CN107608214B - Multi-stage grid point positioning method in three-degree-of-freedom helicopter explicit model predictive control - Google Patents

Multi-stage grid point positioning method in three-degree-of-freedom helicopter explicit model predictive control Download PDF

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CN107608214B
CN107608214B CN201710959276.6A CN201710959276A CN107608214B CN 107608214 B CN107608214 B CN 107608214B CN 201710959276 A CN201710959276 A CN 201710959276A CN 107608214 B CN107608214 B CN 107608214B
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张聚
修晓杰
赵恺伦
周俊
田峥
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Zhijiang College of ZJUT
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Abstract

The method for positioning the multilevel grid points in the three-degree-of-freedom helicopter explicit model predictive control comprises the following steps: step 1, constructing a grid tree structure; step 2, point positioning on-line searching; and 3, applying the optimal control rate obtained in the step 2 to the explicit model prediction control of the three-degree-of-freedom helicopter. The main body part of the method is of a multi-branch tree structure, and compared with the existing direct search method, BST tree construction method and the like, the method greatly reduces the off-line calculation time and the on-line search speed. The three-degree-of-freedom helicopter system developed by the Canada Quanser company is used as a research object, and the effectiveness and superiority of the three-degree-of-freedom helicopter system are verified through a tracking test of the three-degree-of-freedom helicopter and compared with a control effect of a direct search method.

Description

Multi-stage grid point positioning method in three-degree-of-freedom helicopter explicit model predictive control
Technical Field
The invention relates to a three-degree-of-freedom helicopter explicit model predictive control method, which is an online calculation method designed aiming at the problem of point positioning in three-degree-of-freedom helicopter system explicit model predictive control.
Background and meaning
All practical Control problems are limited by various types of constraints, and most systems operate near the constraint boundary for which Model Predictive Control (MPC) yields a typical successful paradigm for modern Control applications. MPC has emerged in the process industry, particularly the petrochemical industry, for as long as 30 years as an effective means of dealing with multivariable constrained control problems, and the theoretical basis of this technology has grown to maturity.
MPC solves the problem of infinite time constraint optimal control based on the idea of rolling time domain online iterative optimization, however, the idea of online iterative optimization causes the controller to be overloaded and inefficient, MPC technology can only be applied to the situation (such as process control system) where the problem scale is not large or the dynamic change of the system is slow, and is difficult to be applied to the system with higher sampling rate and the system with faster dynamic change, such as motor system, power electronic system, mechanical vibration control, automobile electronic control, etc. Before and after 2002, in order to solve the problems of low efficiency and the like in the traditional MPC, scholars such as Manfred Morari and AlbertoBemporad introduce multi-parameter Quadratic Programming (mp-QP) theories into the MPC, perform convex division on the state area of the system, perform off-line calculation to obtain an optimal Explicit Control law of state feedback corresponding to each state partition, and convert a closed-loop Model Predictive Control system based on repeated on-line optimization calculation into an Explicit PWA system equivalent to the closed-loop Model Predictive Control system, namely the EMPC. The main process of explicit model predictive control is shown in fig. 1.
On the basis of off-line calculation, the state partition where the current state point is located is calculated on line, the optimal control quantity of the current moment is obtained according to the optimal control law on the partition, and the process becomes point positioning. The point positioning problem is the key to feel the online efficiency of the explicit model predictive control, and the performance of the point positioning algorithm is directly related to the performance of the explicit model predictive control. When a point positioning algorithm is designed, how to quickly construct an optimal control rate storage structure needs to be considered, and the required storage space is ensured to be as small as possible, so that hardware implementation is facilitated; and at the same time, fast online lookup efficiency is required.
Disclosure of Invention
The invention provides a multistage grid point positioning method in three-degree-of-freedom helicopter explicit model predictive control, aiming at overcoming the defects in the prior art and aiming at the point positioning problem in three-degree-of-freedom helicopter explicit model predictive control.
The method for positioning the multilevel grid points in the three-degree-of-freedom helicopter explicit model predictive control comprises the following steps:
step 1, constructing a grid tree structure
Step 2, point positioning on-line searching;
and 3, applying the optimal control rate obtained in the step 2 to the explicit model prediction control of the three-degree-of-freedom helicopter.
