehicles over time at steady state uncontrolled system;
FIG. 4 is a graph of speed of 1 st, 50 th, 100 th vehicles over time with an uncontrolled system instability;
FIG. 5 is a block diagram of a control system for a jth vehicle of the present invention;
FIG. 6 shows the number of unstable feature roots
Figure BDA0002399527510000081
Discrete plot as a function of time lag τ: (a) for preliminary calculation, the step length is 0.1 s; (b) for accurate calculation, the step length is 0.01 s;
FIG. 7 is a graph of the transfer function norm as a function of ω for a controlled system;
FIG. 8 is a graph of speed over time for the 1 st, 50 th, and 100 th vehicles of the controlled system;
FIG. 9 is a comparison of vehicle speed profiles for an uncontrolled system and a controlled system;
FIG. 10 is a comparison graph of vehicle phase-track distributions for an uncontrolled system and a controlled system;
FIG. 11 is a vehicle space-time patch diagram for an uncontrolled system: (a) the first 300 seconds; (b) last 300 seconds;
FIG. 12 is a vehicle space-time patch diagram of a controlled system;
Detailed Description
For a better understanding of the objects and advantages of the present invention, reference should be made to the following detailed description taken in conjunction with the accompanying drawings and examples.
The method is suitable for closed loop test in the unmanned technology development stage, and in order to verify the feasibility of the method, an annular single-lane road is selected, the total length is L2500 m, the total number of vehicles is N100, and the lane has no overtaking condition, namely the vehicles are all in a following driving mode.
As shown in fig. 1, the speed time lag feedback control method of the traffic flow following model disclosed in this embodiment specifically includes the following steps:
the method comprises the following steps: and establishing a dynamic equation of the optimized speed following model under a traffic flow system, selecting an optimized speed function, and determining the stability of the optimized speed following model.
The first implementation method comprises the following steps:
step 1.1: and establishing a kinetic equation of the traffic flow optimization speed following model, and determining the uniform stable flow of the optimization speed following model by selecting an optimization speed function.
Fig. 2 is a schematic diagram of a circular road for establishing a traffic flow dynamic model. Under the simulated road condition, a dynamic model of an optimized speed following model is established as follows:
Figure BDA0002399527510000091
wherein v isj(t) represents the speed of the jth vehicle at time t; y isj(t)＝xj+1(t)-xj(t) represents the inter-vehicle distance between the jth vehicle and the preceding vehicle; for drivingA driver reaction coefficient associated with a characteristic of the driver; v (y)j(t)) is an optimum speed function with respect to the inter-vehicle distance yj(t) is the optimal speed of the vehicle determined by the driver according to the distance between the vehicle and the preceding vehicle. Optimal speed function V (y)j(t)) may be selected in a variety of forms including formulas that are fitted to or derived empirically from measured road data. Optimal speed function V (y)j(t)) includes an exponential function form, a hyperbolic function form, and preferably, the optimal velocity function V (y) in step 1.1j(t)) takes the form:
V(yj(t))＝16.8[tanh0.0860(yj(t)-25)+0.913] (2)
the optimized speed following model is used for describing the behavior of the traffic flow, including the transition from free flow to congested flow, the density-flux relationship and the stop-and-go traffic wave behavior. When all vehicles run at the same uniform speed, the optimized speed following model is in a uniform stable flow solution state in which the inter-vehicle distances of all vehicles are the same, i.e., the equilibrium solution of the equation is expressed as:
Figure BDA0002399527510000092
step 1.2: and (3) determining the stability of the optimized speed following model in the step 1.1 by utilizing a small disturbance and a Taylor expansion method.
