A two-part consensus method for sampling control of intelligent connected vehicle systems under privacy protection

Abstract

The present invention relates to the field of intelligent connected vehicle systems, and specifically to a two-part consensus method for sampling-controlled intelligent connected vehicle systems under privacy protection. It includes: establishing an intelligent connected vehicle system under non-uniform sampling control; obtaining a hybrid distributed two-part consensus controller and a privacy-preserving noise generator for the intelligent connected vehicle system; using discretization to obtain a new closed-loop system, and constructing a Lyapunov function for the closed-loop system to derive the two-part consensus conditions for the intelligent connected vehicle system to achieve asymptotically unbiased mean square output; quantifying the sensitivity and privacy level of the intelligent connected vehicle system in a finite time domain; and enabling the intelligent connected vehicle system to achieve a two-part consensus that meets the predefined accuracy and the expected privacy upper bound in an infinite time domain. The present invention solves the problem of two-part consensus for non-uniform sampling-controlled intelligent connected vehicle systems in the presence of adversarial information, while ensuring ideal privacy protection.

Description

Technical Field

[0001] The present invention relates to the field of intelligent networked vehicle systems, and specifically to an asymptotically unbiased mean square output bipartite consensus method for sampling and controlling intelligent networked vehicle systems under privacy protection. Background Art

[0002] In the past few decades, intelligent transportation systems have become an effective way to improve the performance of transportation systems and enhance driving safety. Intelligent connected vehicle systems can be regarded as a type of multi-agent system. To date, there have been many studies on consensus control of multi-agent systems, generally aiming to develop a network protocol for a group of interacting agents so that all agents reach an average convergence of the initial values ​​of the variables. However, these essentially assume that the weight of each edge in the topology is greater than or equal to zero, which is restrictive to a certain extent. In many practical scenarios, collaboration and antagonism always coexist. It can be described as a signed graph, that is, the edge weights in a signed graph can also be negative. Bipartite consensus means that all agents converge to the same value except for the sign.

[0003] With the rapid development of Internet technology and wireless communication technology, privacy leakage has become an increasingly serious problem due to malicious eavesdropping attacks on privately available data, and its data set needs to be protected for privacy. In the intelligent connected vehicle system, the exchanged driving information may be eavesdropped, which may damage the privacy of the vehicle and even destroy the communication channel of the fleet, so privacy protection between vehicles is imperative. Many methods have been proposed for data privacy protection, such as cryptography theory, homomorphic encryption, time-varying transformation, state decomposition, etc., but this traditional privacy protection method has the disadvantages of large computational complexity, difficulty in implementation, and incomplete coverage of the scope of use.

[0004] In addition, the high communication cost and huge computational complexity have greatly reduced the communication efficiency of intelligent connected vehicles. Sampling technology can effectively enhance the flexibility and scalability of control. Many periodic sampling technologies used in existing literature are very effective in many application results, but there are problems such as inflexible data processing and limited scope of application. For example, periodic sampling cannot handle packet loss, data confusion, delays and other problems that are more suitable for actual scenarios during data transmission.

[0005] Therefore, how to solve the two-part consensus problem of non-uniform sampling control of intelligent connected vehicle systems under privacy protection is a technical problem that needs to be solved urgently in this field. Summary of the invention

[0006] The present invention discloses a two-part consensus method for sampling-controlled intelligent connected vehicle systems under privacy protection, which solves the two-part consensus problem of non-uniform sampling-controlled intelligent connected vehicle systems under privacy protection. While realizing differential privacy protection of the intelligent connected vehicle system, the present invention provides an asymptotically unbiased mean square output two-part consensus method for sampling-controlled intelligent connected vehicle systems under privacy protection.

[0007] The present invention provides a privacy-preserving down-sampling control intelligent connected vehicle system two-part consensus method, which is characterized by comprising the following steps:

[0008] Step 1: Establish an intelligent connected vehicle system under non-uniform sampling control;

[0009] Step 2: Obtain a hybrid distributed two-part consensus controller and a privacy-preserving noise generator for an intelligent connected vehicle system;

[0010] Step 3: Substitute the hybrid distributed two-part consensus controller and the privacy-preserving noise generator into the intelligent connected vehicle system, use discretization to obtain a new closed-loop system, and construct a Lyapunov function for the closed-loop system to obtain the asymptotically unbiased mean square output two-part consensus condition for the intelligent connected vehicle system.