The main body part of the method is of a multi-branch tree structure, and compared with the existing direct search method, BST tree construction method and the like, the method greatly reduces the off-line calculation time and the on-line search speed. The three-degree-of-freedom helicopter system developed by the Canada Quanser company is used as a research object, and the effectiveness and superiority of the three-degree-of-freedom helicopter system are verified through a tracking test of the three-degree-of-freedom helicopter and compared with a control effect of a direct search method.
The invention relates to a multi-level grid tree-type storage structure based on a k-d tree and a corresponding online searching method. The tree structure integrates the ideas of three data structures of a k-d tree, a hash table (hash table) and a search binary tree (BST), and a built multi-level grid tree, wherein a tree root part is used for searching a next level sub-tree on each layer of the k-d tree by using a hash function, and a tree tip part is built by the search binary tree (BST). Each layer of the grid part is divided equidistantly on a specified coordinate axis, and the hash function of each layer is also corresponding to the specified coordinate axis. The grids are divided equidistantly, a multi-branch tree can be established, a common binary tree structure is replaced, the online search time is greatly reduced, and a control system with multiple state partitions and high dimension can be effectively solved. Setting a threshold as a standard for judging whether a current node needs to be divided again, if the optimal control rate quantity in a space corresponding to the current node is smaller than the threshold, the node is used as a root node for constructing the BST, and the optimal control rate in a subspace is constructed and stored by the BST; otherwise, the grid division is continued. The main body part of the method is of a multi-branch tree structure, and compared with the existing direct search method, BST tree construction method and the like, the method greatly reduces off-line calculation time and on-line search speed.
The invention has the advantages that: (1) the trunk part of the tree-type storage structure adopts a multi-branch tree in a grid form, so that the online searching efficiency is high; (2) in order to avoid dividing a large amount of small redundant subspaces in the grid dividing process, BST is adopted in the treetop part of the tree structure so as to reduce the subspaces generated by the division; (3) the invention is suitable for the control problems of more state partitions and high state space dimension.
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FIG. 1 is a process schematic of explicit model predictive control of the method of the present invention;
FIG. 2 is a force diagram of a three-degree-of-freedom helicopter using the method of the present invention;
FIG. 3 is a diagram of a tracking experiment _ Simulink structure employed in the method of the present invention;
FIG. 4 is an expanded view of a desired corner module employed in the method of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more clear, the technical solutions of the present invention are further described below with reference to the accompanying drawings.
The method for positioning the multilevel grid points in the three-degree-of-freedom helicopter explicit model predictive control comprises the following specific steps of:
step 1, constructing a grid tree structure;
the whole tree structure is composed of two parts, wherein the trunk part of the tree root is composed of a grid structure based on a k-d tree, and the treetop part is composed of BST. When the tree root tree trunk part is constructed, each node corresponds to a partition space formed by a plurality of state partitions, in the partition spaces, a hyperplane is selected and divided at equal intervals on a specified coordinate axis, the partition space is divided into a plurality of sub-partition spaces, and each sub-partition space corresponds to one child node.Let M be the set of the number of lattices divided in time division per layer state division, and M ═ M1,m2,…,mNxWhere Nx is the dimension of the state partition, m1Is the number of the divided lattices m when the division is performed on the 1 st coordinate axis2The number of division lattices is determined in the division on the 2 nd coordinate axis, and so on. In order to make the divided lattices as full as possible, the lattices divided in the two-dimensional space are squares, the lattices divided in the three-dimensional space are right cubes, and so on; m is2,…,mNxCan be composed of1And (4) calculating.
When a grid structure is constructed, the grid structure cannot be divided all the time, otherwise, the obtained division grid becomes smaller and smaller, in the dividing process, the state partitions in the grid are divided into a plurality of small partitions for two times, and the depth of a storage requirement and a tree is increased when redundant partitions are generated, so that a threshold th is set to determine whether the next-level grid needs to be divided again. When the BST tree is used for construction, it is determined by the magnitude relationship between the optimal affine control rate number | F | in the partitioned space and the threshold th. When | F | th, the cells still need to be divided; when | F | is less than or equal to th, constructing by using a BST tree form, and selecting and calculating a segmentation hyperplane in a partition boundary hyperplane at the moment; for a "leaf" node, only one piece of information needs to be included, namely the affine control rate of the partition.