Perturbing the tiny displacement to xijAnd velocity disturbance ηjTo the homogeneous stream: y isj(t)＝h*+ξj(t)，vj(t)＝V(h*)+ηj(t), the linearized equation around the equilibrium point is described as:
Figure BDA0002399527510000093
V′(yj) Is the equation V (y)j) To variable vehicle spacing yjThe derivative function of (a) is,
Figure BDA0002399527510000101
setting etaj＝z·eikj+ztThen, the expression is obtained: xij＝eikj+zt·(eik-1)，
Figure BDA0002399527510000104
Substituting the above-mentioned perturbation expression into equation (4) to obtain:
z2-αΛ(eik-1)+αz＝0 (5)
combining with Taylor expansion method, expanding the form of z into z ═ z1(ik)+z2(ik)2+ …, taken into equation (4) and taking only the first and second expansions of (ik), the following equation is obtained:
Figure BDA0002399527510000102
and respectively calculating the real part and the imaginary part of the separation equation to obtain:
Figure BDA0002399527510000103
z1and z2Corresponding to the imaginary and real parts of z, respectively. According to the stability criterion, when z2When the speed is more than 0, the optimized speed following model described in the equation (1) is stable, and the stability condition of the finally obtained optimized speed following model is as follows:
α＞2Λ(8)
the reaction coefficients alpha and lambda are two important parameters of the optimized speed following model, and the stability of the optimized speed following model is determined by value taking. When the value meets the stability condition alpha is larger than 2 lambda, the optimized speed following model is stable; when the parameter does not satisfy the stability condition, namely alpha is less than or equal to 2 lambda, the optimization speed following model is unstable. After the traffic flow is subjected to micro disturbance, the micro disturbance is amplified along with time, the speed amplitude of each vehicle is continuously increased, the oscillation waves are simultaneously propagated to the upstream and downstream of the traffic flow, and finally the oscillation waves are evolved into a stop-and-go traffic flow form to cause traffic jam.
In this example, two sets of parameters are selected. (3 s) (. alpha.)-1The Λ is 1.448, namely, α is greater than 2 Λ, which indicates that the optimized speed following model is stable, and after the traffic flow is slightly disturbed, the disturbance gradually decreases and disappears with time, and finally the traffic flow returns to a stable traffic flow state; 2s ═ alpha-1And Λ is 1.448, namely α < 2 Λ, which indicates that the optimized speed following model is unstable, and after the traffic flow is slightly disturbed, the traffic flow can evolve into a stop-and-go traffic flow form over time, so that traffic jam is caused.
Fig. 3 and 4 are graphs of the speed of the 1 st, 50 th and 100 th vehicles with time in the stable and unstable states of the classical following model, respectively, and correspond to the alpha being 3s-1And α ═ 2s-1In the case of (1), the time course is 1000s, and the matlab software is used for numerical simulation. When alpha is 3s-1In time, the influence of disturbance on the model disappears quickly, and the vehicle speed tends to be stable and is vj(t) about 15.338 m/s; when alpha is 2s-1And in time, the influence of disturbance on the model is increasingly greater, and the speed is greater and smaller, so that the traffic flow is indicated to be congested corresponding to the stop-and-go state of the vehicle. Then, for α ═ 2s-1The feedback control design is carried out under the condition so as to achieve the aim of traffic flow stabilization.
Step two: and (3) improving the classical optimization speed following model in the step one by a speed time-lag feedback control method, and determining the stability condition of the improved following model containing a time-lag feedback control item by combining a control theory.
The second step is realized as follows:
step 2.1: introducing a time-lag speed feedback control signal Fj(t) improving the original optimized speed following model and improving the stability of the model, FjThe specific form of (t) is expressed as follows:
Fj(t)＝κ(vj(t)-vj(t-τ)) (9)
where κ is the feedback gain coefficient and the unit s-1(ii) a τ is the feedback lag, in units of s. The feedback gain factor k and the feedback time lag tau are feedback controlTwo important parameters in the signal are made, and the stable range interval of alpha and Lambda in the model is enlarged by selecting the numerical values of the two important parameters, so that the original unstable interval alpha is less than or equal to 2 Lambda, and one part of the unstable interval alpha becomes a stable parameter interval, thereby improving the stability of the original classical optimization speed following model.
Fig. 5 is a schematic block diagram of a control system of the jth vehicle, a feedback control signal equation (8) is substituted into an optimized speed following model (1), and a microcosmic traffic flow following model containing a time-lag control item is established, wherein a specific expression is as follows:
Figure BDA0002399527510000111
step 2.2: and (3) determining the stability condition of the 2.1 following model with a time-lag control term by applying a dynamic control principle.
