CN107528659A - The high-accuracy network topology equivalence algorithm of clock nonsynchronous network - Google Patents

The high-accuracy network topology equivalence algorithm of clock nonsynchronous network Download PDF

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CN107528659A
CN107528659A CN201710855694.0A CN201710855694A CN107528659A CN 107528659 A CN107528659 A CN 107528659A CN 201710855694 A CN201710855694 A CN 201710855694A CN 107528659 A CN107528659 A CN 107528659A
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CN107528659B (en
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蒋轶
余宏伟
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Fudan University
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    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04J—MULTIPLEX COMMUNICATION
    • H04J3/00—Time-division multiplex systems
    • H04J3/02—Details
    • H04J3/06—Synchronising arrangements
    • H04J3/0635—Clock or time synchronisation in a network
    • H04J3/0638—Clock or time synchronisation among nodes; Internode synchronisation
    • H04J3/0658—Clock or time synchronisation among packet nodes
    • H04J3/0661—Clock or time synchronisation among packet nodes using timestamps
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04W—WIRELESS COMMUNICATION NETWORKS
    • H04W4/00—Services specially adapted for wireless communication networks; Facilities therefor
    • H04W4/02—Services making use of location information
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04W—WIRELESS COMMUNICATION NETWORKS
    • H04W4/00—Services specially adapted for wireless communication networks; Facilities therefor
    • H04W4/02—Services making use of location information
    • H04W4/023—Services making use of location information using mutual or relative location information between multiple location based services [LBS] targets or of distance thresholds
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04W—WIRELESS COMMUNICATION NETWORKS
    • H04W64/00—Locating users or terminals or network equipment for network management purposes, e.g. mobility management
    • H04W64/003—Locating users or terminals or network equipment for network management purposes, e.g. mobility management locating network equipment

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Abstract

The invention belongs to the topology equivalence technical field of network, the high-accuracy network topology equivalence algorithm of specially a kind of clock nonsynchronous network.The present invention is divided into two large divisions：First, the agreement based on a kind of time division multiple acess (TDMA), each node is collected, sends information, and is diffused into whole network；Second, being based on these information, the high-accuracy network topology equivalence for not needing clock synchronous is drawn.For a single-hop networks, the present invention provides a kind of centralization algorithm, and for an extensive multihop network, the present invention provides a kind of distributed location method.Further, the time delay estimation of each discrete nodes local clock in network can also be drawn.The present invention take into account the incorrectness of anchor point information so that for the algorithm closer to reality, positioning is more accurate, and can reach carat U.S. labor limit.

Description

High-precision network topology positioning algorithm of clock asynchronous network
Technical Field
The invention belongs to the technical field of network topology positioning, and particularly relates to a high-precision network topology positioning algorithm of a clock asynchronous network.
Background
The topological positioning of the network is a key technology in a wireless sensor network, an unmanned aerial vehicle network and a wireless self-organizing network. Especially for an unmanned aerial vehicle network, collision and high-precision positioning control of cluster nodes are avoided, and decimeter-level high-precision positioning is realized on each node of the network. Therefore, we cannot rely on the positioning information of GPS/beidou alone. But not based on the positioning information (GPS/Beidou positioning or other positioning information) of the anchor points, and network topology information with the precision higher than the meter level is cooperatively obtained through self-aggregation and information exchange among nodes.
A great deal of research work has been published on the above problems.
The traditional high-precision network topology positioning assumes clock synchronization of all distributed nodes, and in practice, clock synchronization of network nodes is difficult to achieve due to crystal oscillator frequency deviation. Meanwhile, how to efficiently gather information of each discrete node about distance measurement between nodes and realize optimal topological positioning is also a difficult problem.
Disclosure of Invention
The invention aims to provide a network topology positioning algorithm for realizing high-precision positioning of a clock asynchronous network.
The network topology positioning algorithm provided by the invention is divided into two parts, one is based on a Time Division Multiple Access (TDMA) protocol, and each node collects and sends information and diffuses the information to the whole network; and secondly, based on the information, obtaining high-precision network topology positioning without clock synchronization. And as a byproduct of the positioning algorithm, the time delay estimation of the local clock of each discrete node in the network can be obtained.