Let each node in the tree structure be Nk(k-1, 2, …), node NkThe corresponding partition space is in the ith (i is 1,2, …, N)x) Division on one coordinate axis, xi minAnd xi maxRepresents the minimum value and the maximum value of the partition space on the coordinate axis i, then NkThe corresponding state partition can be represented as { x | xi min≤xi≤xi max,i＝1,2,…,NxIn which xiRepresenting the component of point x in the state partition on the ith coordinate axis.
From the root node N1Initially, the entire partitioned space is equally divided into m on the 1 st coordinate axis1A grid, and a node N1Has m1An individual child node. In thatThe separation distance divided on the 1 st coordinate axis is denoted as e1＝(x1 max-x1 min)/m1Then pass e1Can be calculated to obtain m2,…,mNxRespectively is mi＝floor((xi max-xi min)/e1),(i＝2,…,Nx) M for each nodeiIndividual child nodes numbered 1,2,3, …, mi。
Part of a lattice tree, N for each nodekThe following information needs to be included: minimum set of values, X, for each coordinate axis in a partitioned spacek min＝[x1 min,x2 min,…,xNx min](ii) a Set of maximum values, Xk max＝[x1 max,x2 max,…,xNx max](ii) a Interval to be divided on ith coordinate axis, ei(ii) a State partition number set I in partition spacek. Algorithm 1 describes a specific construction process of a tree structure.
Algorithm 1: construction of a lattice tree structure
Inputting: 1 st coordinate axis division number m1State partition set P ═ { P ═ P1,P2,…,PNrF, the corresponding optimal affine control rate set F ═ F1,F2,…,FKH, state space dimension Nx。
Step S1: initializing a segmented hyperplane dataset
Figure GDA0002415626470000061
And corresponding set of split hyperplane sequence numbers
Figure GDA0002415626470000062
Step S2: calculating the minimum value X of the partition space corresponding to each coordinate axis1 minAnd maximum value X1 maxAnd calculating the 2 nd coordinate axis to the Nth coordinate axis, respectivelyxNumber of divisions m on coordinate axis2,…,mNx。
Step S3: initializing the root node, N1＝{I1,i1,X1 min,X1 max,e1}←{(1,…,Nr),1,X1 min,X 1 max0, and U ← { N ← N }1}。
Step S4: WHILE
Figure GDA0002415626470000063
Selecting a node Nk∈ U and set U ← U \ Nk。
IF|F(Ik)|>th THEN
a. Computing
Figure GDA0002415626470000064
And N isk←e；
b. Creating
Figure GDA0002415626470000074
Child node and is numbered as
Figure GDA0002415626470000071
And identifies the node as Nk|t(ii) a For each child node Nk|tLet X bek|t min＝Xk，Xk|t max＝Xk maxThen update the ithk thThe components on the coordinate axes are such that,
Figure GDA0002415626470000072
c.ik|t＝(ik+1)％Nx；
d.
Figure GDA0002415626470000073
ELSE IF|F(Ik)|>1THEN
a.ik←0；
b. construction using BSTThe method calculates the segmentation hyperplane, adds the hyperplane obtained by calculation to the segmentation hyperplane set H, and simultaneously adds the sequence number jkAdded to J.
c. Completion node Nk←jkAnd creating two child nodes, N±←(I(Jk∪jk ±),Jk∪jk ±,ik) And added to U.
ELSE
Identify the current node as a leaf node, and ik←-1，N±←(F(I±),ik)。
END IF
END WHILEEND
Step 2, positioning and on-line searching;
the performance of the explicit model predictive control is determined by the on-line computation speed, i.e., the performance of the point positioning. And the current state of the control system is used as a state point x, searching is carried out in the constructed grid tree structure, the state partition where the control system is located is found, and the optimal control rate is obtained through calculation. Firstly, starting from a root node, in a grid tree part, a hash function n is utilizedi＝floor((xi-xi min)/ei) +1, calculating the number of the child node, and finding the next node, wherein eiThe size of (a) directly affects the height of the whole tree; in BST part, calculating the partition hyperplane d of current nodej(x)＝aj Tx-bjFinding child nodes; when i is in the current nodekWhen the value of (1) is-1, calculating to obtain the optimal control rate corresponding to the current state partition. See specifically algorithm 2.
