阅读说明：本技术 无人机航迹规划的无损凸优化实现方法 (Lossless convex optimization implementation method for unmanned aerial vehicle flight path planning ) 是由 李建勋 张哲� 于 2018-08-06 设计创作，主要内容包括：一种无人机航迹规划的无损凸优化实现方法,基于无损凸化思想和广义Banders分解算法,利用一系列凸优化和混合整数线性规划获得非凸混合整数非线性规划问题的最优解,在最小成本的基础上大大提高了计算效率。(A lossless convex optimization implementation method for unmanned aerial vehicle track planning is based on a lossless convex idea and a generalized Banders decomposition algorithm, and utilizes a series of convex optimization and mixed integer linear programming to obtain an optimal solution of a non-convex mixed integer nonlinear programming problem, so that the calculation efficiency is greatly improved on the basis of the minimum cost.)

1. A non-destructive convex optimization implementation method for unmanned aerial vehicle flight path planning is characterized by comprising the following steps:

modifying a linear target function of an unmanned aerial vehicle track planning problem into a non-convex function related to energy consumption, modifying a linear state constraint into a convex function state constraint, modifying a control constraint into a non-convex norm range constraint, and then establishing a non-linear optimal control problem model of the unmanned aerial vehicle track planning;

step two, relaxing the nonlinear optimal control problem model obtained in the step one to obtain a convex mixed integer linear programming model, wherein the convex mixed integer linear programming model is obtained by introducing a relaxation variable gamma, converting a non-convex mixed integer nonlinear programming problem into a convex mixed integer nonlinear programming problem, introducing an auxiliary variable β, and eliminating a target function of the relaxation problem to obtain a feasibility problem of the relaxation problem, introducing an auxiliary variable α, and obtaining a generalized Banders decomposition problem by a relaxation method;

step three, according to the starting point position x of the unmanned aerial vehicle₀And the end position x_fMaximum speed V of unmanned aerial vehicle_maxMaximum acceleration A_maxBarrier and the likeAssigning obstacle information to corresponding variables in the nonlinear optimal control problem model of the unmanned aerial vehicle track planning in the step one to obtain an initial planning track;

step four, obtaining an optimal solution through an iterative convex mixed integer nonlinear programming problem, a feasibility problem of a relaxation problem and a generalized Banders decomposition problem;

step five, according to the obtained optimal solution of the convex relaxation problem in the step four, the introduced relaxation variable gamma, the auxiliary variable β and the auxiliary variable α are omitted, and the optimal solution of the nonlinear optimal control problem model of the unmanned aerial vehicle track planning, namely the complete optimal track, is obtained;

the nonlinear optimal control problem model for unmanned aerial vehicle track planning is as follows:

s.t.x(i+1)＝A(i)x(i)+B(i)u(i),i＝0,1,…,T-1

2. The method of claim 1, wherein the convex mixed integer non-linear programming problem is:wherein: Γ is the newly introduced relaxation variable. Setting a new variable z ═ x^T,u^T,Γ)^TObtaining an equivalent simplified model on the right side, wherein f (z) is an objective function of equivalent transformation, g (z, b) is less than or equal to 0 and is a constraint condition of simplification,is the feasible field of the new variable z pair.

3. The method of claim 1, wherein the feasibility problem of the convex relaxation problem is:

4. The method of claim 1, wherein the generalized Banders decomposition problem is:

5. The method of claim 1, wherein said iterating comprises:

1) judging whether the convex relaxation problem is feasible, and executing the step 2) when the convex relaxation problem is feasible, or executing the step 3); wherein the convex relaxation problem is the convex mixed integer nonlinear programming problem in the step two;

the judgment means that: if the feasibility problem of the relaxation problem in the step two is solved, the convex relaxation problem is feasible, otherwise, the convex relaxation problem is not feasible;

2) solving the convex relaxation problem and obtaining an original dual optimal solution pair (z)^k,λ^k) Then, for index set I^kAnd J^kMake a correction, J^k＝J^k-1∪{k},I^k＝I^k-1And sets the upper bound Up of the problem^k＝min{Up^k-1,f(z^k)}；

3) Solving the feasibility problem of the convex relaxation problem and obtaining the original dual optimal solution pair (z)^k,μ^k) Then, for index set I^kAnd J^kMake a correction, J^k＝J^k-1∪{k},I^k＝I^k-1；

4) Solving the generalized Banders decomposition problem and obtaining the original dual optimal solution pair

5) Step 6) is executed when the upper and lower boundaries of the problem are not equal and the error range is not within the error range, otherwise step 7) is executed;

6) by accumulating the metrics for an updated metric set and assigning k to k +1 to the corresponding variable,

7) and combining the iteration of the previous k steps to obtain a solution pair of the problem and outputting the optimal solution of the problem.