The linearized equation around the equilibrium point is described in matrix form as:
Figure BDA0002399527510000112
wherein the content of the first and second substances,
Figure BDA0002399527510000113
is constant, Fj(t)＝κ(vj(t)-vj(t- τ)) is the feedback control signal mentioned in step 2.1. By performing a laplace transform on equation (11):
Figure BDA0002399527510000114
description of the drawings: l (-) represents the laplace transform of the corresponding variable. After laplace transformation, equation (11) is expressed in the form:
Figure BDA0002399527510000115
wherein:
Figure BDA0002399527510000121
Vj(s) and Vj+1(s) the following relationship exists:
Vj(s)＝G(s)Vj+1(s) (15)
wherein G(s) is a transfer function. Derivation of Vj(s) and Vj+1(s) relationship between:
Vj(s)＝G11(s)(1-G12(s)H(s))-1Vj+1(s) (16)
combining equation (13) and equation (14), the form of G(s) is derived as follows:
Figure BDA0002399527510000122
definition d(s) as a characteristic polynomial, d(s) s2+αs+αΛ-sκ(1-e-sτ). According to the principles of control, the stability condition of a follow-up model (9) containing a time-lag control term is given by satisfying two property conditions of a transfer function g(s):
the first condition is as follows: the characteristic polynomial d(s) in g(s) as denominator is asymptotically stable;
and a second condition: h of G(s)∞Norm | g(s) | non-woven phosphor∞≤1。
d(s) is progressively stable and | | | G(s) | non-woven phosphor∞And the stability condition of the follow-up model with the time-lag control item is not more than 1.
Step three: and determining the value range of the feedback gain coefficient according to the first stability condition of the following model containing the time-lag feedback control item in the step two. The specific calculation steps are as follows:
in conjunction with the small gain theorem, the sufficient requirement that the characteristic polynomial d(s) is asymptotically stable is:
||G12(s)||∞·||H(s)||∞＜1 (18)
according to G12(s) and H(s) expression, derivation of G12H of(s) and H(s)∞Norm form is as follows:
Figure BDA0002399527510000123
||H(s)||∞＝||κ(1-e-sτ)||∞＝2|κ| (20)
will | G12(s)||∞And | | H(s) | non-woven phosphor∞The expressions (19) and (20) are substituted into the expression (18), and the value range of the feedback gain coefficient k is deduced:
Figure BDA0002399527510000131
namely:
-1＜κ＜1 (22)
step four: and step two, expressing the microcosmic traffic flow car-following model containing the time-lag control item into a time-lag differential equation, and solving a corresponding stable time-lag interval after determining a feedback gain coefficient according to a stability switching principle of the time-lag differential equation and a fixed integral stability analysis method.
The implementation method of the fourth step is as follows:
step 4.1: according to the stability analysis of a time-lag kinetic system, setting lambda as a characteristic value of equation (9), and solving a characteristic value equation f (lambda) of a following model containing a time-lag feedback control term by combining a stability analysis method of a time-lag differential equation:
Figure BDA0002399527510000132
derivative function f' (λ) of eigenvalue equation for eigenvalue λ:
f′(λ)＝2λ+(α-κ)+κe-λτ-τκλe-λτ (24)
step 4.2: on the basis of selecting a feedback gain coefficient kappa, a constant integral stability analysis method is applied to set a time lag interval tau epsilon [0, T ∈ ]]To countCalculating interval, calculating the number of unstable characteristic roots contained in the characteristic value equation f (lambda) under the selected time lag value
Figure BDA0002399527510000133
And further judging the stability of the kinetic system. The specific method comprises the following steps:
(a) taking the eigenvalue equation expression F (λ) in step 4.1, and calculating F (ω) ═ i-nf (i ω) is as follows:
Figure BDA0002399527510000134
Figure BDA0002399527510000135
n-1, wherein k is 1, 2. Taking the real part of F (omega), and recording as R (omega), the expression is as follows:
Figure BDA0002399527510000136
(b) selecting kappa when the feedback gain coefficient condition obtained in the step three is that-1 is more than kappa and less than 10＝0.7s-1As a feedback gain coefficient, in a time lag interval τ ∈ [0,1 ]]In s, selecting time lag value tau with 0.1s as step length0. At different time lags tau0Then, the equation solution equation R (ω) in step (a) is solved to 0, and the maximum positive real root is taken as ωmaxRandom T0＞ωmaxAs the upper limit of integration for the next step.