[0011] Step 4: quantify the sensitivity and privacy level of the intelligent connected vehicle system in a limited time domain;

[0012] Step 5: Determine the time-varying step size and noise parameters so that the intelligent connected vehicle system can achieve a two-part consensus that meets the predefined accuracy and expected privacy upper bound in the infinite time domain.

[0013] Furthermore, the specific process of step one is, in a two-way lane, A self-driving car is driving on a level road. represents a group of vehicles, each vehicle can communicate with its neighbors to exchange information, so that Vehicle and Distance between vehicles , vehicle speed and vehicle acceleration Stay consistent;

[0014] A simplified nonlinear intelligent connected vehicle dynamics model is given as follows:

[0015] (1)

[0016] in, is a continuous time, It is The quality of the car, It is The location of the car, is the combined aerodynamic drag coefficient, is the acceleration due to gravity, is the rolling resistance coefficient, Indicates the actual driving / braking torque, is the desired driving / braking torque, is the inertial delay of vehicle dynamics, Indicates the tire radius, is the mechanical efficiency of the drive train;

[0017] The nonlinear intelligent connected vehicle dynamics model (1) is linearized using the precise feedback linearization technique.

[0018] (2)

[0019] Then get,

[0020] (3)

[0021] in, It is The control input of a car, the intelligent connected car system is represented as follows:

[0022] (4)

[0023] in, It's the state. , , are the coefficient matrices of the vehicle system, is the measured output.

[0024] Furthermore, the hybrid distributed two-part consensus controller under non-uniform sampling is designed as follows:

[0025] (5)

[0026] The privacy-preserving noise generator is designed as,

[0027] (6)

[0028] in, It is sampling points, From natural numbers, yes The left limit of time, and is the gain matrix, is the time-varying step size, is the noise parameter, noise follow Laplace and are independent of each other, is an element in the weighted adjacency matrix, express The symbol function of It is The reference state of the vehicle and satisfies , is the initial time, It is The reference state transmitted to other vehicles after adding noise. It is The reference state of the vehicle after adding noise is transmitted to other vehicles. , and are two real matrices satisfying and , The dimension is The identity matrix of .

[0029] Furthermore, in step 3, after substituting the hybrid distributed two-part consensus controller (5) and the privacy-preserving noise generator (6) into the intelligent connected vehicle system (4), we define , , get ,

[0030] (7)

[0031] in, express is a diagonal matrix with diagonal elements, , , , discretized to get,

[0032]

[0033] in, is the sampling interval, Moment, get,

[0034] (9)

[0035] (10)

[0036] in, is the Laplacian matrix, is the weighted adjacency matrix, is the Kronecker product, , is a column vector with all elements equal to 1; therefore, and The closed-loop system of is expressed in compact form as follows,

[0037] (11)

[0038] in, ;exist In the case of Hurwitz stability, define two Lyapunov functions and , ; Taking both sides at the same time, we expect , which is equivalent to and , we can conclude ,in, Indicates the expected value, symbol matrix , , It is The sign value of the elements.

[0039] In practical applications, this condition is applied to prove the asymptotically unbiased mean square output average bipartite consensus, that is,

[0040] (12)

[0041] in, is a random variable that satisfies , , Table vector The covariance matrix of Indicates The speed of a car.

[0042] Furthermore, in step 4, the privacy level is described by the sum of sensitivities. Specifically expressed as:

[0043] (13)

[0044] in, is a finite set of sampling points, is the last sampling point, and the sensitivity expression is,

[0045] (14)

[0046] in, is the minimum degree.

[0047] Furthermore, in step 5, a predefined accuracy is given and the expected privacy upper bound , assuming the time-varying step size is , the noise parameter is , where the parameter ,parameter , are all real parameters greater than zero. In the case of Hurwitz stability, we first obtain according to Chebyshev inequality ; Substituting variables, we get the inequality: ,Will After substituting, we can simplify to get The predefined accuracy is ensured ; Set the time-varying step size to And the noise parameters are Substitution get

[0048] (15)

[0049] Using the summation property of decreasing functions , simplifying and scaling it to get if the conditions are met:

[0050] (16)

[0051] The intelligent connected car system will Achieve asymptotically unbiased mean square output bipartite consensus, that is, ;

[0052] For privacy analysis in infinite time domain, In the case of Substituting into equation (14) and taking the logarithm of both sides, we obtain:

[0053] (17)

[0054] because , so we only need to analyze , the expected privacy upper bound is derived from the following four cases:

[0055] Case 1: When , ,get

[0056] (18)

[0057] Case 2: , ,get

[0058] (19)

[0059] Case 3: When , ,get

[0060] (20)

[0061] Case 4: When , ,get

[0062] (twenty one)

[0063] in, is the upper incomplete gamma function.

