CN106483496A - Based on CHAN algorithm with improve Newton iteration combine time difference positioning method - Google Patents

Abstract

The invention discloses a kind of based on CHAN algorithm with improve Newton iteration combine time difference positioning method, using CHAN algorithm, obtain unknown node position estimation value；Estimated value is iterated as the initial value of Newton iterative, in order to limit, the direction of search carries out eigenvalue correction in an iterative process to Hessian matrix, in order to accelerate, search speed cubic interpolation method introduces step factor, the suppression phenomenon that convergence rate especially even lost efficacy slowly near multiple-roots introducing repeated root coefficient improves iterative formula；Complete the solution to unknown node position.The Newton iteration of the present invention is good closer to extreme point, positioning result effect when finding Approximate Extreme Points；Measurement error root-mean-square value is little；Iterationses are significantly less than Newton iteration method and revise Newton iteration method.The amount of calculation that method of the present invention amount of calculation is less than Newton iteration method and revises Newton iteration method, positioning precision is also high than other two kinds of arithmetic accuracy.

Description

Based on CHAN algorithm with improve Newton iteration combine time difference positioning method

Technical field

The invention belongs to field of locating technology, more particularly, to one kind are combined with improving Newton iterative based on CHAN algorithm Reaching time-difference localization method.

Background technology

With the progress of modern science and technology, location technology is increasingly taken seriously.In location technology, the positioning precision of system Lifting be also taken seriously, in addition the calculating time of algorithm be also increasingly taken seriously.Except improving the performance of hardware, It is important that innovatory algorithm.Conventional iterative method has：Taylor algorithm, Kalman filtering algorithm etc., major defect is Its convergence is highly dependent on the precision of initial position.When initial position selects bad, during iteration, it is easily trapped into local Minimum point, and it cannot be guaranteed that convergence.Although Newton iteration method is also required to initial estimated location, its convergence rate compares Hurry up, in the case of meeting certain positioning accuracy request, the iterative calculation time can be solved the problems, such as, but when initial value away from During optimal solution, efficiency is very low, and good optimized algorithm is not relying on accurate search procedure.

In sum, Newton iterative occurs dependence initial value degree of accuracy and iterationses excessively and in multiple-roots The especially slow problem even losing efficacy of convergence rate nearby.For this problem, on the basis of Newton iterative, first to Hai Sen Matrix anon-normal timing iteration direction is incorrect to be led to algorithm not restrained or even lost efficacy carry out eigenvalue improvement to Hessian matrix；Then Initial value from optimal solution far when amount of calculation is bigger, iteration efficiency is very low when introduce step factor；Finally near substance root When fast convergence rate, but near multiple-roots, convergence rate is especially slow when even losing efficacy, and introduces repeated root coefficient at multiple-roots.

Content of the invention

It is an object of the invention to provide a kind of based on CHAN algorithm with improve Newton iteration combine time difference positioning method, Aim to solve the problem that and in Newton iterative, occur that dependence initial value degree of accuracy and iterationses are received excessively and near multiple-roots Hold back the especially slow problem even losing efficacy of speed.

The present invention be achieved in that a kind of based on CHAN algorithm with improve Newton iteration combine time difference positioning method, Described based on CHAN algorithm with improve Newton iteration combine time difference positioning method use CHAN algorithm, obtain unknown node position Estimated value；Estimated value is iterated as the initial value of Newton iterative, in iterative process, feature is carried out to Hessian matrix Value is corrected, is introduced step factor by cubic interpolation method, the iterative formula having multiple root is improved；Complete unknown node position Solve.