(c) Calculating at different time lags tau0Lower, F (0, T)0)kThe formula is as follows:
Figure BDA0002399527510000141
Figure BDA0002399527510000142
(d) calculating at different time lags tau0Lower, number of unstable feature roots
Figure BDA0002399527510000143
Figure BDA0002399527510000144
Step 4.3: imaging the calculation result of each time lag point in the step 4.2 to obtain the number of unstable characteristic roots
Figure BDA0002399527510000145
A discrete point diagram with a change in the time lag τ is shown in fig. 6 (a).
Determining the time lag τ0Whether the time lag value belongs to the stable time lag interval or not. The judging method comprises the following steps: if it is
Figure BDA0002399527510000146
Then τ0Not belonging to the stationary time-lag interval, i.e. in the feedback gain factor k and time-lag τ0Next, the following model containing the time-lag feedback control term is unstable; if it is
Figure BDA0002399527510000147
Then τ0Belonging to a steady-state lag interval, i.e. in the feedback gain factor k0And a time lag τ0Next, the following model including the time-lag feedback control term is stable. As can be seen from FIG. 6(a), it is stable that the time lag interval is at τ0∈(0.4,0.7)s。
In order to obtain more accurate result, in the time-lag interval tau epsilon (0.4,0.7) s, the time-lag value tau is selected by taking 0.01s as a step length0And repeating the steps 4.2(b) - (d) to calculate the time lag tau0Number of lower unstable characteristic roots
Figure BDA0002399527510000148
Imaging the calculation results of each time lag point in the step to obtainNumber of root to unstable feature
Figure BDA0002399527510000149
A discrete point diagram with a change in the time lag τ is shown in fig. 6 (b). As can be seen from FIG. 6(b), it is stable that the time lag interval is τ0∈[0.45,0.67]s。
Step five: obtaining the feedback gain coefficient kappa according to the third step and the fourth step0＝0.7s-1And a settling time lag interval τ0∈[0.45,0.67]s, feedback of gain factor k by introducing a control term0＝0.7s-1And the steady time lag value tau0And designing a time-delay speed feedback controller for 0.5s, and substituting the time-delay speed feedback controller into the original classical speed optimization following model to form an optimized speed following model with a time-delay feedback item. Each parameter alpha is equal to 2s-1,Λ＝1.448,κ0＝0.7s-1,τ0Stability condition two in step two was substituted for 0.5 s:
Figure BDA00023995275100001410
FIG. 7 illustrates | | G(s) | non-woven phosphor∞The value of (A) is always less than 1, namely after the controller is introduced, the new model with the time-lag feedback control item meets the stability condition of the model.
Through the feedback control of the controller, the unstable parameter combination of the classical optimized speed car-following model is changed into stable, the stable parameter combination interval of the original optimized speed car-following model is expanded, namely the original unstable optimized speed car-following model is changed into the stable car-following model containing the time-lag feedback control item, the stability, the control precision and the control efficiency of the car-following model are effectively improved, the purposes of restraining traffic jam, improving traffic trip efficiency and traffic resource utilization rate are achieved, and the stability of a traffic flow system is effectively improved.
Step six: and carrying out numerical simulation on the original classical optimization speed following model and a time-lag feedback control model obtained after introducing a controller. The change conditions of the speed, the position and the phase track of the vehicle on the annular road section along with the time are simulated, the time course is 1000s, the disturbance is generated by an array which is randomly and uniformly distributed, the simulation result is shown in figures 8-12, the numerical simulation results of the speed and the phase track are compared, after a traffic flow system is controlled, the speed and the inter-vehicle distance are converged and recovered to a uniform flow state, and the speed time-lag feedback control method is verified to be effective, namely the traffic jam can be effectively inhibited after the time-lag feedback control is introduced.