And 2, algorithm: traversal algorithm of grid tree structure
Input of arbitrarily Inquiry State Point
Figure GDA0002415626470000082
Step T1, starting from the root node, taking the root node as the current node, Nk←N1；
Step T2: WHILE ik≠-1DO
IF ik>0THEN
Use of
Figure GDA0002415626470000081
Calculating the current node NkThe sequence number of the child node of (1); and taking the child node as the current node;
ELSE
calculating the current node NkIn (d) a segmentation hyperplanej(x)＝aj Tx-bj(j＝jk) And according to dj(x) Selects a child node satisfying the condition and takes the child node as the current node.
END IF
END WHILE
Step T3: and calculating the optimal control rate u (x) of the current node.
END
And 3, applying the optimal control rate obtained in the step 2 to the explicit model prediction control of the three-degree-of-freedom helicopter.
Performance of the algorithm
The multi-level grid tree structure in the invention is composed of two parts, namely a grid tree and a BS. The number of optimal affine control rates in the nodes of the BST part is not more than the threshold th. Optimally, each lattice has the same number of optimal affine control rates, and each lattice is smaller than the number of optimal imitative rates in its parent node. The tree can now be optimally balanced and the height of the tree depends on the parameters m and th. By KsumRepresenting the total number of the best refractive indexes in each lattice, K is less than or equal to KsumWhen K issumThe algorithm has the highest efficiency when K is equal to K, and the height of the multi-level mesh tree is D equal to logm(K/th)+log2(th) + 2. Wherein logm(K/th) +1 is the height of the treelike part, log2(th) +1 is the height of the tree of the BST part.
For the general case, it is assumed that all child nodes corresponding to one node satisfy
Figure GDA0002415626470000091
(when
Figure GDA0002415626470000092
When this is the case), the above-described optimum condition is satisfied). Since for the node at the bottom of the trellis tree portion, there are
Figure GDA0002415626470000093
Due to the non-uniformity of the optimal control rate distribution, the constructed mesh tree is not balanced, its maximum height D1Can be expressed as
Figure GDA0002415626470000094
Since the number of optimal control rates of the first layer of the BST part is less than th, the maximum height D of the BST part2Is composed of
Figure GDA0002415626470000095
So that the maximum height of the constructed multi-level lattice tree is
Figure GDA0002415626470000096
Case analysis
The invention uses a three-degree-of-freedom helicopter system developed by Quanser in Canada as a research object. The three-degree-of-freedom helicopter system is a typical multi-input multi-output system, has the characteristics of strong coupling, nonlinearity and the like, is a relatively troublesome controlled object in the field of automatic control, and can reflect the control effect of the helicopter through the control effects of three degrees of freedom of a height axis, a pitch axis and a rotating axis. The Quanser three-degree-of-freedom helicopter is specifically shown in fig. 2.
Firstly, a state space model of a Quanser three-degree-of-freedom helicopter is established. On the basis of analyzing the stress principle of the three-degree-of-freedom helicopter system, a dynamic model of the height axis, the pitch axis and the rotating shaft and a motion equation of the height axis, the pitch axis and the rotating shaft are obtained, and a state space model of the three-degree-of-freedom helicopter is established. Selecting elevation angle, pitch angle rho, rotation angle gamma and their differentiation
Figure GDA0002415626470000101
And
Figure GDA0002415626470000102
if the state variable is a state matrix, then the state matrix is
Figure GDA0002415626470000103
And establishing a state space partition set through an MPT tool box according to the obtained state equation and the state space model. The partition sets obtained after the state space is divided are all multi-cell shapes, and the partition sets are all closed and bounded convex sets in Euclidean space. And in the obtained state partition space, constructing a grid tree structure by using the offline calculation method.
The effectiveness and superiority of the invention are verified by the tracking test of the three-degree-of-freedom helicopter and simultaneously compared with the control effect of adopting a direct search method.
FIG. 3 is a schematic diagram of the Simulink structure of the tracking experiment. The main structure comprises an expected angle module, an MPT controller, a three-degree-of-freedom helicopter module and an oscilloscope module. The desired angle module of the three-degree-of-freedom helicopter is unfolded as shown in fig. 4.