[0064] The present invention provides a two-part consensus method for a sampling-controlled intelligent connected vehicle system under privacy protection. In order to solve the two-part consensus and privacy protection problems between two-way two-lane intelligent connected vehicles, the present invention designs a hybrid distributed two-part consensus controller and a privacy protection noise generator, which can well control and manage the vehicles while ensuring the privacy of intelligent connected vehicles. In order to achieve a more efficient and wider communication form, the present invention uses a more complex and flexible non-uniform sampling technology and introduces non-uniform control technology into the intelligent connected vehicle system. The introduction of non-uniform sampling technology can be applied to the uncertainty of privacy protection, which can not only minimize information distortion and noise interference, but also reduce the communication burden. In order to achieve optimal privacy protection, noise design is also crucial. In many studies, differential privacy under Laplace noise is considered to be the optimal privacy. The present invention adds Laplace noise with time-varying variance, allowing the noise variance to vary with the sampling time, that is, it can be attenuated, constant or even increased, while ensuring that the information is not distorted, making it more difficult for eavesdroppers to crack the real information, and better balancing the privacy and availability of data. In summary, the present invention solves the two-part consensus problem of non-uniform sampling control of intelligent connected vehicle systems in the presence of adversarial information, while ensuring ideal privacy protection. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] Figure 1 It is a schematic diagram of the implementation process of the present invention;

[0066] Figure 2 A communication topology diagram of sampled data for automated vehicle platoon control in scenario (a) of the present invention;

[0067] Figure 3 A communication topology diagram of sampled data for automated vehicle platoon control in scenario (b) of the present invention;

[0068] Figure 4 A communication topology diagram of sampled data for automated vehicle platoon control in scenario (c) of the present invention;

[0069] Figure 5 The vehicle measurement output trajectory corresponding to scenario (a) of the present invention;

[0070] Figure 6 The vehicle measurement output trajectory corresponding to scenario (b) of the present invention;

[0071] Figure 7 The vehicle measurement output trajectory corresponding to scenario (c) of the present invention;

[0072] Figure 8 is the vehicle reference state trajectory corresponding to scenario (a) of the present invention;

[0073] Fig. 9 is the vehicle reference state trajectory corresponding to scenario (b) of the present invention;

[0074] Fig.10 is the vehicle reference state trajectory corresponding to scenario (c) of the present invention;

[0075] Fig.11 A more complex vehicle topology diagram consisting of 12 vehicles according to the present invention;

[0076] Fig.12 Output state trace for a more complex vehicle topology consisting of 12 vehicles of the present invention;

[0077] Fig.13 A reference state trajectory for a more complex vehicle topology consisting of 12 vehicles of the present invention;

[0078] Fig.14 is the non-attenuated noise trajectory of the present invention;

[0079] Fig.15 is the decaying noise trajectory whose scale parameter is a constant in the existing literature. DETAILED DESCRIPTION

[0080] like Figure 1 As shown, the present invention provides a privacy-preserving down-sampling control intelligent connected vehicle system two-part consensus method, and the specific implementation process mainly includes the following steps.

[0081] Step 1: Establish an intelligent connected vehicle system under non-uniform sampling control. The specific process is as follows: A self-driving car is driving on a level road. represents a group of vehicles, each vehicle can communicate with its neighbors to exchange information, so that Vehicle and Distance between vehicles , vehicle speed and vehicle acceleration Stay consistent;

[0082] A simplified nonlinear intelligent connected vehicle dynamics model is given as follows:

[0083] (1)

[0084] is a continuous time, It is The quality of the car, It is The location of the car, is the combined aerodynamic drag coefficient, is the acceleration due to gravity, is the rolling resistance coefficient, Indicates the actual driving / braking torque, is the desired driving / braking torque, is the inertial delay of vehicle dynamics, Indicates the tire radius, is the mechanical efficiency of the drive train;

[0085] The nonlinear intelligent connected vehicle dynamics model (1) is linearized using the precise feedback linearization technique.