The above detailed description is intended to illustrate the objects, aspects and advantages of the present invention, and it should be understood that the above detailed description is only exemplary of the present invention and is not intended to limit the scope of the present invention, and any modifications, equivalents, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (5)

1. A time lag feedback control method of a traffic flow optimization speed following model is characterized in that: comprises the following steps of (a) carrying out,
the method comprises the following steps: establishing a dynamic equation of an optimized speed following model under a traffic flow system, and then selecting an optimized speed function;
the first step of the realization method is as follows,
step 1.1: establishing a kinetic equation of a traffic flow optimization speed following model, and determining uniform stable flow of the optimization speed following model by selecting an optimization speed function;
the optimized speed following model has the following working conditions: the total length of the single lane is L, the number of vehicles is N, and the overtaking condition does not exist; the dynamic equation of the optimized speed following model is established as follows:
Figure FDA0003225411580000011
wherein v isj(t) represents the speed of the jth vehicle at time t, vj+1(t) represents the speed of the j +1 th vehicle at time t, the j +1 th vehicle being an adjacent preceding vehicle to the j-th vehicle; y isj(t)＝xj+1(t)-xj(t) represents the inter-vehicle distance between the jth vehicle and the preceding vehicle, where xj(t) and xj+1(t) represents the position of the jth vehicle and the jth +1 vehicle, respectively, at time t; alpha is the driver reaction coefficient, related to the characteristics of the driver, and has the unit of s-1；V(yj(t)) is an optimum speed function with respect to the inter-vehicle distance yj(t) a function of the optimal speed of the driver's own vehicle determined according to the distance between the driver's own vehicle and the preceding vehicle; optimal speed function V (y)j(t)) may be selected in a variety of forms including a formula fitted or empirically derived from measured road data; optimal speed function V (y)j(t)) includes exponential functional forms, hyperbolic functional forms;
the optimized speed following model is used for describing the behavior of the traffic flow, wherein the behavior of the traffic flow comprises the transition from free flow to congestion flow, the density-flux relation and the traffic wave behavior of stop-and-go; when all vehicles run at the same uniform speed, the optimized speed following model is in a uniform stable flow solution state in which the inter-vehicle distances of all vehicles are the same, i.e., the equilibrium solution of the equation is expressed as:
Figure FDA0003225411580000012
h*representing an initial vehicle distance;
step 1.2: determining the stability of the optimized speed following model in the step 1.1 by utilizing a micro-disturbance and Taylor expansion method;
will slightly disturb xijAnd ηjTo the homogeneous stream:
yj(t)＝h*+ξj(t)，vj(t)＝V(h*)+ηj(t) (3)
wherein ξj(t) and ηj(t) respectively representing the tiny inter-vehicle distance disturbance and the tiny speed disturbance of the jth vehicle at the moment t;
the linearized equation around the equilibrium point is described as:
Figure FDA0003225411580000013
V′(yj) Is the equation V (y)j) To variable vehicle spacing yjThe derivative function of (a) is,
Figure FDA0003225411580000021
variable etajExponential form of the series expansion to ηj＝z·eikj+ztThen, the expression is obtained: xij＝eikj+zt·(eik-1)，
Figure FDA0003225411580000022
Substituting the above-mentioned perturbation expression into equation (4) to obtain:
z2-αΛ(eik-1)+αz＝0 (5)
combining with Taylor expansion method, expanding the form of z into z ═ z1(ik)+z2(ik)2+ …, taken into equation (5) and taking only the first and second expansions of (ik), the following equation is obtained:
Figure FDA0003225411580000023
and respectively calculating the real part and the imaginary part of the separation equation to obtain:
Figure FDA0003225411580000024
z1and z2Corresponding to the imaginary part and the real part of z, respectively, and Λ is the optimal velocity function V (y)j) Derivative function V' (y)j) At the initial uniform inter-vehicle distance h*Taking the value of (A); according to the stability criterion, when z2When the speed is more than 0, the optimized speed following model described in the equation (1) is stable, and the stability condition of the finally obtained optimized speed following model is as follows:
α＞2Λ (8)
the reaction coefficients alpha and lambda are two important parameters of the optimized speed following model, and the stability of the optimized speed following model is determined by value taking; when the value meets the stability condition alpha is larger than 2 lambda, the optimized speed following model is stable; when the parameter does not meet the stability condition, namely alpha is less than or equal to 2 lambda, the optimized speed following model is unstable; after the traffic flow is subjected to micro disturbance, the micro disturbance is amplified along with time, the speed amplitude of each vehicle is continuously increased, the oscillation waves are simultaneously propagated to the upstream and downstream of the traffic flow, and finally the oscillation waves are evolved into a stop-and-go traffic flow form to cause traffic jam;