The network topology positioning algorithm (also called a centralization algorithm) provided by the invention comprises the following specific steps:
in the first step, a single-hop network of N nodes is constructed. The network employs an information exchange protocol based on Time Division Multiple Access (TDMA) as shown in fig. 1.
Broadcasting a pilot signal (beacon) by each node in the network in turn; assume that the nth node is broadcasting a pilot signal at this time, which contains the following:
(1)1 synchronization sequence for accurately estimating local time of a receiving node when a pilot signal arrives;
(2)1 time stamp TnI.e. the local time when the nth node sends the pilot signal;
(3) n-1 timestamps { RnmAnd m is 1, …, N-1, N +1, …, N }, i.e., the local time when the nth node receives the pilot signals of the other N-1 nodes.
The above information is transmitted in turn from node 1 to node N. Over N rounds, all nodes in these single hop regions will collect { T }nN is 1,2, …, N (i.e., the local time when each of the N nodes in the network transmits a pilot signal) and RnmN, m is 1,2, …, N (i.e., the local time when the N nodes in the network receive the pilots from the other N-1 nodes).
Based on this information, the following relationship can be obtained:
wherein, DeltanIs the clock drift of node n relative to node 1, apparently Δ1＝0，dmnIs the distance between nodes m, n C is the speed of signal propagation, ∈mnDue to measurement of RmnAnd TnIntroduced error, ∈mnC is the measurement error of the node spacing, and is assumed to obey the Gaussian distribution C. ∈mn～N(0,σ2) Wherein σ is the standard deviation;
and secondly, supposing that N nodes with K less than or equal to N are equipped with GPS/Beidou positioning, the nodes start self-positioning firstly, and the positioning information is broadcasted to other nodes. All nodes in the network obtain the GPS/Beidou positioning information of the K anchor points, but have errors. That is:
wherein p iskIs the true position coordinate vector of the kth node,is GPS/Beidou positioning, Q, of node kkIs the 3x3 covariance matrix of the kth node GPS/Beidou measurement error (here we consider three dimensional positioning, if two dimensional, then QkIs a 2x2 matrix),is the set of all nodes with GPS/Beidou;
third, all nodes in the network have obtained the following information so far: { Tn,n＝1,2,…,N}，{Rnm,n,m＝1,2,…,N}，According to the information, each node in the network operates the positioning algorithm provided by the invention, and the positioning of each node in the network can be obtained (or only a certain node with higher battery residual capacity in the network operates the algorithm, and then the positioning result is broadcasted).
In the third step, the used positioning algorithm is a high-performance positioning algorithm, which specifically comprises the following steps:
according to expression (14), define:
the optimal positioning is calculated accordingly:
wherein,is the set of all node pairs directly interconnected with each other, σ being defined as above, pm,pnRepresenting the position coordinate vectors of the m, N nodes, p is a 3 × N matrix formed by transversely splicing the position vectors of the N nodes (here we consider three-dimensional positioning, and a 2 × N matrix formed by transversely splicing the position vectors of the N nodes if two-dimensional positioning),representing the estimate of p produced using the present algorithm.
Firstly, positioning by K nodes with GPS/BeidouAnd (4) obtaining the positioning of all the N nodes for the anchor point by adopting a traditional triangulation positioning method. If the network is multi-hop, some nodes may not be directly connected with the nodes with GPS/Beidou. To solve this problem, nodes that have been triangulated are added to an expanded set of anchor points. Thereby ensuring that all N nodes can be located.
To facilitate our discussion, define:
wherein,
based on the initial estimation of each node position obtained by the method, the following algorithm Sequential Convex Programming (SCP) is used for solving the formula (17):
(1) calculating the first derivative of the function f (p)And a second derivative
(2) ComputingEigenvalue decomposition ofWherein Λ isThe eigenvalues are diagonal matrices of diagonal elements, U isOf the feature vectors of (1), UHIs the conjugate transpose of U.