[0086] (2)

[0087] Then get,

[0088] (3)

[0089] in, It is The control input of a car, the intelligent connected car system is represented as follows:

[0090] (4)

[0091] in, It's the state. , , are the coefficient matrices of the vehicle system, is the measured output.

[0092] Furthermore, the hybrid distributed two-part consensus controller under non-uniform sampling is designed as follows:

[0093] (5)

[0094] The privacy-preserving noise generator is designed as,

[0095] (6)

[0096] in, It is sampling points, From natural numbers, yes The left limit of time, and is the gain matrix, is the time-varying step size, is the noise parameter, noise follow Laplace and are independent of each other, is an element in the weighted adjacency matrix, express The symbol function of It is The reference state of the vehicle and satisfies , is the initial time, It is The reference state transmitted to other vehicles after adding noise. It is The reference state of the vehicle after adding noise is transmitted to other vehicles. , and are two real matrices satisfying and , The dimension is The identity matrix of .

[0097] Furthermore, in step 3, after substituting the hybrid distributed two-part consensus controller (5) and the privacy-preserving noise generator (6) into the intelligent connected vehicle system (4), we define , , get ,

[0098] (7)

[0099] in, express is a diagonal matrix with diagonal elements, , , , discretized to get,

[0100]

[0101] in, is the sampling interval, Moment, get,

[0102] (9)

[0103] (10)

[0104] in, is the Laplacian matrix, is the weighted adjacency matrix, is the Kronecker product, , is a column vector with all elements equal to 1; therefore, and The closed-loop system of is expressed in compact form as follows,

[0105] (11)

[0106] in, ;exist In the case of Hurwitz stability, define two Lyapunov functions and , ; Taking both sides at the same time, we expect , which is equivalent to and , we can conclude ,in, Indicates the expected value, symbol matrix , , It is The sign value of the elements.

[0107] In practical applications, this condition is applied to prove the asymptotically unbiased mean square output average bipartite consensus, that is,

[0108] (12)

[0109] in, is a random variable that satisfies , , Table vector The covariance matrix of Indicates The speed of a car.

[0110] Furthermore, in step 4, the privacy level is described by the sum of sensitivities. Specifically expressed as:

[0111] (13)

[0112] in, is a finite set of sampling points, is the last sampling point, and the sensitivity expression is,

[0113] (14)

[0114] in, is the minimum degree.

[0115] Furthermore, in step 5, a predefined accuracy is given and the expected privacy upper bound , assuming the time-varying step size is , the noise parameter is , where the parameter ,parameter , are all real parameters greater than zero. In the case of Hurwitz stability, we first obtain according to Chebyshev inequality ; Substituting variables, we get the inequality: ,Will After substituting, we can simplify to get The predefined accuracy is ensured ; Set the time-varying step size to And the noise parameters are Substitution get

[0116] (15)

[0117] Using the summation property of decreasing functions , simplifying and scaling it to get if the conditions are met:

[0118] (16)

[0119] The intelligent connected car system will Achieve asymptotically unbiased mean square output bipartite consensus, that is, ;

[0120] For privacy analysis in infinite time domain, In the case of Substituting into equation (14) and taking the logarithm of both sides, we obtain:

[0121] (17)

[0122] because , so we only need to analyze , the expected privacy upper bound is derived from the following four cases:

[0123] Case 1: When , ,get

[0124] (18)

[0125] Case 2: , ,get

[0126] (19)

[0127] Case 3: When , ,get

[0128] (20)

[0129] Case 4: When , ,get

[0130] (twenty one)

[0131] in, is the upper incomplete gamma function.

[0132] In order to verify the validity of the proposed theoretical results, the specific values ​​of the vehicle parameters used in the simulation are shown in Table 1.

[0133] Table 1 Vehicle design parameters

[0134]

[0135] Next, several basic intelligent connected vehicle platoon control sampling data communication topologies are simulated. By using two rows of intelligent connected vehicles for numerical simulation, the reference state and output state trajectories of different conditional topologies are shown in Figure 1. Figure 5-Figure 10 shown.