When a sine wave is tracked, a rectangular wave with a height angle tracking frequency of 0.04HZ and an amplitude of 7.5 degrees is tracked. The rotation angle traces a rectangular wave with a frequency of 0.03HZ and an amplitude of 30 °. The initial value of the height angle is-27.5 degrees, so a constant value module is superposed on the basis of rectangular waves, the initial value is-27.5 degrees, and the tracking experiment is carried out after the height angle can be adjusted to 0 degree through a sliding block. The altitude control effects based on the multi-level network tree structure and the direct lookup method are summarized in table 1, and the rotation angle control effects are summarized in table 2.
Table 1 table of altitude angle adjustment control effect based on multi-stage mesh tree structure search method and direct search method
Figure GDA0002415626470000111
For the tracking of the altitude angle, the method can react faster at the moment of expecting signal change based on a multi-level grid tree structure searching method, has better dynamic performance, has smaller maximum offset, and can track faster and more stably. In time domain, the delay of the searching method based on the multi-level grid tree structure is smaller than that of the direct searching method.
Table 2 list of rotation angle adjustment control effects based on multi-stage lattice tree structure search method and direct search method
Figure GDA0002415626470000112
The direct search method has a poor tracking effect on the rotation angle, only can track a rough outline, and has a fast rise time and good dynamic response based on the multi-level grid tree structure search method.
The embodiments described in this specification are merely illustrative of implementations of the inventive concept and the scope of the present invention should not be considered limited to the specific forms set forth in the embodiments but rather by the equivalents thereof as may occur to those skilled in the art upon consideration of the present inventive concept.

Claims (1)

1. The method for positioning the multilevel grid points in the three-degree-of-freedom helicopter explicit model predictive control comprises the following steps:
step 1, constructing a grid tree structure;
the whole tree structure is composed of two parts, wherein the trunk part of a tree root is composed of a grid structure based on a k-d tree, and the treetop part is composed of BST; when a tree root tree trunk part is constructed, each node corresponds to a partition space formed by a plurality of state partitions, in the partition spaces, a hyperplane is selected and divided equidistantly on a specified coordinate axis, the partition space is divided into a plurality of sub-partition spaces, and each sub-partition space corresponds to one child node; let M be the set of the number of lattices divided in time division per layer state division, and M ═ M1,m2,…,mNxWhere Nx is the dimension of the state partition, m1Is the number of the divided lattices m when the division is performed on the 1 st coordinate axis2Is a division grid for division on the 2 nd coordinate axisCounting, and so on; in order to make the divided lattices as full as possible, the lattices divided in the two-dimensional space are squares, the lattices divided in the three-dimensional space are right cubes, and so on; m is2,…,mNxFrom m1Calculating to obtain;
when a grid structure is constructed, the grid structure cannot be divided all the time, otherwise, the obtained division grid becomes smaller and smaller, in the dividing process, the state partitions in the grid are divided into a plurality of small partitions for the second time, redundant partitions are generated, the storage requirement and the tree depth are increased, and therefore a threshold th is set to determine whether the next-level grid needs to be divided again; when the BST tree is adopted for construction, judging through the size relation between the optimal affine control rate quantity | F | in the partitioned space and the threshold th; when | F | th, the cells still need to be divided; when | F | is less than or equal to th, constructing by using a BST tree form, and selecting and calculating a segmentation hyperplane in a partition boundary hyperplane at the moment; for the leaf node, only one piece of information is needed to be contained, namely the affine control rate of the partition;
let each node in the tree structure be Nk(k-1, 2, …), node NkThe corresponding partition space is in the ith (i is 1,2, …, N)x) Division on one coordinate axis, xi minAnd xi maxRepresents the minimum value and the maximum value of the partition space on the coordinate axis i, then NkThe corresponding state partition is denoted as { x | xi min≤xi≤xi max,i＝1,2,…,NxIn which xiRepresenting the component of the point x in the state partition on the ith coordinate axis;