The step of described iteration is：

1) position initial value u will be solved with CHAN algorithm₀；

2) seek Hessian matrixIn H_uIn the case of may being nonpositive definite matrix, volumes of searchesSearcher To being not necessarily accurately, algorithm may be led to not restrain or even lost efficacy, by matrix H_uCarry out Eigenvalues Decomposition to obtain：H_u=U Λ U^T, U be characterized vector combination matrix, Λ=diag { h₁,h₂}(h₁,h₂Ascending arrangement), h₁,h₂... there are following three kinds Situation is simultaneously modified：

A. just it is, i.e. h₂＞ h₁＞ 0, now H_uFor positive definite matrix, direction of search l is descent direction, the position solution tried to achieve For minimum point,

B. it is negative, i.e. h₁＜ h₂＜ 0, now takes

C. one positive one negative, i.e. h₁＜ 0 ＜ h₂, take

The eigenvalue of the Hessian matrix being corrected is all higher than 0, such that it is able to draw which kind of initial value no matter giving, warp Revised Hessian matrix is all positive definite matrix so that the direction of search of volumes of searches l is descent direction, and then makes Searching point more Close to minimum point, corrected Hessian matrix expression formula is：

3) seek step factor：Add step factor in newton iteration formula, obtainAnd OrderConstruction cubic polynomial p (λ)=a+b λ+c λ²+dλ³, when λ=0, order

In the hope of the value of a, b, c, d, and then the minimum of p (λ) can be tried to achieve.The minimum value point of three multinomial p (λ) is made ForExtreme point approximate.UnderstandThenTherefore step factor is optional For：

4) seek repeated root coefficient：NoteRepeated root coefficient

5) by initial value u₀, substitute into improved Newton iteration relational expressionAfter obtaining iteration Position u_k+1=(x^(k+1),y^(k+1)) (k=1,2 ..., N-1)；

6) by u_k=(x^(k),y^(k)) substitute into redundance type

In, obtainε₀Require specified value for range accuracy；By u_k+1=(x^(k+1), y^(k+1)) substitute into Δ_k+1=max (| x^(k+1)-x^k|,|y^(k+1)-y^k|)；

7) judge whether that satisfaction requires ε_i＜ ε or Δ_k+1＜ Δ, if meeting one of condition, this time iterates to This terminates, and output valve is (x^(k+1),y^(k+1)), otherwise this value is continued iteration as initial value, till meeting requirement.

Further, described MS position estimation value is to minimize quadratic model object function, that is,：

J (u)=(r-f (u))^TQ^-1(r-f(u))；

Q is the positive definite covariance matrix of noise, solves the minima that u must seek J (u), obtained estimator is referred to as A young waiter in a wineshop or an inn takes advantage of estimated value.

Further, the Hessian matrix expression formula of the eigenvalue correction of described Hessian matrix is：

Diagonal on element be just.

Further, described by the improved step factor of cubic interpolation method it is：

Wherein

Further, described repeated root value is improved and is included：

It is the m repeated root of g (u), whenIt is substantial access to u_αWhen, had by Taylor expansion：

In formula,

OrderIt is substantial access to u in u_αWhen (h, k) → (0,0), then have：

Defined functionObviously there is Ω (u) → m (when (h, k) → (0,0)), therefore using iteration Formula：

Another object of the present invention is to provide a kind of using described based on CHAN algorithm and improvement the combining of Newton iteration The alignment system of time difference positioning method.

The present invention provide based on CHAN algorithm with improvement Newton iteration combine time difference positioning method, by nonlinearity Problem dress turns to Least Square Solution, utilizes Newton iterative iteration on the basis of using CHAN Algorithm for Solving initial solution Solve, in iterative process revise Hessian matrix eigenvalue, then accelerate iteration time adopt cubic interpolation method introduce step-length because Son, finally to have multiple take root in a little carry out improve iterative formula.Finally by simulating, verifying, this algorithm has preferable positioning Performance.

In emulation Fig. 3, when criteria for noise difference is for 3m, several corrections based on Newton's algorithm and improved path simulation In it can be seen that basic Newton iterative position error is worst, and the calculation making improvements on the basis of revising Newton iteration Method locating effect is best.This is because improved Newton iteration find Approximate Extreme Points when closer to extreme point, so positioning Result effect is best.

In the diagram, in the case of different noise mean square root values, the measurement error root-mean-square value of basic Newton's algorithm is Greatly, minimum to the measurement error root-mean-square value revising Newton iteration.For every kind of algorithm, measurement error root-mean-square value is with making an uproar The increase of sound error mean square root and be gradually increased.The measurement error being several algorithms between 0.5 to 3 in noise error root-mean-square value Root-mean-square value difference is not very big, and basic Newton iterative is than other two algorithm after noise error root-mean-square value is more than 3 Measurement error root-mean-square substantially increased.

Fig. 5 is the statistical result of the mean iterative number of time that each algorithm changes with measurement error.With the increase of measurement error, The mean iterative number of time of modified hydrothermal process (innovatory algorithm to revising Newton iteration) of the present invention keeps minimum, and iterationses are obvious Less than Newton iteration method with revise Newton iteration method, its measurement error root-mean-square value also keeps minimum in figure 3.This explanation is originally Invent the amount of calculation that improved method amount of calculation is less than Newton iteration method and revises Newton iteration method, and the present invention is improved Method positioning precision is also high than other two kinds of arithmetic accuracy.

Brief description

Fig. 1 be provided in an embodiment of the present invention based on CHAN algorithm with improve Newton iteration combine time difference positioning method stream Cheng Tu.

Fig. 2 is the schematic diagram of positioning using TDOA provided in an embodiment of the present invention.

Fig. 3 is that several algorithm positioning result provided in an embodiment of the present invention emulates schematic diagram.

Fig. 4 is several algorithm Positioning Error Simulation schematic diagram provided in an embodiment of the present invention.

Fig. 5 is that several algorithm provided in an embodiment of the present invention positions mean iterative number of time.

Specific embodiment

In order that the objects, technical solutions and advantages of the present invention become more apparent, with reference to embodiments, to the present invention It is further elaborated.It should be appreciated that specific embodiment described herein, only in order to explain the present invention, is not used to Limit the present invention.

Below in conjunction with the accompanying drawings the application principle of the present invention is explained in detail.

As shown in figure 1, provided in an embodiment of the present invention based on CHAN algorithm with improve Newton iteration combine positioning using TDOA Method comprises the following steps：

S101：Nonlinearity problem dress is turned to Least Square Solution, using CHAN Algorithm for Solving initial solution；

S102：The eigenvalue of Hessian matrix is revised during Newton iteration

S103：Accelerating iteration time adopts cubic interpolation method to introduce step factor；

S104：Finally to there being multiple take root in introduce a repeated root coefficient and improve iterative formula.

S105：Judge whether to meet condition：No satisfaction requires ε_i＜ ε or Δ_k+1＜ Δ.

S106：Require until meeting, output result.

The step of described iteration is：

1) position initial value u will be solved with CHAN algorithm₀；

2) seek Hessian matrixIn H_uIn the case of may being nonpositive definite matrix, volumes of searchesSearcher To being not necessarily accurately, algorithm may be led to not restrain or even lost efficacy, by matrix H_uCarry out Eigenvalues Decomposition to obtain：H_u=U Λ U^T, U be characterized vector combination matrix, Λ=diag { h₁,h₂}(h₁,h₂Ascending arrangement), h₁,h₂... there are following three kinds Situation is simultaneously modified：

A. just it is, i.e. h₂＞ h₁＞ 0, now H_uFor positive definite matrix, direction of search l is descent direction, the position solution tried to achieve For minimum point,

B. it is negative, i.e. h₁＜ h₂＜ 0, now takes

C. one positive one negative, i.e. h₁＜ 0 ＜ h₂, take

The eigenvalue of the Hessian matrix being corrected is all higher than 0, such that it is able to draw which kind of initial value no matter giving, warp Revised Hessian matrix is all positive definite matrix so that the direction of search of volumes of searches l is descent direction, and then makes Searching point more Close to minimum point, corrected Hessian matrix expression formula is：

3) seek step factor：Add step factor in newton iteration formula, obtainAnd OrderConstruction cubic polynomial p (λ)=a+b λ+c λ²+dλ³, when λ=0, order

In the hope of the value of a, b, c, d, and then the minimum of p (λ) can be tried to achieve.The minimum value point of three multinomial p (λ) is made ForExtreme point approximate.UnderstandThenTherefore step factor is optional For：

4) seek repeated root coefficient：NoteRepeated root coefficient

5) by initial value u₀, substitute into improved Newton iteration relational expressionAfter obtaining iteration Position u_k+1=(x^(k+1),y^(k+1)) (k=1,2 ..., N-1)；

6) by u_k=(x^(k),y^(k)) substitute into redundance typeIn, obtain ε=min (ε₀,ε_i) i=3,4 ..., N,ε₀Require specified value for range accuracy；By u_k+1=(x^(k+1),y^(k+1)) substitute into Δ_k+1=max (| x^(k+1)-x^k|,|y^(k+1)-y^k |)；

7) judge whether that satisfaction requires ε_i＜ ε or Δ_k+1＜ Δ, if meeting one of condition, this time iterates to This terminates, and output valve is (x^(k+1),y^(k+1)), otherwise this value is continued iteration as initial value, till meeting requirement.

Below in conjunction with the accompanying drawings the application principle of the present invention is further described.

1 positioning using TDOA mathematical model

Assume have M base station to be in random distribution state, the schematic diagram based on positioning using TDOA is as shown in Figure 1.In figure, u=[x, y]^TFor unknown node MS position coordinateses to be estimated；s_i=[x_i,y_i]^TActual position coordinate for i-th known node BS.MS arrives BS_iThe distance between be

Wherein, | | | | represent mould.

t_iFor the signal of telecommunication from MS to BS_iThe LOS propagation time, t_eiIt is from MS to BS_iNLOS excessive delay, it be one Individual on the occasion of stochastic variable, each t_eiBetween separate, when the signal of telecommunication is from MS to BS_iBe LOS propagate when t_ei=0.Do not lose one As property, choose s₁As with reference to receiving station, MS to BS can be obtained_iPositioning using TDOA equation be：

Wherein,For MS to BS₁、BS_iBetween LOS propagate TDOA true value；t_ei1=t_ei-t_e1It is broadcast to for NLOS The error that TDOA measurement introduces, its average is typically all not zero；n_i1For systematic measurement error, relatively NLOS error t_ei1For very Little can correct, and it and t_ei1Separate.It is multiplied by formula (2) two ends with c, can obtain location from range-difference measurements equation is simultaneously：

Wherein, c is propagation of electrical signals speed；It is from MS to BS₁、BS_iLOS propagation distance poor；η_i1= ct_ei1+cn_i1It is broadcast to range error and the measurement error sum that TDOA measurement introduces for NLOS, satisfaction is independently distributed.Make R_i1 (u)=r_i ⁰-r₁ ⁰, formula (3) is write as vector form is：

R=R (u)+η (4)

Wherein, r=[r₂₁,…,r_M1]^T；R (u)=[R₂₁(u),…,R_M1(u)]^T；η=[η₂₁,…,η_M1]^T, rise for convenience See it is believed that this noise η obey average be zero, positive definite covariance matrix be Q Gauss distribution.

Q=E [(η-E [η]) (η-E [η])^T] (5)

Wherein, E [] represents expected value.

Likelihood function can be obtained according to formula (4) is：

Wherein, Q^-1It is the inverse matrix of Q, because Q is symmetric positive definite matrix, its inverse presence.Maximal possibility estimation is to maximize (6) value, therefore can be converted into minimum quadratic model object function, that is,：

J (u)=(r-R (u))^TQ^-1(r-R(u)) (7)

Understood according to formula (7), solve the minima that u must seek J (u), obtained estimator is referred to as least-squares estimation Value, and N^-1A weighting coefficient matrix can be considered as.But it is extremely difficult with analytical method solving nonlinearity function.Therefore, The present invention solves this nonlinear equation using overall offset method, determines the position coordinate value of unknown node MS.

2 Newton iterative

The essence of Newton iteration is nonlinear equation progressively to develop into a kind of linear equation to solve, and method is to use letter Before the Taylor series of number several finding equation root.

According to the condition of Newton iteration method, according to formula (3) and (4) establishing equation positioning equation group：

G (u)=R (u)-r=0 (8)

If assume u for single argument, the method seeking non trivial solution is in iteration point u₀Place uses two grades of Taylor expansions, then Remove linear segment, that is,：

G (u)=g (u₀)+(u-u₀)g'(u₀) (9)

Make formula (9) be equal to 0, then have：

Through continuous iteration, iterative formula can be obtained：

Solve in the present invention is minimum problem it is possible to be converted into seeking its derived function Zeroes, that is, seek g' The solution of (u)=0.

G (u) is deployed into second order with Taylor's formula, that is,：

The left side and g (u) approximately equal, order in formula (12)：

Then to Δ u derivation, obtain：

G'(u)+g " (u) Δ u=0 (14)

Solve：

Then recursion can obtain iterative formula：

Inferred above is to discuss for single argument, and sets initial value u=(xy)^TShi Zewei multivariate, is solving cattle Jacobian matrix and Hessian matrix is introduced during the optimization problem of iteration of pausing.

Choose front 2 establishing equations as follows from formula (8)：

From formula (9), the gradient vector tried to achieve and Hessian matrix are respectively：

The gradient vector of formula (8) and Hessian matrix are respectively：

Wushu G_uAnd H_uIn substitution formula (16), the equation obtaining Newton iteration is：

By this inference, set initial value u₀=(x₀,y₀)^TWhen newton iteration equation be：

In formula,WithIt is respectively function in given initial value u₀Under gradient vector and Hessian matrix

3 improved Newton iteration method algorithms

Newton iterative has the problem easily dissipating, and to convergence rate near multiple-roots, especially slow even inefficacy shows As improving, improved Newton iterative is proposed.

3.1 correction Hessian matrix

Newton iterative, substitutes into the relational expression of Newton iteration after estimating initial value, until meeting given precision Require.Assume that function g (u) Second Order Continuous at u can be led, then its Hessian matrix is likely to occur three kinds of situations：

1) if H_uIt is positive definite matrix, then now u is in local minizing point；

2) if H_uIt is negative definite matrix, then now u is in Local modulus maxima；

3) if H_uIt is indefinite matrix, then be not now extreme point at u.

In H_uIn the case of may being nonpositive definite matrix, volumes of searches in formula (21)The direction of search not necessarily It is accurate, algorithm may be led to not restrain or even lost efficacy.For this problem, the author proposes the correction cattle based on CHAN algorithm Pause iteration Localization Estimate Algorithm of TDOA.

By matrix H_uCarry out Eigenvalues Decomposition to obtain：

H_u=U Λ U^T(22)

In formula, U is characterized the matrix of vector combination, Λ=diag { h₁,h₂}(h₁,h₂Ascending arrangement).

H in formula (21)₁,h₂... there are following three kinds of situations and be modified：

1) just it is, i.e. h₂＞ h₁＞ 0, now H_uFor positive definite matrix, direction of search l is descent direction, the position solution tried to achieve For minimum point,

2) it is negative, i.e. h₁＜ h₂＜ 0, now takes

3) one positive one negative, i.e. h₁＜ 0 ＜ h₂, take

The eigenvalue of the Hessian matrix being corrected is all higher than 0, such that it is able to draw which kind of initial value no matter giving, warp Revised Hessian matrix is all positive definite matrix so that the direction of search of volumes of searches l is descent direction, and then makes Searching point more Close to minimum point, corrected Hessian matrix expression formula is：

From formula (23) it can be seen that although given arbitrary initial value can make Hessian matrix positive definite through revising and make to search Rope point is closer to minimum point, but is intended to precise search and amount of calculation is little, also relies on the accurate of initial value.When initial value with When excellent solution differs greatly, it is iterated solving, computationally intensive efficiency is low.

3.2 introducing step factors

In the correct direction of search, the direction of search depends on volumes of searches with sizeThrough Hessian matrix The direction of search after correction is it has been determined that and search for the value that size additionally depends on volumes of searches l.This searching method is in order to accurately search Rope needs for substantial amounts of iterative calculation, especially initial value from optimal solution far when amount of calculation bigger, this search efficiency Very low.In order that the direction of search, closer to minimum, introduces step factor λ, so that volumes of searches is revised as in each iterative processAnd the selection of step factor λ can be calculated using cubic interpolation method.

Iterative equation (21) is added step factor, obtains：

(22) are substituted into function g (u), and makesConstruction cubic polynomial：

P (λ)=a+b λ+c λ²+dλ³(25)

When λ=0, order：

In the hope of the value of a, b, c, d, and then the minimum of p (λ) can be tried to achieve by (26)～(29).By three multinomial p's (λ) The point conduct of minimum valueExtreme point approximate.

Understand in conjunction with (19)：ThenTherefore step factor is chosen as：

3.3 repeated root values are improved

Newton iteration method is in fact a kind of special fixed point iteration, fast convergence rate when near substance root, but Near multiple-roots, convergence rate is especially slow even lost efficacy.

Seek the convergence rate of nonlinear equation repeated root in order to improve Newton iteration method, the Newton iterations of common two kinds of corrections Form is as follows：

1) rememberTo f (u) Newton iteration method, that is,Wherein G_f(u)、H_f(u) It is respectively gradient vector and Hessian matrix；

2) if knowing multiplicity of a root m (m >=2) in advance,

Both forms all have second order convergence speed, but have certain limitation in the application.Because form 1) Require to calculate gradient vector and the Hessian matrix of f (u), this g (u) three rank local derviation to be calculated increased amount of calculation；And form 2) then must know in advance that multiplicity of a root, be difficult in actual applications accomplish.Make following improvement for this：

IfIt is the m repeated root of g (u), whenIt is substantial access to u_αWhen, launched by Taylor Have：

In formula,

OrderIt is substantial access to u in u_αWhen (h, k) → (0,0), then have：

Defined functionObviously there is Ω (u) → m (when (h, k) → (0,0)), therefore using iteration Formula：

Convergence rate can be improved with formula (33), during general Newton iteration, with formula (21) calculate g'(u), g " After (u), then calculate Ω (u) very big cost will not be spent, it is worth for therefore calculating Ω (u).

4 arthmetic statements

Revise Newton iterative process：

1) position will be solved with CHAN algorithm and judge whether to meet allowable error, if meeting iteration stopping；If being unsatisfactory for, It is set to initial value u₀, substitute into Newton iteration relational expression, construct volumes of searches, Hessian matrix is tried to achieve according to formula (23)According to Formula (30) material calculation factor lambda, goes out the position u after iteration according to seeking formula (33)_k+1=(x^(k+1),y^(k+1)) (k=1,2 ..., N- 1)；

2) by u_k=(x^(k),y^(k)) substitute into redundance typeIn, obtain ε=min (ε₀,ε_i) i=³,4,..., N,ε₀Require specified value for range accuracy.By u_k+1=(x^(k+1),y^(k+1)) substitute into Δ_k+1=max (| x^(k+1)-x^k|,|y^(k+1)-y^k |)；

3) judge whether that satisfaction requires ε_i＜ ε or Δ_k+1＜ Δ, if meeting one of condition, this time iterates to This terminates, and output valve is (x^(k+1),y^(k+1)), the iterative step otherwise this value being continued the above as initial value, will until meeting Till asking.

The application effect of the present invention is explained in detail with performance evaluation with reference to algorithm simulating.

In the present invention, the positioning performance of each algorithm is weighed using the root-mean-square error of location estimation, it is fixed Adopted formula is：

The present invention emulates under Matlab R2010a environment.Assume the region that cell is 100 × 100, participate in the base of positioning Stand as 4, base station is respectively (0,0), (0,100), (100,100), (100,0).When mobile station is done at the uniform velocity with the speed of 4m/s Linear motion, the sampling time is 1s, and when noise criteria difference is 3m, the result emulation of several algorithm path orientations based on newton Figure such as Fig. 3.

In emulation Fig. 3, when criteria for noise difference is for 3m, several corrections based on Newton's algorithm and improved path simulation In it can be seen that basic Newton iterative position error is worst, and the calculation making improvements on the basis of revising Newton iteration Method locating effect is best.This is because improved Newton iteration find Approximate Extreme Points when closer to extreme point, so positioning Result effect is best.

In order to will become apparent from the locating effect of several algorithms further, then divided in different noise mean square root values with several algorithms Wei not emulated in the case of 0.5,1,1.5,2,2.5,3,3.5,4,4.5,5,5.5,6, experiment every time independently carries out 200 Secondary.Statistical result analogous diagram such as Fig. 4 that measurement error compares.

In the diagram, in the case of different noise mean square root values, the measurement error root-mean-square value of basic Newton's algorithm is Greatly, minimum to the measurement error root-mean-square value revising Newton iteration.For every kind of algorithm, measurement error root-mean-square value is with making an uproar The increase of sound error mean square root and be gradually increased.The measurement error being several algorithms between 0.5 to 3 in noise error root-mean-square value Root-mean-square value difference is not very big, and basic Newton iterative is than other two algorithm after noise error root-mean-square value is more than 3 Measurement error root-mean-square substantially increased.

These three algorithms not only have certain difference in positioning precision, also have difference, Average Iteration on iterationses The statistical result of number of times such as Fig. 5.

Fig. 5 is the statistical result of the mean iterative number of time that each algorithm changes with measurement error.With the increase of measurement error, The mean iterative number of time of modified hydrothermal process (innovatory algorithm to revising Newton iteration) of the present invention keeps minimum, and iterationses are obvious Less than Newton iteration method with revise Newton iteration method, its measurement error root-mean-square value also keeps minimum in figure 3.This explanation is originally Invent the amount of calculation that improved method amount of calculation is less than Newton iteration method and revises Newton iteration method, and the present invention is improved Method positioning precision is also high than other two kinds of arithmetic accuracy.

The foregoing is only presently preferred embodiments of the present invention, not in order to limit the present invention, all essences in the present invention Any modification, equivalent and improvement made within god and principle etc., should be included within the scope of the present invention.

Claims (7)

1. a kind of based on CHAN algorithm with improve Newton iteration combine time difference positioning method it is characterised in that described be based on CHAN algorithm uses CHAN algorithm with the time difference positioning method of combining improving Newton iteration, obtains unknown node position estimation value； Estimated value is iterated as the initial value of Newton iterative, in order to limit the direction of search in iterative process in iterative process In Hessian matrix carried out with eigenvalue correction, introduce step factor, suppression many to accelerate search speed cubic interpolation method Nearby the especially slow phenomenon introducing repeated root coefficient even losing efficacy of convergence rate improves iterative formula to repeated root；Complete to unknown node position The solution put.

2. as claimed in claim 1 time difference positioning method combined based on CHAN algorithm and improvement Newton iteration, its feature exists In the step of described iteration is：

1) position initial value u will be solved with CHAN algorithm₀；

2) seek Hessian matrixIn H_uIn the case of nonpositive definite matrix, by matrix H_uCarry out Eigenvalues Decomposition to obtain：H_u=U Λ U^T, U be characterized vector combination matrix, Λ=diag { h₁,h₂, h₁,h₂Ascending arrangement, h₁,h₂... there are following three kinds of feelings Condition is simultaneously modified：

Just it is, i.e. h₂＞ h₁＞ 0, now H_uFor positive definite matrix, direction of search l is descent direction, and the position tried to achieve solves as minimum Value point,

It is negative, i.e. h₁＜ h₂＜ 0, now takes

One positive one negative, i.e. h₁＜ 0 ＜ h₂, take

Corrected Hessian matrix expression formula is：

3) seek step factor：Add step factor in newton iteration formula, obtainAnd makeConstruction cubic polynomial p (λ)=a+b λ+c λ²+dλ³, when λ=0, order：

Try to achieve the value of a, b, c, d, and then try to achieve the minimum of p (λ)；Using the minimum value point of three multinomial p (λ) asPole It is worth the approximate of point；UnderstandThenTherefore step factor is chosen as：

4) seek repeated root coefficient：NoteRepeated root coefficient

5) by initial value u₀, substitute into improved Newton iteration relational expressionObtain the position after iteration u_k+1=(x^(k+1),y^(k+1)) (k=1,2 ..., N-1)；

6) by u_k=(x^(k), y^(k)) substitute into redundance type In, obtain ε=min (ε₀, ε_i) i=3,4 ..., N, ε₀Require specified value for range accuracy；By u_k+1=(x^(k+1),y^(k+1)) Substitute into Δ_k+1=max (| x^(k+1)-x^k|,|y^(k+1)-y^k|)；

7) judge whether that satisfaction requires ε_i＜ ε or Δ_k+1＜ Δ, if meeting one of condition, this time iterates to this knot Bundle, output valve is (x^(k+1),y^(k+1)), otherwise this value is continued iteration as initial value, till meeting requirement.

3. as claimed in claim 1 time difference positioning method combined based on CHAN algorithm and improvement Newton iteration, its feature exists In described MS position estimation value is to minimize quadratic model object function, that is,：

J (u)=(r-f (u))^TQ^-1(r-f(u))；

Q is the positive definite covariance matrix of noise, solves u and must seek the minima of J (u), an obtained estimator referred to as young waiter in a wineshop or an inn Take advantage of estimated value.

4. as claimed in claim 1 time difference positioning method combined based on CHAN algorithm and improvement Newton iteration, its feature exists In the Hessian matrix expression formula of the eigenvalue correction of described Hessian matrix is：

Diagonal on element be just.

5. as claimed in claim 1 time difference positioning method combined based on CHAN algorithm and improvement Newton iteration, its feature exists In described by the improved step factor of cubic interpolation method being：

Wherein

6. as claimed in claim 1 time difference positioning method combined based on CHAN algorithm and improvement Newton iteration, its feature exists In described repeated root value is improved and included：

It is the m repeated root of g (u), whenIt is substantial access to u_αWhen, had by Taylor expansion：

$g^{\′}\left(u\right)=\frac{1}{m!}\underset{p=0}{\overset{m}{\Σ}}{C}_m^p{h}^p{k}^{m-p}\frac{{\∂}^{m}f}{\∂x^p\∂y^{m-p}}{|}_{(x_{\mathrm{\α}},y_{\mathrm{\α}})}+o\left({\mathrm{\ρ}}^{m+1}\right);$

$g^{\′\′}\left(u\right)=\frac{1}{(m-1)!}\underset{p=0}{\overset{m-1}{\Σ}}{C}_{m-1}^p{h}^p{k}^{m-1-p}\frac{{\∂}^{m}f}{\∂x^p\∂y^{m-p}}{|}_{(x_{\mathrm{\α}},y_{\mathrm{\α}})}+o\left({\mathrm{\ρ}}^m\right);$

In formula,

OrderIt is substantial access to u in u_αWhen (h, k) → (0,0), then have：

$\frac{ln\left|f\left(u\right)\right|}{ln\left|g\left(u\right)\right|}\→m;$

Defined functionObviously there is Ω (u) → m (when (h, k) → (0,0)), therefore using iterative formula：

$u=u_0-\mathrm{\λ}\mathrm{\Ω}\left(u\right){\tilde{H}}_{u_0}^{-1}{G}_{u_0}.$

7. determined with the time difference of combining improving Newton iteration based on CHAN algorithm described in a kind of utilization claim 1～6 any one The alignment system of method for position.