[0136] In this example, our goal is to achieve a predefined accuracy and the desired privacy upper bound ,in , , , each vehicle meets the requirements of the intelligent connected vehicle system (4) and the vehicle design parameters shown in Table 1, and sets the noise parameters and time-varying step size , the upper bound of non-uniform sampling is ,in , the control gain is solved by using the Riccati equation: ;

[0137] By comparing Figure 2 , attached Figure 3 and attached Figure 4 , it is found that when the topology changes, the information exchange between intelligent vehicles may become more difficult or easier, resulting in changes in the time it takes for the output to reach consensus; corresponding scenarios arrive , make the output state trajectory of the corresponding topology as Figure 5-Figure 7 As shown in Figure 2, and the reference state trajectory of the corresponding topology is shown in Figure 2 Figure 8-Figure 10 As shown, it can be seen that the convergence speeds of the output state trajectories and reference state trajectories of different communication topologies are different. For a vehicle system with a longer average communication distance traveling in opposite directions, the time required to achieve a two-part consensus is also longer.

[0138] Next, a more complex sample data communication topology for intelligent connected vehicle platoon control is simulated. Fig.11 Random topology between 12 connected cars in the communication diagram. The simulation goal is to achieve a predefined accuracy. and the desired privacy upper bound ,in , , , each vehicle meets the requirements of the intelligent connected vehicle system (4) and the values ​​in Table 1, and sets the noise parameters and time-varying step size , compared with the noise parameter design in the existing literature , similarly, non-periodic sampling is also performed within the stability threshold, and the control gain is solved by the Riccati equation as , .

[0139] Fig.12 The output status trajectories of 12 smart connected cars are displayed. Fig.13 The reference state trajectories of 12 smart connected cars are displayed. It can be seen that under the hybrid distributed two-part consensus controller and privacy-preserving noise generator designed in the present invention, the output state trajectory and the reference state trajectory of the two-lane smart connected car can reach a two-part consensus. Fig.14 The trajectory of the non-attenuated time-varying noise designed by the present invention is shown, Fig.15 The decaying time-varying noise trajectory with a constant scale parameter in the existing literature can be observed in more complex platoon topologies. The noise in the existing literature eventually decays to zero. In contrast, the information protected by noise cannot be directly inferred over time in this paper, which verifies that after obtaining suitable noise, the intelligent connected vehicle system can achieve asymptotically unbiased mean square output bipartite consensus with predefined accuracy and desired privacy upper bound, thereby realizing inter-vehicle communication information protection in the case of more vehicles and more complex topologies.

[0140] The above results show that the privacy-preserving down-sampling control two-part consensus method for intelligent connected vehicle systems proposed in the present invention effectively solves the two-part consensus problem of intelligent connected vehicle systems in the presence of adversarial information, while ensuring ideal privacy protection. The present invention uses a hybrid distributed two-part consensus controller and a privacy-preserving noise generator to achieve differential privacy protection of the asymptotically unbiased mean square output two-part consensus of the intelligent connected vehicle system. The added noise has a time-varying variance, and the noise variance is allowed to vary with the sampling time, that is, it can be attenuated, constant, or even increased. While ensuring that the information is not distorted, it makes it more difficult for eavesdroppers to crack the real information, and better balances the privacy and availability of data.

[0141] In summary, the present invention establishes an intelligent connected vehicle system by using precise feedback linearization technology. Then, a hybrid distributed two-part consensus controller is proposed to allow vehicles in the same or opposite directions to exchange information at discrete moments, and a privacy-preserving noise generator is proposed to add Laplace noise with time-varying variance to protect the sensitive information of each vehicle. Finally, the convergence accuracy range is given, the differential privacy level is quantified, and the design of time-varying step size and noise parameters is proposed. It is proved that the intelligent connected vehicle system can achieve asymptotically unbiased mean square output two-part consensus with predefined accuracy and expected privacy upper bound, and the trade-off between system performance and privacy protection is explained.