6. A system for implementing the method of any preceding claim, comprising: the input module is connected with the modeling module and transmits input information, the modeling module is connected with the resolving module and transmits model information, the resolving module is connected with the output module and transmits resolving information, and the input module is connected with the modeling module and transmits the input information, wherein: and inputting various conditions and constraints of actual problems to the input module, standardizing the actual problems by the input module and then transmitting the standardized actual problems to the modeling module, establishing a nonlinear optimal control problem of the unmanned aerial vehicle flight path planning by the modeling module according to input information and transmitting the nonlinear optimal control problem to the resolving module, transmitting the resolving module to the output module according to the model information and an iterative resolving method in the fourth step, solving the problem result, and outputting the obtained resolving information after the output module standardizes the resolving result.

Technical Field

The invention relates to a technology in the field of unmanned aerial vehicle control, in particular to a lossless convex optimization method for a non-convex unmanned aerial vehicle flight path planning problem with control constraint.

Background

Unmanned aerial vehicle route planning refers to planning one or more safe flight routes based on the factors such as the self performance of the unmanned aerial vehicle, the task time, the energy consumption, the enemy information, the terrain environment, the weather conditions and the like. Due to the fact that the factors to be considered are multiple and complex, a plurality of technical problems need to be solved and perfected in unmanned aerial vehicle track research. In order to describe specific constraints, a 0-1 variable is introduced into a model to express a logical relationship, so that the original linear programming problem becomes a mixed integer nonlinear programming problem and even a non-convex mixed integer nonlinear programming problem.

In most cases, the solution obtained by directly solving the non-convex optimization problem is a feasible solution or a local minimum solution rather than a global optimal solution. The non-convex mixed integer programming problem is characterized in that integer variables are added on the basis of the non-convex problem, so that the difficulty of the problem is further increased, and the problem becomes one of the most difficult problems to solve in the optimization field.

Disclosure of Invention

Aiming at the defects in the prior art, the invention provides a lossless convex optimization realization method for unmanned aerial vehicle track planning, which is based on a lossless convex idea and a generalized Banders decomposition algorithm and utilizes a series of convex optimization and mixed integer linear programming to obtain an optimal solution of a non-convex mixed integer nonlinear programming problem, thereby greatly improving the calculation efficiency on the basis of the minimum cost.

The invention is realized by the following technical scheme:

the invention relates to a lossless convex optimization implementation method for unmanned aerial vehicle track planning, which comprises the following steps:

the method comprises the steps of firstly, correcting a linear target function of an unmanned aerial vehicle track planning problem into a non-convex function related to energy consumption, correcting a linear state constraint into a convex function state constraint, correcting a control constraint into a non-convex norm range constraint, and then establishing a non-linear optimal control problem model of the unmanned aerial vehicle track planning.

The nonlinear optimal control problem model for unmanned aerial vehicle track planning is as follows:

wherein: j is an objective function, x is a state variable, u is a control input, and the current time step is i belongs to [0, T ∈]，

For a convex function of the control variable u, l is

A, B are constant matrices to describe the system equation, M, p, q, r are constant matrices or constant vectors to describe the constraints of the state variable x, p₁,ρ₂Two constants are used to describe the upper and lower bounds of the control variable u within a certain range, and δ is the buffer time domain.

Is an affine function with respect to x and b, where b is a 0-1 variable used to describe the logical relationship of a particular constraint, C is a constant matrix used to describe the logical variable b, x₀And x_fRespectively an initial state and a terminal state,

is a feasible field of the state variable,for control of the feasible fields of the input, n_bThe dimension of the logical variable b.

And step two, relaxing the nonlinear optimal control problem model obtained in the step one to obtain a convex mixed integer linear programming model, specifically, converting the non-convex mixed integer nonlinear programming problem into a convex mixed integer nonlinear programming problem by introducing a relaxation variable gamma, introducing an auxiliary variable β, eliminating a target function of the relaxation problem to obtain a feasibility problem of the relaxation problem, and introducing an auxiliary variable α to obtain a generalized Banders decomposition problem by a relaxation method.