(3) And (4) if all diagonal elements of the lambda are positive numbers, entering the step (4), and otherwise, jumping to the step (5).
(4) ComputingAnd (6) skipping.
(5) ComputingΛ therein-Is to Λ-1The non-positive diagonal elements in (1) are replaced with 0 to get a new diagonal matrix.
(6) Using a back-tracking linear search method or any other linear search method, the energy minimization f (p + su) is calculatednt) S, where s is the step size, f (p + su)nt) The position coordinates p of all nodes in the network are updated to p + suntThen, the cost function value of equation (5).
(7) Determine whether a stopping criterion is met (e.g.: f (p + su)nt) -f (p) l ≦ η, where η is a self-setting small positive number), updating p to p + sunt，p←p+sunt. If the stopping criterion is met, finishing the algorithm to obtain the position coordinate estimation p of the nodes in the network; otherwise, returning to the step (1).
In the above algorithm process, a first derivativeComprises the following steps:
wherein,is a set of one-hop neighbor nodes of node n, andis the set of all nodes with GPS/Beidou.
Second order leadIs composed of
Wherein:
further, the invention also deduces the theoretical optimal performance limit of the positioning algorithm, namely the Claimelau limit (CRB), as follows:
wherein Q ═ diag { Q ═ Q1,Q2,…,QKIs a 3K × 3K block diagonal matrix, and the diagonal sub-blocks of T are:
wherein,is a set of neighbor nodes for node n. And the non-diagonal sub-blocks of T are
The invention also provides a distributed positioning algorithm, which is evolved from a centralized algorithm, basically divides a large network into a plurality of small sub-networks, and executes a third step in each sub-network; but some information exchange between neighbouring subnets is required. The specific introduction is as follows:
in the network topology positioning algorithm, the network is assumed to be a single-hop network, the network scale is small, and a centralized processing mode is adopted for all information collected by all nodes in the network, namely, all information collected by all nodes in the whole network is processed together at one time (namely, a centralized algorithm). If the network is a multi-hop network and the network scale is large, a distributed positioning algorithm evolved from a centralized positioning algorithm can be adopted, specifically, in each local part of the network, the local information is firstly utilized to perform positioning respectively, and then the local parts feed back positioning results mutually and perform mutual calibration. Therefore, the method avoids processing all information of the whole network at one time, thereby avoiding ultrahigh calculation complexity and keeping the positioning precision of a centralized positioning algorithm.
The basic idea of the distributed positioning algorithm is shown in fig. 3. Assume that a large-scale multi-hop network is clustered into several sub-networks by some self-aggregation method (self-clustering), where any two adjacent sub-networks, cluster a and cluster B, contain several common nodes, called bridge nodes. The distributed positioning algorithm comprises the following specific steps:
(1) respectively performing the first step and the second step in the centralized positioning algorithm process in the cluster A and the cluster B, and collecting cluster information: these notations are the same as above, A, B identifying intra-cluster information pertaining to cluster a and cluster B, respectively.
(2) In cluster a, the information collected in cluster a is processed using the positioning algorithm described earlier for intra-cluster node positioning. Retrieving positioning results of bridge nodes therein(the column vectors formed by splicing the location vectors located by the bridge nodes end-to-end). Calculating a node positioning CRB matrix in the cluster, and taking out a CRB matrix sub-block C corresponding to the bridge nodeA. Will be provided withCATo cluster B.
(3) In cluster B, the bridge nodes are regarded asAs anchor points (like nodes equipped with GPS/Beidou) and willAs their anchor point location (as a result of GPS/beidou location), whereThe column vector is formed by splicing the real position vectors of the bridge nodes from beginning to end. The information collected in cluster B is processed for intra-cluster node localization using the localization algorithm described earlier. Retrieving positioning results of bridge nodes thereinCalculating a node positioning CRB matrix in the cluster, and taking out a CRB matrix sub-block C corresponding to the bridge nodeB. Will be provided withCBTo cluster a.