step two: improving the optimized speed following model in the first step by a speed time-lag feedback control method, and determining a first stability condition and a second stability condition of the improved following model with a time-lag feedback control item by combining a control theory;
the second step is realized by the following method,
step 2.1: introducing a time-lag speed feedback control signal Fj(t) improving the original optimized speed following model and improving the stability of the model, FjThe specific form of (t) is expressed as follows:
Fj(t)＝κ(vj(t)-vj(t-τ)) (9)
where κ is the feedback gain coefficient and the unit s-1(ii) a τ is feedback time lag, in units of s; the feedback gain coefficient kappa and the feedback time lag tau are two important parameters in a feedback control signal, and the stable range interval of alpha and Lambda in the model is expanded by selecting the values of the two important parameters, so that the original unstable interval alpha is less than or equal to 2 Lambda, and one part of the unstable interval alpha is a stable parameter interval, thereby improving the stability of the altitude optimization speed following model;
substituting a feedback control signal equation (9) into the optimal speed following model (1), and establishing a microcosmic traffic flow following model containing a time-lag control item, wherein the specific expression is as follows:
Figure FDA0003225411580000031
step 2.2: determining the stability condition of a 2.1 microcosmic traffic flow car-following model with a time-lag control item by applying a dynamics control principle;
the linearized equation around the equilibrium point is described in matrix form as:
Figure FDA0003225411580000032
wherein the content of the first and second substances,
Figure FDA0003225411580000033
is a constant; fj(t)＝κ(vj(t)-vj(t- τ)) is the feedback control signal mentioned in step 2.1; by performing a laplace transform on equation (11):
Figure FDA0003225411580000034
description of the drawings: l (-) represents the Laplace transform of the corresponding variable; s is a complex variable generated after Laplace transform; v, Y, gamma correspond to the variable after V, Y, F transform respectively; after laplace transformation, equation (11) is expressed in the form:
Figure FDA0003225411580000035
wherein each coefficient expression is as follows:
Figure FDA0003225411580000036
Vj(s) and Vj+1(s) the following relationship exists:
Vj(s)＝G(s)Vj+1(s) (15)
wherein G(s) is a transfer function; derivation of Vj(s) and Vj+1(s) relationship between:
Vj(s)＝G11(s)(1-G12(s)H(s))-1Vj+1(s) (16)
combining equation (13) and equation (14), the form of G(s) is derived as follows:
Figure FDA0003225411580000041
definition d(s) as a characteristic polynomial, d(s) s2+αs+αΛ-sκ(1-e-sτ) (ii) a According to the principles of control, the stability condition of a follow-up model (10) containing a time-lag control term is given by satisfying two property conditions of a transfer function g(s):
the stability condition is as follows: the characteristic polynomial d(s) in g(s) as denominator is asymptotically stable;
and (2) stability condition II: h of G(s)∞Norm | g(s) | non-woven phosphor∞≤1；
d(s) is progressively stable and | | | G(s) | non-woven phosphor∞The stability condition of the improved follow-up model with the time-lag control item is not more than 1;
step three: determining the value range of the feedback gain coefficient according to the first stability condition of the following model containing the time-lag feedback control item in the step two;
the third step is realized by the following method,
in conjunction with the small gain theorem, the sufficient requirement that the characteristic polynomial d(s) is asymptotically stable is:
||G12(s)||∞·||H(s)||∞＜1 (18)
according to H∞Norm definition, taking s as pure virtual root s ═ i ω according to G12(s) and expressions of H(s), deriving G12H of(s) and H(s)∞Norm form is as follows:
Figure FDA0003225411580000042
Figure FDA0003225411580000043
will | G12(s)||∞And | | H(s) | non-woven phosphor∞The expressions (19) and (20) are substituted into the expression (18), and the value range of the feedback gain coefficient k is deduced:
Figure FDA0003225411580000044
step four: the microcosmic traffic flow car-following model containing the time-lag control item is expressed as a time-lag differential equation, and a corresponding stable time-lag interval is obtained after a feedback gain coefficient is determined according to a stability switching principle of the time-lag differential equation and a fixed integral stability analysis method;
the implementation method of the step four is as follows,
step 4.1: according to the stability analysis of a time-lag kinetic system, setting lambda as a characteristic value of an equation (9), and solving a characteristic value equation f (lambda) of a following model containing a time-lag feedback control term and a derivative function f' (lambda) of the characteristic value equation to the characteristic value lambda by combining a stability analysis method of a time-lag differential equation;
