CN105654187A - Grid binary tree method of control system midpoint locating method - Google Patents

Abstract

A grid binary tree method is divided into two main phases-an offline preprocessing phase and an online computation phase. The theory of multi-parameter secondary programming is introduced in the offline preprocessing phase. A computer can autonomously divide the state space of a control system into convex partitions and obtains control rate corresponding to each partition through computation. Then we construct a hash table grid area polytope according to division parameters and construct a binary tree in a grid area in which a conflict exists. In the online computation phase, firstly the located grid area is rapidly determined according to the coordinates of state points, the control rate of the state points is obtained through screening of the constructed binary tree or is directly obtained, and control output quantity of the system is obtained through simple linear operation.

Description

Grid Binary Tree Method for Central Point Locating Method in Control System

technical field

The present invention is aimed at the optimization of the midpoint positioning method of explicit model predictive control. Compared with the traditional binary tree method, the grid binary tree method only needs to deal with fewer polytopic partition scales, reduces the complexity of preprocessing, and greatly reduces preprocessing time. At the same time, it also solves the conflict problem in the hash table method, and improves the online calculation time of the hash table method. In terms of the performance of storage space requirements, it also meets our requirements for the control system.

Background technique

In the traditional model predictive control, there are repeated online optimization calculations, which cause the controller to be overloaded and inefficient. In order to solve these problems, scholars such as Manfred Morari and Alberto Bemporad introduced the multi-parameter quadratic programming theory around 2002, and established an explicit model predictive control method system. It mainly uses the inherent piecewise affine law of the model predictive control system, according to the model, constraints, performance requirements and other information of the control object, and divides the system state space through multi-parametric Quadratic Program (mp-QP). For each convex partition and pre-calculate the corresponding optimal control rate on each partition. This means that the complex and time-consuming online optimization process in traditional model predictive control is completed before the actual operation of the control system, while online control only needs to determine the partition where the current state point of the system is located, and the corresponding optimal control rate can be obtained. The efficiency of this search operation is much higher than repeated online optimization calculation, the real-time performance of the control system is greatly improved, and the requirements for the software and hardware of the control system are also reduced.

According to the above introduction, we can know that the main task of the explicit model predictive control online control stage is to solve the problem of point positioning. As the name suggests, the point location problem refers to judging which partition the state point in the space is in. The partition here refers to the convex partitions that divide the state space into convex partitions through multi-parameter quadratic programming (mp-QP). Excellent control. The performance of the point location method we adopt is directly related to the performance of the explicit model predictive control system. The performance of the point location method here refers to three aspects: the storage space occupied by the data, the offline computing time and the online computing time.

Traditional point location methods include direct search method, reachable partition method, hash table method, binary tree method, etc. There are many public documents that introduce these methods in detail, so I won’t repeat them here. Although they can actually and effectively solve the point positioning problem, they can no longer meet our control needs in terms of performance. Compared with other point positioning methods, the traditional binary tree method has unmatched advantages in storage space requirements and online search efficiency, but its preprocessing time cannot meet the requirements of our control system. The traditional hash table method The online efficiency exhibited also leaves us speechless. Here we hope to propose a new point positioning method, which needs to retain the advantages of the traditional binary tree method and hash table method, and also has a good performance in terms of storage space requirements.

Contents of the invention

The present invention overcomes the above-mentioned shortcomings of the traditional point positioning method, and provides a grid binary tree method. It not only fully retains the low preprocessing time and high online efficiency exhibited by the traditional binary tree method and hash table method, but also somewhat inherits the advantages of the binary tree method in terms of storage space requirements.

The essence of point positioning is to determine the partition of a certain point in the space, and then obtain the control rate of this partition to achieve the control effect. The most complicated and time-consuming operation in the preprocessing process of the binary tree method is to select a group of the most suitable combinations from a large number of partition boundary hyperplanes to build a binary tree. This process requires repeated operations when building each node of the binary tree. For calculation and comparison, the amount of calculation increases exponentially with the dimension and number of partitions. The most time-consuming step in the hash table method is to determine the partition where the state point is located by directly searching during the online stage. So here we combine the two methods, take the essence and discard the dross, and propose the grid binary tree method.

