CN106372649A - Quantization-based set value Kalman filtering algorithm - Google Patents

Abstract

The invention discloses a quantization-based set value Kalman filtering algorithm. Compared with the prior art, the algorithm has the advantages that the problem in remote state estimation of a dynamic random system in a wireless sensor limited in bandwidth is solved; original measurement values are quantized into messages, and the messages are transmitted to a remote estimator from a local sensor; and a set region of the original measurement values is represented by utilizing information contained in a quantization policy, and a closest ellipsoid approximation method of the region is given. Therefore, the quantization-based set value Kalman filtering algorithm is proposed. Three algorithms are compared and analyzed in computer simulation to explain the validity of the algorithm proposed by the invention.

Description

Based on the set value Kalman filtering algorithm quantifying

Technical field

The present invention relates to microelectronics technology, more particularly, to a kind of set value Kalman filtering algorithm based on quantization.

Background technology

With the development of microelectric technique, wireless communication technology and embedded technology, wireless sensor network has obtained soon The development of speed.Wireless sensor network is a kind of distributed sensor, and the sensor because being arranged is cheap, and position is moved Flexibly, network settings are variable, and fault-tolerant ability is strong, therefore widely apply in national defense and military, smart home, biologic medical, environment prison The various fields such as survey, space exploration, industrial business.

Because the energy of the sensor of wireless sensor network is limited, and the energy of sensor is to be provided by battery , change battery and expend very greatly, sometimes highly difficult, change battery sometimes even due to due to working sensor environment Impossible, that is, battery exhaust the end meaning sensor life-time.The pertinent literature of prior art is pointed out to send one The energy consumption of individual bit data is 800 times consumed of energy of one instruction of computing device.Therefore, quantifying metrical information is one Effectively method, the data of measurement can be processed in net, decrease volume of transmitted data, can be effectively saved energy.Communication It is also the subject matter that wireless sensor network is faced limited by resource, the data that sensor side obtains tends not to directly Connected cordless communication network and passed to estimation end, each sensor can only transmit limited bit every time, and this allows for observation Must be quantified before being transmitted.Therefore quantification problem is all a study hotspot all the time.Another document proposes one Plant the Optimal Distribution formula estimator of random parameter, and give the dispatching method of sensor node of system distribution bit number.Another document It is also proposed the adaptive quantizing strategy under a kind of multidimensional observation, thus the centralized many sensings under the conditions of devising Bandwidth-Constrained Device Fusion Estimation Algorithm.Another document adopts the strategy of uniform quantization, reduces the traffic by the length reducing packet.

All at present is all point value based on the estimated value that the estimation quantifying finally is given, depositing yet with quantization error Actual observation is uncertain, is carried out estimating clearly irrational with the One-Point-Value after quantifying.Another document is directed to When Initial state estimation is distributed as a convex set, propose set value Kalman filtering algorithm first, but this algorithm is still only Only consideration be single point value measurement.When another document considers measured value for set value, by loosening Posterior probability distribution only One property vacation gives state estimation algorithm, and another document further this algorithm is generalized in Multi-sensor Fusion problem.Recently One document assumes that measuring assembly can be represented with spheroid, gives state estimation using between spheroid and Operation Nature equal The computational methods of value set, that is, set value Kalman filtering algorithm.Author demonstrates blending algorithm in detail with sensing data Fusion sequence is unrelated, and demonstrates the gradual of estimation average set sizes.

Content of the invention

The purpose of the present invention is that provides a kind of set value Kalman filtering based on quantization to solve the above problems Algorithm.

The present invention is achieved through the following technical solutions above-mentioned purpose:

The present invention includes quantization strategy and set value Kalman filtering algorithm；

Quantization strategy:

The quantizing bit number of hypothesis quantizer i has distributed and has been l_iPosition, that is, message m_iK () has l_iPosition, according to Assume 2, obtain

$\underset{i=1}{\overset{m}{\σ}}{l}_i=l---\left(3\right)$

And, quantizer i is in intervalUpper totalIndividual point of quantificationThese Point of quantification uniform or heterogeneous be distributed in quantized interval, that is, meet