(4) In cluster a, again, the bridge node is taken as the anchor point andas their anchor locations. The information collected in cluster a is processed for intra-cluster node localization using the localization algorithm described earlier. And (5) if the positioning result in the cluster A has no important change compared with the previous round of result, namely the updated positioning estimation is compared with the original estimation, and the change is smaller than a certain optional threshold, usually set to be 0.05 m. Entering the step (5); otherwise, the positioning result of the bridge node in the network is taken outAnd calculating a node positioning CRB matrix in the cluster, and taking out a sub-block C corresponding to the bridge nodeAWill beCATo cluster B. And (4) returning to the step (3).
(5) And merging the positioning results of the cluster A and the cluster B to obtain the node positioning of the whole network: p ← pA,pB. Wherein p isA,pBThe final positioning results in cluster a and cluster B, respectively.
Under the condition that the number of the subnets is more than or equal to 3, the steps are executed between any pair of adjacent clusters, such as between AB, AC and BC under the condition of three subnets; or four subnets between AB, AC, AD, BC, BD, CD, etc. And executing the distributed cooperation algorithm to finally obtain the optimal positioning of the whole large network.
Further, the invention can obtain an optimal estimate of the clock drift of each node relative to the clock of the first node (reference node):
wherein y ismn(see the formula (16) one on top of anotherVector (c):
y＝(y21,y31，…,yN1,y12,y32，…,yN2,……,y1N,y2N,…,y(N-1)N)T
a is a matrix obtained by removing the first column;
wherein A is1,A2,A3,…,ANAre all (N-1) × N dimensional matrices:
by the way of analogy, the method can be used,
from this, the theoretical optimum performance limit, the cramer limit (CRB), is also derived as:
the estimates derived by this algorithm may also reach the Claume limits.
Advantages of the method of the invention
(1) Considering the incorrectness of the anchor point information, the algorithm is closer to reality, the positioning is more accurate, and the Clarithrome limit can be reached.
(2) The method is easy to evolve into a distributed positioning algorithm, maintains the positioning precision thereof, and is suitable for positioning in a large-scale network.
(3) The delay estimate of the local clock of each node in the network can be further derived, and the estimate can also reach the Clamanow limit.
Drawings
Fig. 1 is a diagram of node rotation broadcasting.
Fig. 2 is a graph of the performance of a centralized positioning algorithm. The performance of the algorithm reaches the optimal limit-Cramerilou limit (CRB)
Fig. 3 is a diagram of a positioning algorithm that is centralized in two subnets, respectively, and then iterated back and forth through bridge nodes.
FIG. 4 is a comparison of distributed and centralized processing performance.
Fig. 5 shows the performance of the delay estimation.
Detailed Description
The invention is further described below by means of specific examples.
By way of example, the present invention emulates a 20-node single-hop network with a computer. Each node is randomly distributed in a 200m area, wherein 5 nodes have GPS/Beidou, and the three-dimensional measurement error is Gaussian distributionThe unit is meters. The measurement error of the node distance is N (0, sigma)2). The Monte Carlo experiment is carried out for 200 times by using a triangulation positioning method as an initial point and utilizing SCP algorithm iteration provided by the invention. The performance of the resulting positioning error is shown in fig. 2. Wherein the x-axis is the variance σ of the distance error between the measurement nodes2(ii) a The y-axis is the average error in node location. The average error is calculated as follows: in each Monte Carlo experiment, the average value of the square errors of all node positioning is calculated Wherein (x)i,yi,zi) Represents the true position vector of the ith node,is an estimate thereof; and averaging the calculation results in 200 Monte Carlo experiments, and then obtaining the average error of node positioning represented by the y-axis. For the numerical calculations on the cramer limit curve in the figure: the CRB matrix of equation (9) is calculated first, the diagonal elements are taken to calculate the mean value and multiplied byIt can be seen that the performance of the algorithm provided by the invention is far superior to that of the classical MDS algorithm [1 ]]Can beTo reach the theoretical optimum performance limit-caramella limit (CRB).