step 4.2: on the basis of selecting a feedback gain coefficient kappa, a constant integral stability analysis method is applied to set a time lag interval tau epsilon [0, T ∈ ]]For calculating the interval, the number of unstable characteristic roots contained in the characteristic value equation f (lambda) under the selected time lag value is calculated
Figure FDA0003225411580000051
Further judging the stability of the kinetic system;
step 4.3: imaging the calculation result of each time lag point in the step 4.2 to obtain the number of unstable characteristic roots
Figure FDA0003225411580000052
Discrete point diagram, in the figure, as a function of time lag τ
Figure FDA0003225411580000053
The corresponding time lag interval is the feedback gain coefficient kappa of the power system0A lower corresponding stability time lag interval;
step five: designing a time-lag speed feedback controller by introducing a control item feedback gain coefficient and a stable time-lag value according to the feedback gain coefficient and the stable time-lag interval obtained in the third step and the fourth step, and bringing the time-lag speed feedback controller into an original optimized speed following model to form an optimized speed following model with a time-lag feedback item; substituting each parameter into the stability condition II in the step II to enable the model to meet the stability condition of the model; through the feedback control of the controller, the unstable parameter combination of the optimized speed following model becomes stable, and the stable parameter combination interval of the original optimized speed following model is expanded.
2. The time lag feedback control method of the traffic flow optimization speed following model according to claim 1, characterized in that: carrying out numerical simulation on the original speed-optimized following model and a new following model obtained after introducing a controller; and (3) simulating the change conditions of the speed and the position of the vehicle on the annular road section along with time, comparing the numerical simulation results of the speed and the position of the vehicle, and verifying the speed time-lag feedback control method.
3. The time lag feedback control method of the traffic flow optimization speed following model according to claim 1, characterized in that: the specific method for judging the stability of the kinetic system in step 4.2 is as follows,
(a) taking the eigenvalue equation expression F (λ) in step 4.1, and calculating F (ω) ═ i-nF (i omega), wherein n is the power of the highest term of the characteristic value equation, and the real part of F (omega) is taken and is recorded as R (omega);
(b) selecting a feedback gain coefficient kappa based on the feedback gain coefficient range obtained in the step three0In the interval [0, T]In selecting the time lag value tau0Solving equation R (omega) to 0, and taking the maximum positive real root as omegamaxRandom T0＞ωmaxAs the upper limit of integration for the next step;
(c) calculation of F (0, T) according to the following equation0) Is/are as followsNumerical values:
Figure FDA0003225411580000054
f '(λ) is a derivative function of the characteristic equation f (λ) with respect to the characteristic value λ, and f' (i ω) is a derivative function value when the characteristic root is a pure imaginary root λ ═ i ω;
(d) calculating at time lag τ0Number of lower unstable characteristic roots
Figure FDA0003225411580000055
Figure FDA0003225411580000056
4. The time lag feedback control method of the traffic flow optimization speed following model according to claim 3, characterized in that: step five shows the method that the feedback gain coefficient k is obtained according to the step three and the step four0And a steady time lag interval [ tau ]1,τ2]By introducing a control term to feed back the gain factor k0And the steady time lag value tau0Designing a time-lag speed feedback controller, and bringing the time-lag speed feedback controller into an original speed optimization following model to form an optimized speed following model with a time-lag feedback item; the parameters alpha, lambda and kappa are respectively measured0,τ0And substituting a stability condition II in the step II to meet the stability condition of the model:
Figure FDA0003225411580000061
namely, the following conditions are satisfied:
Figure FDA0003225411580000062
through the feedback control of the controller, the unstable parameter combination of the optimized speed following model becomes stable, and the stable parameter combination interval of the original optimized speed following model is expanded, namely the original unstable optimized speed following model becomes a stable following model containing a time-lag feedback control item.
5. The time lag feedback control method of the traffic flow optimization speed following model according to claim 4, characterized in that: optimal speed function V (y) in step 1.1j(t)) takes the form of,
V(yj(t))＝16.8[tanh0.0860(yj(t)-25)+0.913] (26)。
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