${\underset{\&overbar;}{u}}_i=a_i^1\left(k\right)<a_i^2\left(k\right)<\mathrm{...}<a_i^{t_i}\left(k\right)={\stackrel{\&overbar;}{u}}_i---\left(4\right)$

$a_i^{j+1}\left(k\right)-a_i^j\left(k\right)={\mathrm{\&delta;}}_i^j\left(k\right)---\left(5\right)$

Conventional quantification manner be generally based on probability it is assumed that measurement componentThen y_i(k) root It is quantized to according to equation belowOr

$p\{m_i\left(k\right)=a_i^j\left(k\right)\}=\frac{\&lsqb;a_i^{j+1}\left(k\right)-y_i\left(k\right)\&rsqb;}{{\mathrm{\&delta;}}_i^j\left(k\right)}---\left(6\right)$

$p\{m_i\left(k\right)=a_i^{j+1}\left(k\right)\}=\frac{\&lsqb;y_i\left(k\right)-a_i^j\left(k\right)\&rsqb;}{{\mathrm{\&delta;}}_i^j\left(k\right)}---\left(7\right)$

Set value Kalman filtering:

Because original measured value is after quantifying, long-range estimates that end is not aware that original observation y (k), but Each amount can be utilized by exploring according to the interval range knowing each measurement component place of quantizing rule formula (5) part Change the quantizing rule information of device, provide set value Kalman filtering algorithm:

Assume that measuring assembly yd (k) is oval set and can be turned to parameter:

Yd (k) :=ε (c (k), y (k)) (8)

Wherein c (k) is elliptical center, and y (k) is to represent oval matrix；Represent ellipse with the mark tr (y (k)) of matrix y (k) The size of circle collection yd (k)；

Lemma 1: ε (c₁,x₁), ε (c₂,x₂) represent two oval set, then

$\mathrm{\&epsiv;}(c_1,x_1)\&circleplus;\mathrm{\&epsiv;}(c_2,x_2)\&subsetequal;\mathrm{\&epsiv;}(c_1+c_2,(1+p^{-1})x_1+(1+p\left)x_2\right)---\left(9\right)$

Wherein, p ＞ 0 and p=(tr x₁)^1/2/(tr x₂)^1/2So that matrix (1+p^-1)x₁+(1+p)x₂) mark minimum；

Assume to record original measured value y (k)=[y in k moment sensor₁(k),y₂(k),...,y_m(k)]^t, it every Individual component y_iK () forms quantization message m after the quantizing rule of quantizer i_i(k), then be sent to far through radio communication channel Journey estimates end, and quantizing rule can be known a priori by estimation end, so can according to the message receiving and quantizing rule To judge the interval range that actual measured value is located；

If the message that remote port receivesI=1,2 ..., m, j ∈ { 1,2 ..., t_iQuantization then can be utilized The quantizing rule information of device judges actual measurement component y_iK () is located at intervalInterior, this area Between withCentered on point, withFor interval radius；Due to point of quantification Can be known in advance by quantizing rule, therefore each measures component y_iK interval central point c that () is located_i(k) and interval radius r_i(k) Can real-time online calculate；

Message m after estimating that termination receives all quantizers quantizations_iWhen (k), by using the information of quantizing rule, can To calculate the interval range that each actual measurement component is located, it is now assumed that c_i(k) and r_i(k), i=1,2 ..., m all calculate Finish, and for set value Kalman filtering algorithm, an important step is to provide the spheroid description of observation set, and survey is given below The spheroid of value y (k) describes method；

Foregoing present and estimating each measurement of end component y_i(k) actual interval range being located, and actual measurement The collection area of value y (k) is not given, definition

ω :={ y (k) ∈ r^m|||y_i(k)-c_i(k)||≤r_i(k), i=1,2 ..., m },

It is a m dimension cuboid, a length of 2r on each side_i(k), the set area of this region i.e. actual observed value Domain, present target is to find an outside spheroid the tightest to comprise ω；DefinitionFor comprising the minimum spheroid of ω, and Meet

$\stackrel{\&overbar;}{\mathrm{\&omega;}}:=\{y\left(k\right)\&element;r^m|{\left(y\right(k)-c(k\left)\right)}^t{y}^{-1}\left(k\right)\left(y\right(k)-c(k\left)\right)\&le;{\mathrm{\&delta;}}^2\left(k\right)\},$