The present invention also simulates the use of a distributed algorithm, considering a 35-node network. The network is divided into two 20-node subnets, of which 5 are bridges, shared by the two subnets. With the adoption of the distributed positioning algorithm, 200 Monte Carlo experiments are carried out, the positioning errors of the nodes are shown in figure 4, wherein the x-axis is the standard deviation sigma of the distance errors between the measured nodes, the y-axis is the average error of the positioning of the nodes, and the calculation mode is the same as that in the previous simulation. It can be seen that the performance of the distributed algorithm (line with small dots on the top) is very close to the centralised algorithm (line with square dots in the middle), both close to the caramella limit (CRB, line with large dots on the bottom).
For the time delay estimation, the invention simulates a single-hop network by using a computer. The nodes are randomly distributed in a region of 200 m. The measurement error of the node distance is N (0, sigma)2). For the case that the network contains 5,10,20 nodes, respectively, 200 monte carlo experiments are performed, and the final delay estimation result is shown in fig. 5, where the x-axis is the variance σ of the distance error between the measured nodes2The y-axis is the mean square error of the node delay estimate (the delay misestimate error has been multiplied by the signal propagation speed C and converted to a range error). The mean square error of the node delay estimation is calculated as follows: in each Monte Carlo experiment, the average value of the square errors of the time delay estimation of all the nodes except the node 1 is calculatedWhereiniRepresenting the true delay of the ith node multiplied by C,is an estimate thereof; and then averaging the calculation results in 200 Monte Carlo experiments to obtain the mean square error of the node time delay estimation represented by the y-axis. For the numerical calculations on the cramer limit curve in the figure: the CRB matrix of equation (13) is calculated first, and then the average value is calculated by taking the diagonal elements. Can seeThe time delay estimation provided by the invention can also reach the theoretical optimal performance limit, namely Cramer labor limit (CRB), and the estimation effect is better along with the increase of the number of nodes in the network, so that the network cooperation effect is reflected.
Reference to the literature
[1]I.Dokmanic,R.Parhizkar,J.Ranieri and M.Vetterli,"EuclideanDistance Matrices:Essential theory,algorithms,and applications,"in IEEESignal Processing Magazine,vol.32,no.6,pp.12-30,Nov.2015。

Claims (7)

1. A high-precision network topology positioning algorithm of a clock asynchronous network is characterized by comprising the following specific steps:
step one, for a single-hop network with N nodes, the network adopts an information exchange protocol based on time division multiple access; each node in the network broadcasts the guide signal in turn; assume that the nth node is broadcasting a pilot signal at this time, which contains the following:
(1)1 synchronization sequence for accurately estimating local time of a receiving node when a pilot signal arrives;
(2)1 time stamp TnI.e. the local time when the nth node sends the pilot signal;
(3) n-1 timestamps { RnmM is 1, …, N-1, N +1, …, N }, i.e., the local time when the nth node receives the pilot signals of the other N-1 nodes;
the information is transmitted from the node 1 to the node N in turn, and after N rounds, all the nodes in the single-hop region collect the { T }nN-1, 2, …, N, i.e. the local time when the N nodes in the network each send a pilot signal, and RnmN, m is 1,2, …, N, i.e. the local time when N nodes in the network receive the pilot signals sent by other N-1 nodes;
based on this information, the following relationship is obtained:
<mrow> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mi>n</mi> </msub> <mo>+</mo> <msub> <mi>&amp;Delta;</mi> <mi>m</mi> </msub> <mo>-</mo> <msub> <mi>&amp;Delta;</mi> <mi>n</mi> </msub> <mo>+</mo> <mfrac> <msub> <mi>d</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mi>C</mi> </mfrac> <mo>+</mo> <msub> <mo>&amp;Element;</mo> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
wherein, DeltanIs the clock drift of node n relative to node 1, apparently Δ1＝0，dmnIs the distance between nodes m, n, C is the velocity of signal propagation, ∈mnDue to measurement of RmnAnd TnError due to C. ∈mnI.e. the measurement error of the node spacing, assuming that it obeys the gaussian distribution C · ∈mn～N(0,σ2)，Wherein σ is the standard deviation;
secondly, supposing that N nodes with K less than or equal to N are equipped with GPS/Beidou positioning, the nodes start self-positioning at first and broadcast positioning information to other nodes; so far, all nodes in the network obtain the GPS/Beidou positioning information of the K anchor points, but have errors, namely:
wherein p iskIs the true position coordinate vector of the kth node,is GPS/Beidou positioning, Q, of node kkIs a covariance matrix of a k node GPS/Beidou measurement error, the order is 3x3 or 2x2,is the set of all nodes with GPS/Beidou;
third, all nodes in the network have obtained the following information so far: { Tn,n＝1,2,…,N}，{Rnm,n,m＝1,2,…,N}，According to the information, each node in the network applies a positioning algorithm to obtain the positioning of each node in the network.