Claims

1. A two-part consensus method for sampling control of intelligent connected vehicle system under privacy protection, characterized in that: The following steps are included: Step 1: Establish an intelligent connected vehicle system under non-uniform sampling control; Step 2: Obtain a hybrid distributed two-part consensus controller and a privacy-preserving noise generator for an intelligent connected vehicle system; Step 3: Substitute the hybrid distributed two-part consensus controller and the privacy-preserving noise generator into the intelligent connected vehicle system, use discretization to obtain a new closed-loop system, and construct a Lyapunov function for the closed-loop system to obtain the asymptotically unbiased mean square output two-part consensus condition for the intelligent connected vehicle system. Step 4: quantify the sensitivity and privacy level of the intelligent connected vehicle system in a limited time domain; Step 5: Determine the time-varying step size and noise parameters so that the intelligent connected vehicle system can achieve a two-part consensus that meets the predefined accuracy and expected privacy upper bound in the infinite time domain; The specific process of step one is, in a two-way lane, A self-driving car is driving on a level road. represents a group of vehicles, each vehicle can communicate with its neighbors to exchange information, so that Vehicle and Distance between vehicles , vehicle speed and vehicle acceleration Stay consistent; A simplified nonlinear intelligent connected vehicle dynamics model is given as follows: in, is a continuous time, It is The quality of the car, It is The location of the car, is the combined aerodynamic drag coefficient, is the acceleration due to gravity, is the rolling resistance coefficient, Indicates the actual driving / braking torque, is the desired driving / braking torque, is the inertial delay of vehicle dynamics, Indicates the tire radius, is the mechanical efficiency of the drive train; The nonlinear intelligent connected vehicle dynamics model (1) is linearized using the precise feedback linearization technique. Then get, （3） in, It is The control input of a car, the intelligent connected car system is represented as follows: （4） in, It's the state. , , are the coefficient matrices of the vehicle system, is the measured output; The hybrid distributed two-part consensus controller under non-uniform sampling is designed as follows: （5） The privacy-preserving noise generator is designed as, （6） in, It is sampling points, From natural numbers, yes The left limit of time, and is the gain matrix, is the time-varying step size, is the noise parameter, noise follow Laplace and are independent of each other, is an element in the weighted adjacency matrix, express The symbol function of It is The reference state of the vehicle satisfies , is the initial time, It is The reference state transmitted to other vehicles after adding noise. It is The reference state of the vehicle after adding noise is transmitted to other vehicles. , and are two real matrices satisfying and , The dimension is The identity matrix of In step 3, after substituting the hybrid distributed two-part consensus controller (5) and the privacy-preserving noise generator (6) into the intelligent connected vehicle system (4), we define , , get , （7） in, express is a diagonal matrix with diagonal elements, , , , discretized to get, in, is the sampling interval, Moment, get, （9） （10） in, is the Laplacian matrix, is the weighted adjacency matrix, is the Kronecker product, , is a column vector with all elements equal to 1; therefore, and The closed-loop system of is expressed in compact form as follows, （11） in, ;exist In the case of Hurwitz stability, define two Lyapunov functions and , ; Taking both sides at the same time, we expect , which is equivalent to and , we can conclude ,in, Indicates the expected value, symbol matrix , , It is The sign value of the elements.

2. According to claim 1, the privacy-preserving down-sampling control intelligent connected vehicle system two-part consensus method is also characterized in that: In step 4, the privacy level is described by the sum of sensitivities. Specifically expressed as: （12） in, is a finite set of sampling points, is the last sampling point, and the sensitivity expression is, （13） in, Indicates The speed of a car, is the minimum degree.

3. According to claim 2, the privacy-preserving down-sampling control intelligent connected vehicle system two-part consensus method is also characterized in that: In step 5, given the predefined accuracy and the expected privacy upper bound , assuming the time-varying step size is , the noise parameter is , where the parameter ,parameter , are all real parameters greater than zero. In the case of Hurwitz stability, we first obtain according to Chebyshev inequality ,in is a random variable that satisfies , , Table vector The covariance matrix of ; Substituting the variables gives the inequality ,Will After substituting, we can simplify to get The predefined accuracy is ensured ; Set the time-varying step size to And the noise parameters are Substitution get （14） Using the summation property of decreasing functions , simplifying and scaling it to get if the condition is met （15） The intelligent connected car system will Achieve asymptotically unbiased mean square output bipartite consensus, that is, ; In infinite time domain, the expected privacy upper bound is given in the following four cases: Case 1: When , ,get （16） Case 2: , ,get （17） Case 3: When , ,get （18） Case 4: When , ,get （19） in, is the upper incomplete gamma function.

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