Wherein, c (k)=[c₁(k),c₂(k),...,c_m(k)], Diag represent take diagonal matrix operate；

As m=2, ω withBetween relation as shown in figure 3, in fact, elliptical center is exactly in rectangle Heart point, that is, each interval central value, oval matrix is closely related with interval radius；

Due to convexity, m dimension at least 2 summits of cuboid are in spheroidBorder on, other summits or be included inInterior orBorder on；So the value of δ (k) may be calculated as

$\mathrm{\&delta;}\left(k\right)=\underset{y_i\left(k\right)\&element;\{{\mathrm{\&delta;}}_i^1\left(k\right),{\mathrm{\&delta;}}_i^2\left(k\right)\},i\&element;\{1,2,\mathrm{...},m\}}{\max }\sqrt{{\left(y\right(k)-c(k\left)\right)}^t{y}^{-1}\left(k\right)\left(y\right(k)-c\left(k\right))}---\left(10\right)$

Wherein,

Therefore, in each sampling instant, the tightest ellipsoid collection area that observation is comprised is ε (c (k), δ²(k)y (k)).

The beneficial effects of the present invention is:

The present invention is a kind of set value Kalman filtering algorithm based on quantization, and compared with prior art, the present invention is in bandwidth The remote status estimation problem of dynamic stochastic systems in limited wireless senser.Original measured value is quantized into message from office Portion's sensor is sent to long-range estimator.Illustrate the set area of original measurement value by using the information that quantization strategy is comprised Domain, and give the tightest ellipsoid approximate method in this region.And then it is proposed that based on the set value Kalman filtering algorithm quantifying. In Computer Simulation, three kinds of algorithms are compared and analyzes, illustrate the effectiveness of carried algorithm herein.

Brief description

Fig. 1 is the distributed architecture figure of the quantizer of the present invention；

Fig. 2 is the quantized interval figure of the present invention

Fig. 3 is in the present invention, as m=2,For comprising the minimum ellipse figure of ω；

Fig. 4 is the state estimation interval curve figure of the present invention；

Fig. 5 is the state estimation error burst curve chart of the present invention.

Specific embodiment

The invention will be further described below in conjunction with the accompanying drawings:

Problem describes:

System description:

Consider a following class linear time invariant system

X (k+1)=ax (k)+w (k) (1)

Y (k)=cx (k)+v (k) (2)

Wherein, k is time index,It is system mode vector,It is the observation vector to state x (k),WithIt is systematic procedure noise and observation noise respectively, state transfer matrix a and observing matrix c has suitable When dimension.

Some are presented herein below reasonably necessary assume:

Assume that 1:w (k), v (k) they are separate zero mean Gaussian white noise, variance is respectively q (k) and r (k), and Meet

$e\left\{\left(\begin{array}{c}w\left(k\right)\\ v\left(k\right)\end{array}\right)\left(w^t\right(l\left)v^t\right(l\left)\right)\right\}=\left[\begin{array}{cc}q\left(k\right)& 0\\ 0& r\left(k\right)\end{array}\right]{\mathrm{\&delta;}}_{kl}$

The average of original state x (0) is x₀, variance is p₀, and independent of w (k), v (k).

Assume 2: assume a width of l bit of the band of radio communication channel under bandwidth constraint, and from sensor to fusion The radio communication channel at center is preferable, does not have bit-errors.

Hypothesis 3:1≤i≤m, wherein y_iK () is i-th element of observation vector y (k),u _iWithIt is Observation y_iThe upper bound of (k) and lower bound and be known constant.

Problem is set up:

Consider estimation problem as shown in Figure 1, due to limited communications resources, network environment is complicated, and we usually face no The problem of line sensor network Bandwidth-Constrained.At process end, sensor can only transmit limited bit data estimates end to long-range, So measured value must be quantified before transmission.Measurement component y_iK () is quantized into a quantization message by corresponding quantizer i m_iK () is transferred to long-range estimation end by cordless communication network afterwards.Last estimator is according to the quantization message receiving in optimum The estimation of state is given under criterion.

Due to actual observation y_iK (), when quantizer, quantizer will fall certain according to corresponding quantizing rule What the observation in interval was all approximate is quantized into fixed value m_i(k), that is, quantization message m (k) m (k) receiving=[m₁ (k),m₂(k),...,m_m(k)]^tIt is not original observation y (k).If we know that quantizing rule is it is possible to binding capacity Change the range of convergence that message m (k) determines actual observed value, and quantizing rule is often obtained with advance in practice.If It is clearly irrational, because observation is uncertain that long-range estimator only provides an estimated value after receiving message , it is in a range of convergence rather than One-Point-Value, and particularly, if quantization error is larger, single estimated value may be produced The bigger estimation difference of life.Therefore one natural idea is exactly that estimator is given when observation is a set Estimated value is also a set.

The target of this paper be exactly explore quantizer quantizing rule information, provide set value Kalman filtering algorithm.

The present invention includes quantization strategy and set value Kalman filtering algorithm；

Quantization strategy:

Each measurement component will be quantized into message through corresponding quantizer, certain in total channel bandwidth bit number l Under conditions of, have different estimation performances to the different bit number of each quantizer distribution.Prove using in measurement equation (2) The ratio (signal to noise ratio) of signal and noise can improves the performance of estimator to each quantizer istributes bit number.Herein not to band Wide allocation strategy is studied it is assumed here that the quantizing bit number of quantizer i has distributed and has been l_iPosition, that is, message m_iK () has l_iPosition, according to assuming 2, obtains

$\underset{i=1}{\overset{m}{\&sigma;}}{l}_i=l---\left(3\right)$

And, the quantized interval of single quantizer is as shown in Fig. 2 quantizer i is in intervalUpper totalIndividual amount Change pointThese point of quantification uniform or heterogeneous be distributed in quantized interval, that is, full Foot

${\underset{\&overbar;}{u}}_i=a_i^1\left(k\right)<a_i^2\left(k\right)<\mathrm{...}<a_i^{t_i}\left(k\right)={\stackrel{\&overbar;}{u}}_i---\left(4\right)$

$a_i^{j+1}\left(k\right)-a_i^j\left(k\right)={\mathrm{\&delta;}}_i^j\left(k\right)---\left(5\right)$

Conventional quantification manner be generally based on probability it is assumed that measurement componentThen y_i(k) root It is quantized to according to equation belowOr

$p\{m_i\left(k\right)=a_i^j\left(k\right)\}=\frac{\&lsqb;a_i^{j+1}\left(k\right)-y_i\left(k\right)\&rsqb;}{{\mathrm{\&delta;}}_i^j\left(k\right)}---\left(6\right)$

$p\{m_i\left(k\right)=a_i^{j+1}\left(k\right)\}=\frac{\&lsqb;y_i\left(k\right)-a_i^j\left(k\right)\&rsqb;}{{\mathrm{\&delta;}}_i^j\left(k\right)}---\left(7\right)$

Note 1: these point of quantification shown in formula (4) can arbitrarily be distributed in interval by some rulesInterior. WhenWhen, quantization strategy degenerates to common uniform quantization.It is based on herein more typically Quantization strategy such as formula (4), (5), (6), shown in (7).

Set value Kalman filtering:

Because original measured value is after quantifying, long-range estimates that end is not aware that original observation y (k), but Each amount can be utilized by exploring according to the interval range knowing each measurement component place of quantizing rule formula (5) part Change the quantizing rule information of device, provide set value Kalman filtering algorithm:

Assume that measuring assembly yd (k) is oval set and can be turned to parameter

Yd (k) :=ε (c (k), y (k)) (8)

Wherein c (k) is elliptical center, and y (k) is to represent oval matrix.Represent ellipse with the mark tr (y (k)) of matrix y (k) The size of circle collection yd (k)；

Lemma 1: ε (c₁,x₁), ε (c₂,x₂) represent two oval set, then

$\mathrm{\&epsiv;}(c_1,x_1)\&circleplus;\mathrm{\&epsiv;}(c_2,x_2)\&subsetequal;\mathrm{\&epsiv;}(c_1+c_2,(1+p^{-1})x_1+(1+p\left)x_2\right)---\left(9\right)$

Wherein, p ＞ 0 and p=(tr x₁)^1/2/(tr x₂)^1/2So that matrix (1+p^-1)x₁+(1+p)x₂) mark minimum；

Note 2: due to two oval set through with computing after may be no longer oval collection, so accurately oval collection and The analytical expression of operation can not keep.Calculate the sum of ellipse using lemma 1 with approximate exterior elliptical.Oval in formula (9) Collection ε (c₁+c₂,(1+p^-1)x₁+(1+p)x₂) it is to comprise oval collection ε (c₁,x₁) and ε (c₂,x₂) oval.

Assume to record original measured value y (k)=[y in k moment sensor₁(k),y₂(k),...,y_m(k)]^t, it every Individual component y_iK () forms quantization message m after the quantizing rule of quantizer i_i(k), then be sent to far through radio communication channel Journey estimates end, and quantizing rule can be known a priori by estimation end, so can according to the message receiving and quantizing rule To judge the interval range that actual measured value is located；

If the message that remote port receivesI=1,2 ..., m, j ∈ { 1,2 ..., t_iQuantizer then can be utilized Quantizing rule information judge actual measurement component y_iK () is located at intervalInterior, this interval WithCentered on point, withFor interval radius.Because point of quantification can Known in advance by quantizing rule, therefore each measures component y_iK interval central point c that () is located_i(k) and interval radius r_iK () can Real-time online calculates.

Annotation 3: when the quantization strategy of quantizer i is uniform quantization, that is,So in interval Heart pointInterval radiusNamely interval central point is just to receive Quantization message value, interval radius be fixed constant, double counting need not be carried out.

Message m after estimating that termination receives all quantizers quantizations_iWhen (k), by using the information of quantizing rule, can To calculate the interval range that each actual measurement component is located, it is now assumed that c_i(k) and r_i(k), i=1,2 ..., m all calculate Finish, and for set value Kalman filtering algorithm, an important step is to provide the spheroid description of observation set, and survey is given below The spheroid of value y (k) describes method.

Foregoing present and estimating each measurement of end component y_i(k) actual interval range being located, and actual measurement The collection area of value y (k) is not given, definition

ω :={ y (k) ∈ r^m|||y_i(k)-c_i(k)||≤r_i(k), i=1,2 ..., m },

It is a m dimension cuboid, a length of 2r on each side_i(k), the set area of this region i.e. actual observed value Domain, present target is to find an outside spheroid the tightest to comprise ω.DefinitionFor comprising the minimum spheroid of ω, and Meet

$\stackrel{\&overbar;}{\mathrm{\&omega;}}:=\{y\left(k\right)\&element;r^m|{\left(y\right(k)-c(k\left)\right)}^t{y}^{-1}\left(k\right)\left(y\right(k)-c(k\left)\right)\&le;{\mathrm{\&delta;}}^2\left(k\right)\},$

Wherein, c (k)=[c₁(k),c₂(k),...,c_m(k)], Diag represent take diagonal matrix operate.

As m=2, ω withBetween relation as shown in figure 3, in fact, elliptical center is exactly in rectangle Heart point, that is, each interval central value, oval matrix is closely related with interval radius.

Due to convexity, m dimension at least 2 summits of cuboid are in spheroidBorder on, other summits or be included in Interior orBorder on.So the value of δ (k) may be calculated as

$\mathrm{\&delta;}\left(k\right)=\underset{y_i\left(k\right)\&element;\{{\mathrm{\&delta;}}_i^1\left(k\right),{\mathrm{\&delta;}}_i^2\left(k\right)\},i\&element;\{1,2,\mathrm{...},m\}}{\max }\sqrt{{\left(y\right(k)-c(k\left)\right)}^t{y}^{-1}\left(k\right)\left(y\right(k)-c\left(k\right))}---\left(10\right)$

Wherein,

Therefore, in each sampling instant, the tightest ellipsoid collection area that observation is comprised is ε (c (k), δ²(k)y (k)).

Note 4: by exploring the information in quantizing rule, the m obtaining set of measurements ties up rectangular body region ω, Jin Erli Obtain the outside spheroid of the tightest inclusion region ω with formula (10), finally obtain state using set value Kalman filtering algorithm Estimate collection.It is true that state estimation collection is highly useful, it not only provides the excursion of virtual condition, and to quantization The design of device also provides reference information.Such as, when there being quantization error impact, user is known that the upper of virtual condition change Lower bound, can distribute the bit number of each measurement component simultaneously according to actual needs by reference to the size of state set.

Specifically detailed set value Kalman filtering algorithm is as shown in table 1:

Table 1

Emulation example:

In this section, the examples of simulation of a target following is used to verify the effectiveness of carried algorithm herein.It is based on The Kalman filtering algorithm of pseudo-measurement noise and being used to and carried calculation herein based on the Kalman filtering algorithm of original measurement value Method is compared.Under statistical significance, with two norms of averaged power spectrum errorTo represent the estimation essence of state Degree.T represents emulation step number, and x (k) is actual state,It is the state estimated.

For convenience of description, following filtering algorithm is carried out with simple marking:

Algorithm 1: based on the set value Kalman filtering algorithm quantifying

Algorithm 2: the Kalman filtering algorithm based on pseudo-measurement noise

Algorithm 3: using the Kalman filtering algorithm of original measurement value

Consider following linear second-order system:

$\left[\begin{array}{c}{x}_1(k+1)\\ x_2(k+1)\end{array}\right]=\left[\begin{array}{cc}0.5& 0.3\\ -0.1& 0.8\end{array}\right]\left[\begin{array}{c}{x}_1\left(k\right)\\ x_2\left(k\right)\end{array}\right]+w\left(k\right)---\left(11\right)$

$y\left(k\right)=\left[\begin{array}{cc}1.5& 1\end{array}\right]\left[\begin{array}{c}{x}_1\left(k\right)\\ x_2\left(k\right)\end{array}\right]+v\left(k\right)---\left(12\right)$

The covariance matrix of process noise and observation noise is respectivelyR (k)=0.2.Initial value x₀=[0.01 0.01]^t, initial covariance matrixBandwidth l=16 of radio communication channel is easy for emulation This example uses uniform quantization strategy.Shown in simulation result such as Fig. 4, Fig. 5 and table 2.

Table 2

The statistical result of estimation difference

After being quantized tolerance due to measurement component value, estimator can not obtain original measurement value, but in each sampling Moment can calculate, by using the information that quantization strategy is comprised, the interval that original measurement value is located.In this case, I Can not say estimating which point that average is concentrated has the estimation difference of minimum, but we can be with set value estimator Center value is as point value estimated result.From the point of view of the estimation curve of Fig. 4 state, the estimation interval value of algorithm 1 is all covered at any time Cover virtual condition, three kinds of algorithms substantially can follow the tracks of the actual state 1 of target and state 2, from Fig. 5 state estimation error Interval curve figure and table 2 are seen, algorithm 3 averaged power spectrum error is minimum, and this is original mainly due to the measured value used in algorithm 3 Measured value, and original measurement value is can not be getable under the conditions of Bandwidth-Constrained, is used primarily in and makees algorithm contrast herein.Calculate The estimation difference of method 2 is significantly greater than algorithm 1, and this is primarily due to algorithm 2 and assumes the incoherent white Gaussian noise of quantizing noise simultaneously And using the upper bound of pseudo noise covariance as measurement noise covariance, this excessive demand assume and overly conservative noise at Reason will necessarily affect the precision of algorithm, and algorithm 1 has not only used the message value receiving, and used simultaneously and be included in quantization plan Extraneous information in slightly, also exactly we adopt advantage and the motivation of set value Kalman filtering algorithm for these.

Conclusion:

The present invention discusses first under the conditions of having bandwidth constraint wireless sensor network, based on the set value karr quantifying Graceful state estimation problem.After quantizer bit number for given each measurement component and quantizing rule, using the message quantifying, visit The information of rope quantizing rule provides the hypercube set region of actual measured value.Comprise this overlength side with proximate exterior spheroid Body collection area, and provide the method for solving of the tightest spheroid, and then give based on the set value kalman filtering algorithm quantifying. Finally compared for the kalman filtering algorithm state estimation interval based on covariance matrix with computer examples of simulation, special based on covering The kalman filtering statistic behavior estimation interval of Carlow and set value kalman filtering algorithm state estimation interval size.Demonstrate institute Carry the effectiveness of algorithm.

Ultimate principle and principal character and the advantages of the present invention of the present invention have been shown and described above.The technology of the industry , it should be appreciated that the present invention is not restricted to the described embodiments, the simply explanation described in above-described embodiment and description is originally for personnel The principle of invention, without departing from the spirit and scope of the present invention, the present invention also has various changes and modifications, these changes Change and improvement both falls within scope of the claimed invention.Claimed scope by appending claims and its Equivalent thereof.

Claims (1)

1. a kind of based on quantify set value Kalman filtering algorithm it is characterised in that: include quantization strategy and set value Kalman filter Ripple algorithm；

Quantization strategy:

The quantizing bit number of hypothesis quantizer i has distributed and has been l_iPosition, that is, message m_iK () has l_iPosition, according to hypothesis 2, obtain

And, quantizer i is in intervalUpper totalIndividual point of quantificationThese quantify Point uniform or heterogeneous be distributed in quantized interval, that is, meet

Conventional quantification manner be generally based on probability it is assumed that measurement componentThen y_iK () is according to such as Lower formula is quantized toOr

Set value Kalman filtering:

Because original measured value is after quantifying, long-range estimates that end is not aware that original observation y (k), but permissible According to the interval range knowing each measurement component place of quantizing rule formula (5) part, utilize each quantizer by exploring Quantizing rule information, provide set value Kalman filtering algorithm:

Assume that measuring assembly yd (k) is oval set and can be turned to parameter:

Yd (k) :=ε (c (k), y (k)) (8)

Wherein c (k) is elliptical center, and y (k) is to represent oval matrix；Represent oval collection with the mark tr (y (k)) of matrix y (k) The size of yd (k)；

Lemma 1: ε (c₁,x₁), ε (c₂,x₂) represent two oval set, then

Wherein, p ＞ 0 and p=(trx₁)^1/2/(trx₂)^1/2So that matrix (1+p^-1)x₁+(1+p)x₂) mark minimum；

Assume to record original measured value y (k)=[y in k moment sensor₁(k),y₂(k),...,y_m(k)]^t, it each point Amount y_iK () forms quantization message m after the quantizing rule of quantizer i_i(k), then be sent to through radio communication channel and remotely estimate Meter end, and quantizing rule can be known a priori by estimation end, so can be sentenced according to the message receiving and quantizing rule The interval range that disconnected actual measured value is located；

If the message that remote port receivesI=1,2 ..., m, j ∈ { 1,2 ..., t_iQuantizer then can be utilized Quantizing rule information judges actual measurement component y_iK () is located at intervalInterior, this interval withCentered on point, withFor interval radius；Because point of quantification can be by Quantizing rule is known in advance, and therefore each measures component y_iK interval central point c that () is located_i(k) and interval radius r_iK () can be real When in line computation；

Message m after estimating that termination receives all quantizers quantizations_iWhen (k), by using the information of quantizing rule, Ke Yiji Calculate the interval range that each actual measurement component is located, it is now assumed that c_i(k) and r_i(k), i=1,2 ..., m have all calculated Finish, and for set value Kalman filtering algorithm, an important step is to provide the spheroid description of observation set, and measurement is given below The spheroid of value y (k) describes method；

Foregoing present and estimating each measurement of end component y_i(k) actual interval range being located, and actual measured value y (k) Collection area be not given, definition

ω :={ y (k) ∈ r^m|||y_i(k)-c_i(k)||≤r_i(k), i=1,2 ..., m },

It is a m dimension cuboid, a length of 2r on each side_i(k), the collection area of this region i.e. actual observed value, now Target be to find an outside spheroid the tightest to comprise ω；DefinitionFor comprising the minimum spheroid of ω, and meet

Wherein, c (k)=[c₁(k),c₂(k),...,c_m(k)],diag Represent and take diagonal matrix to operate；

As m=2, ω withBetween relation as shown in figure 3, in fact, elliptical center is exactly the central point of rectangle, Namely each interval central value, oval matrix is closely related with interval radius；

Due to convexity, m dimension at least 2 summits of cuboid are in spheroidBorder on, other summits or be included inInterior or Person existsBorder on；So the value of δ (k) may be calculated as

Wherein,

Therefore, in each sampling instant, the tightest ellipsoid collection area that observation is comprised is ε (c (k), δ²(k)y(k)).