2. A high-precision network topology positioning algorithm of clock non-synchronous network according to claim 1, characterized in that in the third step, said positioning algorithm adopts a high-performance positioning algorithm, specifically as follows:
according to expression (14), define:
<mrow> <msub> <mi>y</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mover> <mo>=</mo> <mi>&amp;Delta;</mi> </mover> <mi>C</mi> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mrow> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>T</mi> <mi>n</mi> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
the optimal positioning is calculated accordingly:
wherein,is a collection of all "node pairs" directly interconnected to each other, σ being defined as before, pm,pnRepresenting the position coordinate vectors of the m, N nodes, p is a 3 × N or 2 × N matrix formed by transversely splicing the position vectors of the N nodes,represents an estimate of p generated using the present algorithm;
firstly, positioning by K nodes with GPS/BeidouAs an anchor point, obtaining the positioning of all N nodes by adopting a traditional triangulation positioning method; adding the triangulated nodes into the expanded anchor point set, thereby ensuring that all N nodes can be located;
defining:
wherein,
based on the initial estimation of each node position obtained by the method, the following algorithm sequential convex programming method (SCP) is used for solving the formula (17):
(1) calculating the first derivative of the function f (p)And a second derivative
(2) ComputingEigenvalue decomposition ofWherein Λ isThe eigenvalues are diagonal matrices of diagonal elements, U isOf the feature vectors of (1), UHIs the conjugate transpose of U;
(3) if all diagonal elements of the lambda are positive numbers, entering the step (4), and otherwise, jumping to the step (5);
(4) computingSkipping to the step (6);
(5) computingΛ therein-Is to Λ-1Replacing the non-positive diagonal elements with 0 to obtain a new diagonal matrix;
(6) using a back-tracking linear search method or any other linear search method, the energy minimization f (p + su) is calculatednt) S, where s is the step size, f (p + su)nt) The position coordinates p of all nodes in the network are updated to p + suntThen, the cost function value of the formula (5);
(7) determining whether a stopping criterion is met: if (p + su)nt) -f (p) l ≦ η, where η is a self-setting small positive number, updating p to p + sunt，p←p+sunt(ii) a If the stopping criterion is met, finishing the algorithm to obtain the position coordinate estimation p of the nodes in the network; otherwise, returning to the step (1);
in the above algorithm process, the first derivative is:
whereinIs a set of one-hop neighbor nodes of node n, andis the set of all nodes with GPS/Beidou;
the second derivative is:
wherein:
3. a high accuracy network topology location algorithm for clock-unsynchronized networks according to claim 1, characterized by further deriving the theoretical optimal performance limit-caramellea limit:
<mrow> <msub> <mi>C</mi> <mi>p</mi> </msub> <mo>=</mo> <msup> <mrow> <mo>&amp;lsqb;</mo> <msubsup> <mi>&amp;sigma;</mi> <mi>n</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msubsup> <mi>T</mi> <mo>+</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <msup> <mi>Q</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>&amp;rsqb;</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow>
wherein Q ═ diag { Q ═ Q1,Q2,…,QKIs a 3K × 3K block diagonal matrix, and the diagonal sub-blocks of T are:
wherein,is a set of neighbor nodes for node n, and the off-diagonal sub-blocks of T are:
4. a high-precision network topology positioning algorithm of a clock asynchronous network is characterized in that for a large-scale multi-hop network, a distributed positioning algorithm is adopted, namely, the large-scale multi-hop network is clustered into a plurality of sub-networks by a certain self-aggregation method, wherein two adjacent sub-network clusters A and B contain a plurality of common nodes called bridge nodes, and the distributed positioning algorithm comprises the following specific steps:
(1) performing the first and second steps of claim 1 in cluster a and cluster B, respectively, and collecting intra-cluster information: these notations are the same as above, and superscript A, B identifies intra-cluster information pertaining to cluster a and cluster B, respectively;
(2) in the cluster A, processing the information collected in the cluster A by using the positioning algorithm of claim 2 to perform intra-cluster node positioning; retrieving positioning results of bridge nodes thereinNamely column vectors formed by splicing the head and the tail of the position vectors positioned by the bridge nodes; calculating a node positioning CRB matrix in the cluster, and taking out a CRB matrix sub-block C corresponding to the bridge nodeA(ii) a Will be provided withCATransmitting to cluster B;
(3) in cluster B, the bridge node is taken as the anchor point, andas their anchor point locations, whereinThe column vector is formed by splicing the real position vectors of the bridge nodes from beginning to end; application claimSolving 2, the positioning algorithm processes the information collected in the cluster B to perform cluster node positioning; retrieving positioning results of bridge nodes thereinCalculating a node positioning CRB matrix in the cluster, and taking out a CRB matrix sub-block C corresponding to the bridge nodeBWill beCBTransmitting to the cluster A;
(4) in cluster a, again, the bridge node is taken as the anchor point andpositioned as their anchor points; processing the collected information in the cluster A for intra-cluster node positioning by using the positioning algorithm of claim 2; if the positioning result in the cluster A has no important change compared with the previous result, namely the updated positioning estimation is compared with the original estimation, and the change is smaller than a certain optional threshold, entering the step (5); otherwise, the positioning result of the bridge node in the network is taken outAnd calculating a node positioning CRB matrix in the cluster, and taking out a sub-block C corresponding to the bridge nodeAWill beCATransmitting to cluster B; returning to the step (3);
(5) and merging the positioning results of the cluster A and the cluster B to obtain the node positioning of the whole network: p ← pA,pB(ii) a Wherein p isA,pBThe final positioning results in cluster a and cluster B, respectively.
5. The algorithm for locating the network topology of the clock-asynchronous network in high precision according to claim 4, wherein the steps (1) - (5) are performed between any pair of adjacent clusters under the condition that the number of subnets is greater than or equal to 3, so as to finally obtain the optimal location of the whole large network.
6. A high accuracy network topology location algorithm for clock-unsynchronized networks according to claim 4, characterized by further obtaining an optimal estimate of clock drift of each node with respect to the first, reference node:
<mrow> <msub> <mover> <mi>&amp;delta;</mi> <mo>^</mo> </mover> <mrow> <mi>M</mi> <mi>L</mi> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mover> <mi>A</mi> <mo>&amp;OverBar;</mo> </mover> <mi>T</mi> </msup> <mover> <mi>A</mi> <mo>&amp;OverBar;</mo> </mover> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mover> <mi>A</mi> <mo>&amp;OverBar;</mo> </mover> <mi>T</mi> </msup> <mi>y</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>
wherein y ismnOne after anotherVector (c):
y＝(y21，y31...，yN1，y12，y32...，yN2，......，y1N，y2N，...，y(N-1)N)T
a is a matrix obtained by removing the first column;
wherein A is1，A2，A3，...，ANAre all (N-1) × N dimensional matrices:
by the way of analogy, the method can be used,
7. a high accuracy network topology location algorithm for clock non-synchronous networks according to claim 6, characterized by further deriving a theoretical optimal performance limit-caramellea limit:
<mrow> <msub> <mi>C</mi> <mi>&amp;sigma;</mi> </msub> <mo>=</mo> <msup> <mi>&amp;sigma;</mi> <mn>2</mn> </msup> <msup> <mrow> <mo>(</mo> <msup> <mover> <mi>A</mi> <mo>&amp;OverBar;</mo> </mover> <mi>T</mi> </msup> <mover> <mi>A</mi> <mo>&amp;OverBar;</mo> </mover> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow>
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