The grid binary tree method is divided into two main stages - offline preprocessing stage and online calculation stage. The multi-parameter quadratic programming theory is introduced in the offline preprocessing stage. The computer can divide the state space of the control system into convex partitions and calculate the control rate corresponding to each partition. Then we construct a hash table according to the partition parameters. The grid area is polytopic, and a binary tree is constructed in the conflicting grid area. In the online calculation stage, the grid area is quickly determined according to the coordinates of the state points, and the control rate of the state points is obtained through the established binary tree screening or directly, and the control output of the system is obtained through simple linear operations.

The grid binary tree method of the midpoint location problem of the control system of the present invention specifically comprises the following steps:

Step 1. Offline preprocessing process of grid binary tree method

1.1. Introduce multi-parameter quadratic programming into the control system, divide the system state space into convex partitions, and calculate the control rate corresponding to each partition, and save it in the FG array.

1.2. The synonymous partitions are calculated and grouped by the formula for determining the synonymous partitions, and only one eigenvalue data is reserved for each group of synonymous partitions, thus eliminating redundant data in the eigenvalue array FG.

The division of partitions in space is based on eigenvalues—all points in the same partition have the same eigenvalues. We refer to partitions with equal eigenvalues as synonymous partitions. The partition eigenvalues in explicit model predictive control (here, the control rate of explicit model predictive control) are called FG matrix. For example, the FG matrix of a two-dimensional state space partition P with an explicit model predictive control output dimension of 1 is:

FG ₁ =[f ₁₁ f ₁₂ g ₁ ](1)

The FG matrix of another partition Q adjacent to it is:

FG ₂ ＝[f ₂₁ f ₂₂ g ₂ ](2)

If satisfied:

(f ₁₁ -f ₂₁ ) ² +(f ₁₂ -f ₂₂ ) ² +(g ₁ -g ₂ ) ² ≤δ(3)

Among them, the formula (3) is the formula for determining the synonymous partition, f and g are the elements that make up the eigenvalue matrix, and the control output we need is calculated from the eigenvalue and the state vector. When δ is a very small positive number, it is considered that P and Q are synonymous, and they are mutually synonymous partitions.

1.3. Calculate the hash function according to the partition parameters, and record the obtained data in an array. We name the array Fhash. The hash function is as follows:

$\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; f</span>}\mathrm{<span\; class="notranslate">\; l</span>}\mathrm{<span\; class="notranslate">\; o</span>}\mathrm{<span\; class="notranslate">\; o</span>}\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\frac{\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

N here represents the division parameter, a and b are the boundary coordinates on a certain dimension, and the data we need to record in the array are -a and In the online calculation stage, the grid area of the hash table can be quickly determined by the coordinate X of the state point.

1.4. Construct the polytope of the grid area of the hash table according to the partition parameters, select the first grid area to intersect with the partition in order, and count the number of eigenvalues in this grid area.

1.5. Determine whether there is a conflict in the grid area. If the number of eigenvalues in the grid area is greater than 1, there is a conflict. If there is a conflict, skip to 1.7 and start building a binary tree in the grid area. If there is no conflict, go to the next step.

1.6, judge whether the grid area is completely outside the object space. If the grid area does not intersect with any partition, it will be directly recorded as 0 in the corresponding position in the Hash array. If there is only one kind of eigenvalue in the grid area, record the eigenvalue number directly in the corresponding position in the Hash array. Skip to 1.19 after completing this step.

1.7, first fill in the address of the root node of the binary tree in the Hash array, and mark it as a negative value. Then remove the linearly dependent hyperplanes in the grid region and the outer boundaries of the object space, and deselect them as candidate hyperplanes. Here, each partition in the object space is divided by a hyperplane. The principle of the binary tree method is to judge the positional relationship between the point and the hyperplane at each node, determine which side the state point is on the hyperplane, and exclude the near After half of the partitions, enter the next node to continue to judge, and finally get the partition where the status point is located. Therefore, it is redundant to use the outer boundary of the object space as the basis for node judgment, and one side of it is still the entire object space, which cannot be excluded.

1.8, group the partitions by eigenvalues (the eigenvalues here are the control rates), group the partitions with the same eigenvalues, and combine the same eigenvalues into one data.

1.9. Calculate the pole coordinates in each group of partitions, and eliminate the duplicate coordinates in each group of pole coordinates, and enter the root node.

1.10, extract the first hyperplane from the hyperplanes to be selected at the current node.

1.11, count the number of eigenvalues on both sides of the hyperplane. This statistical method is an important step in building a binary tree. It can count the number of eigenvalues on both sides of the hyperplane without judging all partition poles, which greatly shortens the preprocessing time. The main steps are as follows:

a, Load partitioned pole data grouped by properties with equal eigenvalues.

b. Load the hyperplane to be judged, and extract the coordinates of the first pole of the first group. We define the two sides of the hyperplane as Hp- and Hp+ respectively, and the number of eigenvalues on both sides is m and n respectively, and shilling m and n is all 0.

c, we use Lf and Rf as the markers of whether the pole is in Hp- and Hp+, the value is 0 for false, and the value for 1 is true. Both Lf and Rf are shilled.

d, Judging the positional relationship between the pole and the hyperplane. For the hyperplane Hp={x|hx=k}, if the point x satisfies hx≤k, then the point x is considered to be located at Hp-, otherwise it is located at Hp+. Among them, h and k are hyperplane expression parameters, and x is the coordinate of the state point to be judged.

e, if the pole is at Hp-, set Lf=1, go to g, otherwise go to the next step.

f, judge whether the pole is located at Hp+, if true, set Rf=1. Otherwise jump to h.

g, judge whether Rf=1 and Lf=1 are true at the same time, if it is false, go to the next step, if it is true, jump to i.

h, judge whether it is the last pole of this group, if it is false, extract the coordinates of the next pole of this group, and jump to d. If true, the value of m is incremented by 1 if Lf=1, and the value of n is incremented by 1 if Rf=1.

i, judge whether this group is the last group of pole data, if it is false, extract the first pole coordinates of the next group, and jump to c, otherwise the statistics of the eigenvalues on both sides of the hyperplane are completed.

1.12. Determine whether this is the last hyperplane to be selected. If it is false, extract the next hyperplane to be selected. Go to 1.11 to count the number of eigenvalues on both sides of the hyperplane. If it is true, go to the next step.

1.13, determine the reference hyperplane according to the index. We hope to build a binary tree that is deep and has few nodes. It is impossible for us to try all combinations to build all possible binary trees and select the best one. We only need to consider that the number of eigenvalues on both sides of the node (that is, on both sides of the hyperplane) is roughly the same, and we think that the hyperplane is more suitable as a reference hyperplane. The description indicators are as follows:

J＝(m+n) ² +(mn) ²

m and n are the number of eigenvalues at Hp- and Hp+ respectively, and the smaller J is, the more suitable the hyperplane is to be called the reference hyperplane. The sum of the number of eigenvalues on both sides describes the expectation of the number of nodes of the binary tree, and the difference of the eigenvalues on both sides describes the expectation of the balance of the left and right subtrees of the binary tree.

1.14. Determine whether the left subtree is established. If true, jump to 1.16, otherwise go to the next step.

1.15. Pass the pole on the Hp-side of the reference hyperplane to the left child node, remove the reference hyperplane from the candidate hyperplane and pass it to the left child node, and then jump to 1.10 after entering the left child node.

1.16. Determine whether the right subtree is established. If true, jump to 1.18, if false, go to the next step.

1.17, transfer the pole located on the reference hyperplane Hp+ side to the right child node, remove the reference hyperplane from the candidate hyperplane and pass it to the right child node, and jump to 1.10 after entering the right child node.

1.18, return to the parent node, and judge whether the binary tree has been established, if it is false, jump to 1.12, if it is true, save the data, and the binary tree in the grid area has been established.

1.19. Determine whether the current operating grid area is the last grid area. If not, select the next grid area and intersect it with the partition, count the number of eigenvalues in it, and jump to 1.5. If it is the last grid area, it means that the preprocessing step has been completed, and the operation ends.

Step 2. Online calculation process of grid binary tree method

2.1. Read the coordinates of the target point, quickly locate the grid area according to the hash function, and read the corresponding records in the Hash array.

2.2, judge whether the recorded value is negative. If the record value is negative, skip to 2.4 and perform a binary tree search. If the recorded value is not negative, proceed to the next step.

2.3, judge whether the recorded value is positive or not. If it is not positive, it means that the record value is 0, the grid area is completely outside the object space, and the state point is not in the object space, and the online search stage is directly exited. If the record value is positive, it means that there is only one kind of eigenvalue in the grid area, and the record value is the eigenvalue number, and the eigenvalue can be obtained directly, and the online search phase ends.

2.4, the reverse of the recorded value is the address of the root node, and enter the root node.

2.5. Determine the relationship between the target point and the reference hyperplane at the node. If the target point is on the Hp- side, enter the left child node, if the target point is on the Hp+ side, enter the right child node.

2.6. Determine whether the node is the last binary tree node. If it is false, go to 2.5. If it is true, go to the next step.

2.7. Determine the positional relationship between the target point and the last reference hyperplane. If it is located at Hp-, select the left cotyledon; if it is located at Hp+, select the right cotyledon. According to the corresponding eigenvalue number on the leaf node, the eigenvalue is extracted from the eigenvalue matrix FG, and the point positioning is completed in the online calculation stage.

The advantage of the present invention is that a very good trade-off has been achieved in terms of storage space requirements, offline preprocessing time and online search time. Compared with the traditional binary tree method and hash table method, the present invention has low off-line preprocessing time and high on-line search efficiency, and at the same time has achieved certain improvements in storage space requirements.

Description of drawings

Fig. 1 is a schematic diagram of the state space partition of the present invention

Fig. 2 is a schematic diagram of grid areas where conflict exists in the present invention

Fig. 3 is the binary tree schematic diagram that the present invention establishes

Fig. 4 is the flow chart of judging the number of eigenvalues on both sides of the hyperplane of the present invention

Fig. 5 is the off-line preprocessing flowchart of the present invention

Fig. 6 is a flow chart of the online computing stage of the present invention

Table 1 is the performance comparison of the grid binary tree method of the present invention with the classical point location method

detailed description

The steps of the grid binary tree method of the present invention will be further described below in conjunction with the accompanying drawings. Referring to accompanying drawings 1-6, table 1.

Grid binary tree method of the present invention, concrete steps are as follows:

Step 1. The offline preprocessing process of the grid binary tree method, the flow chart is shown in Figure 4 and Figure 5

1.1. Introduce multi-parameter quadratic programming into the control system, divide the system state space into convex partitions, and calculate the control rate corresponding to each partition, and save it in the FG array. The schematic diagram of the state space partition is shown in Figure 1.

1.2. Calculate and group the synonymous partitions by the formula used to determine the synonymous partitions. Only one eigenvalue data is reserved for each group of synonymous partitions.

1.3. Calculate the hash function according to the partition parameters, and record the obtained data in an array.

1.4. Construct the polytope of the grid area of the hash table according to the partition parameters, select the first grid area to intersect with the partition in order, and count the number of eigenvalues in this grid area.

1.5, to determine whether there is a conflict in the grid area, the schematic diagram is shown in Figure 2. If the number of eigenvalues in the grid area is greater than 1, there is a conflict. If there is a conflict, skip to 1.7 and start building a binary tree in the grid area. If there is no conflict, go to the next step.

1.6, judge whether the grid area is completely outside the object space. If the grid area does not intersect with any partition, it will be directly recorded as 0 in the corresponding position in the Hash array. If there is only one kind of eigenvalue in the grid area, record the eigenvalue number directly in the corresponding position in the Hash array. Skip to 1.19 after completing this step.

1.7, fill in the address of the root node of the binary tree in the Hash array, and mark it as a negative value. A binary tree is established in the grid area, and the schematic diagram of the established binary tree is shown in FIG. 3 .

1.8, group partitions by eigenvalues, partitions with the same eigenvalues are grouped together, and combine the same eigenvalues into one data.

1.9. Calculate the pole coordinates in each group of partitions, and eliminate the duplicate coordinates in each group of pole coordinates, and enter the root node.