The convex mixed integer nonlinear programming problem is as follows:

wherein: Γ is the newly introduced relaxation variable. Setting a new variable z ═ x^T,u^T,Γ)^TObtaining an equivalent simplified model on the right side, wherein f (z) is an objective function of equivalent transformation, g (z, b) is less than or equal to 0 and is a constraint condition of simplification,

is the feasible field of the new variable z pair.

The feasibility problem of the convex relaxation problem is that:

β is an introduced auxiliary variable used to judge the feasibility of the problem, g_j(z, b) is a concrete representation of each line of the function vector g (z, b), n_qIs the number of rows. n is_bIs the dimension of the logical variable b.

The generalized Banders decomposition problem refers to:

wherein α is an introduced auxiliary variable, λ is a Lagrangian variable, and L (z)ⁿ,bⁿ,λⁿ) A simplified form of lagrangian function for the convex mixed integer nonlinear programming problem in step two,

is the gradient of the Lagrangian function with respect to the variable b, n being the index variable, I^kAnd J^kAnd mu is a dual variable of z, which is an index set of the kth iteration.

Step three, according to the starting point position x of the unmanned aerial vehicle₀And the end position x_fMaximum speed V of unmanned aerial vehicle_maxMaximum acceleration A_maxAssigning the barrier information to corresponding variables in the nonlinear optimal control problem model of the unmanned aerial vehicle flight path planning in the step one to obtain an initial planning flight path;

step four, obtaining an optimal solution through an iterative convex mixed integer nonlinear programming problem, a feasibility problem of a relaxation problem and a generalized Banders decomposition problem, and specifically comprising the following steps of:

1) judging whether the convex relaxation problem is feasible, and executing the step 2) when the convex relaxation problem is feasible, or executing the step 3); wherein the convex relaxation problem is the convex mixed integer nonlinear programming problem in the step two.

The judgment means that: and if the feasibility problem of the relaxation problem in the step two is solved, the convex relaxation problem is feasible, otherwise, the convex relaxation problem is not feasible.

2) Solving the convex relaxation problem and obtaining an original dual optimal solution pair (z)^k,λ^k) Then, for index set I^kAnd J^kMake a correction, J^k＝J^k-1∪{k},I^k＝I^k-1And sets the upper bound Up of the problem^k＝min{Up^k-1,f(z^k)}；

3) Solving the feasibility problem of the convex relaxation problem and obtaining the original dual optimal solution pair (z)^k,μ^k) Then, for index set I^kAnd J^kMake a correction, J^k＝J^k-1∪{k},I^k＝I^k-1；

4) Solving the generalized Banders decomposition problem and obtaining an original dual optimal solution pair (α)^k,

) Further setting the lower limit Low of the problem^k＝α^k；

5) Step 6) is executed when the upper and lower boundaries of the problem are not equal and the error range is not within the error range, otherwise step 7) is executed;

6) by accumulating the metrics for an updated metric set and assigning k to k +1 to the corresponding variable,

return to step 2

7) And combining the iteration of the previous k steps to obtain a solution pair of the problem and outputting the optimal solution of the problem.

And step five, according to the obtained optimal solution of the convex relaxation problem in the step four, eliminating the introduced relaxation variable gamma, the auxiliary variable β and the auxiliary variable α to obtain the optimal solution of the nonlinear optimal control problem model of the unmanned aerial vehicle track planning, namely the complete optimal track.

The invention relates to a system for realizing the method, which comprises the following steps: the input module is connected with the modeling module and transmits input information, the modeling module is connected with the resolving module and transmits model information, the resolving module is connected with the output module and transmits resolving information, and the input module is connected with the modeling module and transmits the input information, wherein: and inputting various conditions and constraints of actual problems to the input module, standardizing the actual problems by the input module and then transmitting the standardized actual problems to the modeling module, establishing a nonlinear optimal control problem of the unmanned aerial vehicle flight path planning by the modeling module according to input information and transmitting the nonlinear optimal control problem to the resolving module, transmitting the resolving module to the output module according to the model information and an iterative resolving method in the fourth step, solving the problem result, and outputting the obtained resolving information after the output module standardizes the resolving result.

Technical effects

Compared with the prior art, the linear model of the traditional unmanned aerial vehicle flight path planning is modified according to the actual situation, the non-convex flight path planning problem with control constraint is formed, the non-convex mixed integer non-linear planning problem is solved, the optimal solution of the original problem is obtained by utilizing convex optimization and mixed integer linear planning through the ideas of generalized Banders decomposition and lossless convexity, and the calculation time is greatly reduced on the basis of obtaining the minimum cost.