from the root node N1Initially, the entire partitioned space is equally divided into m on the 1 st coordinate axis1A grid, and a node N1Has m1An individual child node; the separation distance divided on the 1 st coordinate axis is denoted as e1＝(x1 max-x1 min)/m1Then pass e1Is calculated to obtain m2,…,mNxRespectively is mi＝floor((xi max-xi min)/e1),(i＝2,…,Nx) M for each nodeiIndividual child nodes numbered 1,2,3, …, mi；
Part of a lattice tree, N for each nodekThe following information needs to be included: minimum set of values, X, for each coordinate axis in a partitioned spacek min＝[x1 min,x2 min,…,xNx min](ii) a Set of maximum values, Xk max＝[x1 max,x2 max,…,xNx max](ii) a Interval to be divided on ith coordinate axis, ei(ii) a State partition number set I in partition spacek(ii) a The algorithm 1 describes a specific construction process of a tree structure;
algorithm 1: construction of a lattice tree structure
Inputting: 1 st coordinate axis division number m1State partition set P ═ { P ═ P1,P2,…,PNrF, the corresponding optimal affine control rate set F ═ F1,F2,…,FKH, state space dimension Nx；
Step S1: initializing a segmented hyperplane dataset
Figure FDA0002528945610000021
And corresponding set of split hyperplane sequence numbers
Figure FDA0002528945610000022
Step S2: calculating the minimum value X of the partition space corresponding to each coordinate axis1 minAnd maximum value X1 maxAnd calculating the 2 nd coordinate axis to the Nth coordinate axis, respectivelyxNumber of divisions m on coordinate axis2,…,mNx；
Step S3: initializing the root node, N1＝{I1,i1,X1 min,X1 max,e1}←{(1,…,Nr),1,X1 min,X1 max0, and U ← { N ← N }1}；
Step S4: WHILE
Figure FDA0002528945610000023
DO
Selecting a node Nk∈ U and set U ← U \ Nk；
IF|F(Ik)|>th THEN
a. Computing
Figure FDA0002528945610000024
And N isk←e；
b. Creating
Figure FDA0002528945610000025
Child node and is numbered as
Figure FDA0002528945610000026
And identifies the node as Nk|t(ii) a For each child node Nk|tLet X bek|t min＝Xk，Xk|t max＝Xk maxThen update the ithk thThe components on the coordinate axes are such that,
Figure FDA0002528945610000027
c.ik|t＝(ik+1)％Nx；
d.Nk|t←(I(P(Xk|t min+,Xk|t max-)),Xk|t min,Xk|t max,ik|t,0)；
ELSE IF|F(Ik)|>1THEN
a.ik←0；
b. calculating a segmentation hyperplane by using a BST construction method, adding the calculated hyperplane to a segmentation hyperplane set H, and simultaneously adding a serial number jkAdding to J;
c. completion node Nk←jkAnd createTwo child nodes, N±←(I(Jk∪jk ±),Jk∪jk ±,ik) And added to U;
ELSE
identify the current node as a leaf node, and ik←-1，N±←(F(I±),ik)；
END IF
END WHILE
END
Step 2, point positioning on-line searching;
the online computing speed, namely the performance of point positioning determines the performance of the explicit model predictive control; the current state of the control system is used as a state point x, searching is carried out in the constructed grid tree structure, the state partition where the control system is located is found, and the optimal control rate is obtained through calculation; firstly, starting from a root node, in a grid tree part, a hash function n is utilizedi＝floor((xi-xi min)/ei) +1, calculating the number of the child node, and finding the next node, wherein eiThe size of (a) directly affects the height of the whole tree; in BST part, calculating the partition hyperplane d of current nodej(x)＝aj Tx-bjFinding child nodes; when i is in the current nodekWhen the value of (1) is-1, calculating to obtain the optimal control rate corresponding to the current state partition; see specifically algorithm 2;
and 2, algorithm: traversal algorithm of grid tree structure
Input of arbitrarily Inquiry State Point
Figure FDA0002528945610000031
Step T1: starting from the root node, the root node is taken as the current node, Nk←N1；
Step T2: WHILE ik≠-1DO
IF ik>0THEN
Use of
Figure FDA0002528945610000032
Calculating the current node NkThe sequence number of the child node of (1);
and taking the child node as the current node;
ELSE
calculating the current node NkIn (d) a segmentation hyperplanej(x)＝aj Tx-bj(j＝jk) And according to dj(x) Selecting a child node meeting the condition by the symbol of (1), and taking the child node as a current node;
END IF
END WHILE
step T3: calculating the optimal control rate u (x) of the current node;
END
and 3, applying the optimal control rate obtained in the step 2 to the explicit model prediction control of the three-degree-of-freedom helicopter.
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