1.10, extract the first hyperplane from the hyperplanes to be selected at the current node.

1.11, count the number of eigenvalues on both sides of the hyperplane.

1.12. Determine whether this is the last hyperplane to be selected. If it is false, extract the next hyperplane to be selected. Go to 1.11 to count the number of eigenvalues on both sides of the hyperplane. If it is true, go to the next step.

1.13, determine the reference hyperplane according to the index. Considering that the number of eigenvalues on both sides of the node (that is, on both sides of the hyperplane) is roughly the same, it is considered that the hyperplane is more suitable as a reference hyperplane.

1.14. Determine whether the left subtree is established. If true, jump to 1.16, otherwise go to the next step.

1.15, pass the pole on the side of the reference hyperplane to the left child node, remove the reference hyperplane from the candidate hyperplane and pass it to the left child node, and then jump to 2.10 after entering the left child node.

1.16. Determine whether the right subtree is established. If true, jump to 1.18, if false, go to the next step.

1.17, pass the pole on the other side of the reference hyperplane to the right child node, remove the reference hyperplane from the candidate hyperplane and pass it to the right child node, and then jump to 1.10 after entering the right child node.

1.18, return to the parent node, and judge whether the binary tree has been established, if it is false, jump to 1.12, if it is true, save the data, and the binary tree in the grid area has been established.

1.19. Determine whether the current operating grid area is the last grid area. If not, select the next grid area and intersect it with the partition, count the number of eigenvalues in it, and jump to 1.5. If it is the last grid area, it means that the preprocessing step has been completed, and the operation ends.

Step 2. The online calculation process of the two-level grid method, the flow chart is shown in Figure 5

2.1. Read the coordinates of the target point, quickly locate the grid area according to the hash function, and read the corresponding records in the Hash array.

2.2, judge whether the recorded value is negative. If the record value is negative, skip to 2.4 and perform a binary tree search. If the recorded value is not negative, proceed to the next step.

2.3, judge whether the recorded value is positive or not. If it is not positive, it means that the record value is 0, the grid area is completely outside the object space, and the state point is not in the object space, and the online search stage is directly exited. If the record value is positive, it means that there is only one eigenvalue in the grid area, and the record value is the eigenvalue number, and the eigenvalue can be obtained directly, and the online search stage is exited.

2.4, the reverse of the recorded value is the address of the root node, and enter the root node.

2.5. Determine the relationship between the target point and the reference hyperplane at the node.

2.6. Determine whether the node is the last binary tree node. If it is false, go to 2.5. If it is true, go to the next step.

2.7. Judging the positional relationship between the target point and the last reference hyperplane and obtaining the eigenvalue of the partition where the state point is located accordingly, exiting the online search stage.

case analysis

The present invention compares the performance of the grid binary tree method and the classic point positioning method through a second-order example, and demonstrates its superiority in storage space requirements, offline preprocessing time, and online calculation time.

Table 1 is the performance comparison between the grid binary tree method of the present invention and the classical point positioning method. Table 1 Performance comparison between fast binary tree method and other algorithms

From the table, it is not difficult to find that the grid binary tree method has greatly improved the preprocessing time compared with the traditional binary tree method, and it is also comparable to other point positioning methods. At the same time, compared with the hash table method, its average online calculation time is greatly shortened, and its performance in storage space requirements is not bad, which can meet the needs of our control system.

Claims (1)

1. The grid binary tree method for the midpoint location problem of the control system, which specifically includes the following steps: Step 1. Grid binary tree method offline preprocessing; 1.1. Introduce multi-parameter quadratic programming into the control system, divide the system state space into convex partitions, and calculate the control rate corresponding to each partition, and save it in the FG array; 1.2. The synonymous partitions are calculated and grouped by the formula for determining the synonymous partitions, and only one eigenvalue data is reserved for each group of synonymous partitions, thus eliminating redundant data in the eigenvalue array FG; The division of partitions in space is based on eigenvalues—all points in the same partition have the same eigenvalues; we call partitions with equal eigenvalues synonymous partitions; partition eigenvalues in explicit model predictive control (here The control rate of the explicit model predictive control) is called the FG matrix; for example, the FG matrix of a two-dimensional state space partition P with an explicit model predictive control output dimension of 1 is: FG ₁ =[f ₁₁ f ₁₂ g ₁ ](1) The FG matrix of another partition Q adjacent to it is: FG ₂ ＝[f ₂₁ f ₂₂ g ₂ ](2) If satisfied: (f ₁₁ -f ₂₁ ) ² +(f ₁₂ -f ₂₂ ) ² +(g ₁ -g ₂ ) ² ≤δ(3) Among them, formula (3) is the formula for determining the synonymous partition, f _ij and g _i (i,j=1,2) are the elements that make up the eigenvalue matrix, and the control output we need is calculated by the eigenvalue and the state vector Obtained; define δ as a very small positive number, then P and Q are considered to be synonymous, and they are mutually synonymous partitions; 1.3. Calculate the hash function according to the partition parameters, and record the obtained data in an array. We name the array Fhash; the hash function is as follows: $\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; f</span>}\mathrm{<span\; class="notranslate">\; l</span>}\mathrm{<span\; class="notranslate">\; o</span>}\mathrm{<span\; class="notranslate">\; o</span>}\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; a</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\frac{\mathrm{<span\; class="notranslate">\; N</span>}}{\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; a</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$ N here represents the division parameter, a and b are the boundary coordinates on a certain dimension, and the data we need to record in the array are -a and In the online calculation stage, the grid area of the hash table can be quickly determined by the coordinate X of the state point; 1.4. According to the partition parameters, construct the polytope of the grid area of the hash table, select the first grid area and partition in order to intersect, and count the number of eigenvalues in this grid area; 1.5, determine whether there is a conflict in the grid area; if the number of eigenvalues in the grid area is greater than 1, there is a conflict; if there is a conflict, jump to 1.7 and start building a binary tree in the grid area; if there is no conflict, go to the next step step; 1.6. Determine whether the grid area is completely outside the object space; if the grid area does not intersect with any partition, directly record the corresponding position in the Hash array as 0; if there is only one eigenvalue in the grid area, Then directly record the characteristic value number at the corresponding position in the Hash array; after completing this step, jump to 1.19; 1.7, first fill in the address of the root node of the binary tree in the Hash array, and mark it as a negative value; then remove the linearly related hyperplane and the outer boundary of the object space in the grid area, and do not select them as the hyperplane to be selected; here Each partition in the object space is divided by the hyperplane. The principle of the binary tree method is to judge the positional relationship between the point and the hyperplane at each node, determine which side of the hyperplane the state point is on, and exclude nearly half of the hyperplane. After the partition, enter the next node to continue to judge, and finally get the partition where the state point is located; therefore, it is redundant to use the outer boundary of the object space as the basis for node judgment, and one side of it is still the entire object space, which cannot be excluded; 1.8, group the partitions by eigenvalues (here, the eigenvalues are the control rates), group the partitions with the same eigenvalues, and combine the same eigenvalues into one data; 1.9, calculate the pole coordinates in each group of partitions, and eliminate the duplicate coordinates in each group of pole coordinates, and enter the root node; 1.10, extract the first hyperplane from the hyperplanes to be selected at the current node; 1.11, count the number of eigenvalues on both sides of the hyperplane; this statistical method is an important step in building a binary tree, it can count the number of eigenvalues on both sides of the hyperplane without judging all partition poles, greatly shortening the preprocessing time; the main steps as follows: a, load partitioned pole data grouped by properties with equal eigenvalues; b. Load the hyperplane to be judged, extract the coordinates of the first pole of the first group, define the two sides of the hyperplane as Hp- and Hp+ respectively, and the number of eigenvalues on both sides are m and n respectively, shilling m and n Both are 0. c, use Lf and Rf as the mark of whether the pole is in Hp- and Hp+, the value is 0 for false, and the value for 1 is true; shilling Lf and Rf are both 0; d. Determine the positional relationship between the pole and the hyperplane; for the hyperplane Hp={x|hx=k}, if point x satisfies hx≤k, then point x is considered to be located at Hp-, otherwise it is located at Hp+; where h and k are hyperplanes The plane expression parameter, x is the coordinate of the state point to be judged; e, if the pole is at Hp-, set Lf=1, jump to g, otherwise proceed to the next step; f, judge whether the pole is located at Hp+, if true, set Rf=1; otherwise, jump to h; g, judge whether Rf=1 and Lf=1 are established at the same time, if it is false, go to the next step, if it is true, jump to i; h, to judge whether it is the last pole of this group, if it is false, extract the coordinates of the next pole of this group, and jump to d; if it is true, if Lf=1, add 1 to the value of m, if Rf=1, n Add 1 to the value; i, judge whether this group is the last group of pole data, if it is false, extract the first pole coordinates of the next group, and jump to c, otherwise the number of eigenvalues on both sides of the hyperplane is counted; 1.12. Determine whether this is the last hyperplane to be selected. If it is false, extract the next hyperplane to be selected. Jump to 1.11 to count the number of eigenvalues on both sides of the hyperplane. If it is true, go to the next step; 1.13, determine the reference hyperplane according to the index; the binary tree that is expected to be built is deep and has few nodes, it is impossible to try all combinations to build all possible binary trees, and then select the best one; only need to consider the two sides of the node (that is, the two sides of the hyperplane side) the number of eigenvalues is roughly the same, it is considered that the hyperplane is more suitable as a reference hyperplane; the description index is as follows: J＝(m+n) ² +(mn) ² m and n are the number of eigenvalues at Hp- and Hp+ respectively. The smaller J is, the more suitable the hyperplane is to be called the reference hyperplane; the sum of the eigenvalues on both sides describes the expected number of nodes in the binary tree, and the two sides The difference between the eigenvalues describes the balance expectation of the left and right subtrees of the binary tree; 1.14, judge whether the left subtree is established; if it is true, jump to 1.16, otherwise go to the next step; 1.15, transfer the pole located on the Hp-side of the reference hyperplane to the left child node, remove the reference hyperplane from the hyperplane to be selected and pass it to the left child node, and jump to 1.10 after entering the left child node; 1.16, judge whether the right subtree is established; if it is true, jump to 1.18, if it is false, go to the next step; 1.17, transfer the pole located on the reference hyperplane Hp+ side to the right child node, remove the reference hyperplane from the hyperplane to be selected and pass it to the right child node, and jump to 1.10 after entering the right child node; 1.18, return to the parent node, and judge whether the establishment of the binary tree is completed, if it is false, jump to 1.12, if it is true, save the data, and the establishment of the binary tree in the grid area is completed; 1.19, judge whether the current operation grid area is the last grid area; if not, select the next grid area, and intersect it with the partition, count the number of eigenvalues in it, and jump to 1.5; if it is the last Grid area, indicating that the preprocessing step has been completed, and the operation is ended; Step 2. Grid binary tree method online calculation; 2.1, read the coordinates of the target point, quickly locate the grid area according to the hash function, and read the corresponding records in the Hash array; 2.2. Determine whether the record value is negative; if the record value is negative, jump to 2.4 and perform a binary tree search; if the record value is not negative, go to the next step; 2.3. Determine whether the record value is positive; if it is not positive, it means that the record value is 0, the grid area is completely outside the object space, and the state point is not in the object space, and directly exit the online search stage; if the record value is positive It shows that there is only one kind of eigenvalue in the grid area, and the recorded value is the eigenvalue number, which can directly obtain the eigenvalue, and the online search phase is over; 2.4, the reverse of the record value is the address of the root node, and enter the root node; 2.5. Determine the relationship between the target point and the reference hyperplane at the node; if the target point is on the Hp- side, enter the left child node; if the target point is on the Hp+ side, enter the right child node; 2.6, judge whether the node is the last binary tree node, if it is false, jump to 2.5, if it is true, go to the next step; 2.7. Determine the positional relationship between the target point and the last reference hyperplane. If it is located at Hp-, select the left cotyledon; if it is located at Hp+, select the right cotyledon; according to the corresponding eigenvalue number on the leaf node, extract it from the eigenvalue matrix FG Eigenvalues, point positioning are done in the online calculation phase.