Drawings

FIG. 1 is a schematic flow diagram of the present invention;

FIG. 2(a) is a three-dimensional track chart of an example in which the size of obstacles is 6;

FIG. 2(b) is a three-dimensional track chart in the case where the obstacle size is 11 in the embodiment;

FIG. 3(a) is a top two-dimensional track chart of the embodiment with 6 obstacle scales;

FIG. 3(b) is a top two-dimensional track chart of the embodiment where the obstacle size is 11;

FIG. 4(a) is a graph showing simulation results in the example in which the obstacle size is 6;

fig. 4(b) is a graph of 11 simulation results of the obstacle size in the example.

Detailed Description

As shown in fig. 1, a non-destructive convex optimization method for a non-convex unmanned aerial vehicle flight path planning problem with control constraints according to this embodiment includes the specific steps of:

step 1, initialization: initial conditions of the input question, arbitrary

An error epsilon;

step 2, judging the convex relaxation problem PR (b)^k) If it is feasible, execute step 3 when feasibleOtherwise, executing step 4;

step 3, solving the convex relaxation problem PR (b)^k) Obtaining the optimal original couple (z)^k,λ^k) Setting I^k＝I^k-1∪{k},J^k＝J^k-1，Up^k＝min{Up^k-1,f(z^k) When Up^k＝f(z^k) Then (z)^*,b^*)＝(z^k,b^k)；

Step 4, solving the convex relaxation problem PRF (b)^k) Obtaining the optimal original couple (z)^k,λ^k) Setting J^k＝J^k-1∪{k},I^k＝I^k-1；

Step 5, solving the generalized Banders decomposition problem PR-GBD_kObtaining the optimal original pair (α)^k,

)，Low^k＝α^k；

Step 6, when Up^k-Low^kLess than or equal to epsilon or k is more than k_maxIf not, executing step 8, otherwise, executing step 7;

step 7, updating k to k +1,

and executing the step 2;

step 8, when k is less than or equal to k_maxOutput z^*The first two items ofOtherwise the output is not feasible.

Real-time data

Firstly, an unmanned aerial vehicle is considered to carry out track planning in urban environment in implementation, building modeling with different sizes is realized by taking a cuboid as an obstacle, two map environment scenes with different sizes and different obstacle numbers are specifically implemented, and specific scene parameters are shown in table 1

Table 1 specific scene parameter settings

Second, setting is implemented

The algorithm operating environment is as follows: MATLAB2014a, CPU model Intel Core i7, main frequency 6.4GHz and 8GB memory. The convex optimization problem solver is CVX, and the non-convex optimization problem solver is CPLEX. The specific implementation settings are shown in Table 2

Table 2 implementation environment settings

Third, implement the content

In the implementation, CPLEX is adopted to directly solve the mixed integer programming problem and the lossless convex generalized Banders decomposition algorithm is adopted to carry out comparison, wherein the direct solving method is marked as MINLP, the algorithm of the invention is marked as LC-GBD, the implementation result is shown in table 1, and the path comparison graph generated by the implementation is shown in fig. 2-3.

TABLE 3 implementation results for different scenarios

As shown in fig. 2, a 3D flight path display diagram including 6 obstacle scenes and a 3D flight path display diagram including 11 obstacle scenes are shown, as can be seen: the direct solving method is compared with the result track 3d effect of the method, and the track effect of the method is better.

Fig. 3 shows a 2D track top view containing 6 obstacle scenes and a 2D track top view containing 11 obstacle scenes, as can be seen: the direct solving method is compared with the result track 2d effect of the method, and the track effect of the method is better. .

As shown in fig. 4, the histogram and the line chart of the implementation result of scene containing 6 obstacles and the histogram and the line chart of the implementation result of scene 2 containing 11 obstacles are shown as follows: the comparison between the direct solving method and the objective function value and solving time of the method shows that the method has better effect.

From the implementation results of the above tables and graphs, it can be seen that when the lossless saliency method is used, the obtained solution is better than the solution for directly solving the non-convex problem, so that the cost is lower. Therefore, the method provided by the invention can obtain the optimal solution of the original problem, and simultaneously greatly reduces the calculation time.

The foregoing embodiments may be modified in many different ways by those skilled in the art without departing from the spirit and scope of the invention, which is defined by the appended claims and all changes that come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein.