In the paper, a new approach to the estimation of the localization of flying objects on the basis of multiple radiolocation sources has been proposed. The basic purpose of the radiolocation is the detection and tracking of objects. The main task of a tracking system is the estimation and prediction of localization and motion of tracked objects. In the classical tracking systems, the estimation of localization and motion of tracked objects is based on the series of measurements performed by a single radar. Measurements obtained from other sources are not taken directly into account. It is possible to assess the estimate of a tracking object on the basis of the different sources of information, applying, for example, least squares method (LSM). However, such solution is seldom applied in practice, because necessary formulas are rather complicated. In this paper, a new approach is proposed. The key idea of the proposed approach is so called matrix of precision. This approach makes possible tracking not only on the basis of radar signals, but also on the basis of bearings. It makes also possible the tracking of objects on the basis of multiple information sources. Simplicity is the main attribute of proposed estimators. Their iterative form corresponds well with the prediction-correction tracking model, commonly applied in radiolocation. In the paper numerical examples presenting advantages of the proposed approach are shown.
The paper consists of seven parts. Introduction is the first one.
In the second part the idea of the matrix of precision is presented and it is demonstrated how it can be used to uniformly describe the dispersion of measurement. Traditionally, the measurement dispersion is described by a matrix of covariance. Formally, the matrix of precision is an inverse of the matrix of covariance. However, these two ways of description are not interchangeable. If an error distribution is described by the singular matrix of precision then the corresponding matrix of covariance does not exist, more precisely, it contains infinite values. It means in practice that some components of a measured vector are not measurable, in other words, an error of measurement can be arbitrarily large. The application of the matrix of precision makes possible an uniform description of measurements taken from various sources of information, even if measurements come from different devices and measure different components. Zero precision corresponds to components which are not measured.
In the third part, the problem of estimation of stationary parameters is formulated. Using the matrix of precision, the simple solution of the problem is presented. It appears that the best estimate is a weighted mean of measurements, where the weights are the matrices of the precision of measurements. It has been proved that the proposed solution is equivalent to the least squares method (LSM). Additionally it is simple and scalable.
In the fourth part of this paper the problem of the estimation of states of dynamic systems, such as a flying aircraft, is formulated. Traditionally, for such an estimation the Kalman filter is applied. In this case the uncertainty of measurement is described by the error matrix of covariance. If the matrix of precision is singular, it is impossible to determine the corresponding matrix of covariance and utilize the classical equation of Kalman filter. In the presented approach this situation is typical. It appears that there is such a transformation of Kalman filter equations, that the estimation based on the measurements with error described by the singular matrix of precision, can be performed. Such a transformation is presented and its correctness is proved.
In the fifth part, numerical examples are presented. They show the usability of the concept of matrix of precision.
In the sixth part, the summary and conclusions are shown, as well as the practical application of presented idea is discussed. The practical problems which are not considered in the paper are also pointed out.
2. The concept of precision matrix
Traditionally in order to describe the degree of the dispersion distribution the covariant matrix is used. There exist another statistics which can be used to characterize dispersion of the distribution. However, the covariance matrix is the most popular one and in principle the precision matrix is not used. Formally the precision matrix is the inverse of covariance matrix.
The consideration may take much simpler form if the precision matrix is used. Note that precision matrix is frequently used indirectly, as inverse of covariance matrix. For example, in the well-known equation for the density of multi-dimensional Gauss distribution:
The equation (1) can not be used if the covariance matrix is singular. As long as the covariance (precision) matrix is not singular, the discussion which statistics is better to describe degree of the dispersion distribution is as meagre as discussion about superiority of Christmas above Easter. Our interest is in analysis of extreme cases (e.g. singular matrix).
At first, the singular covariance matrix will be considered. Its singularity means that some of the member variables (or their linear combinations) are measured with error equal to zero. Further measurement of this member variable makes no sense. The result can be either the same, or we shall obtain contradiction if the result would be different. The proper action in such a case is modifying the problem in such a manner that there is less degrees of freedom and the components of the vector are linearly independent. The missing components computation is based on the linear dependence of the variables. In practice, the presence of the singular covariance matrix means that the linear constraints are imposed on the components of the measured vectors and that the problem should be modified. The second possibility (infinitely accurate measurements) is impossible in the real world.
In distinction from the case of the singular covariance matrix, when the measurements are described by the singular precision matrix, measurements can be continued. The singularity of the precision matrix means that either the measurement does not provide any information of one of components (infinitive variation) or there exist linear combinations of member variables which are not measured. This can results from the wrong formulation of the problem. In such a case, the system of coordinates should be changed in such a manner that the variables which are not measured should be separated from these which are measured. The separation can be obtained by choosing the coordination system based on the eigenvectors of precision matrix. The variables corresponding to non-zero eigenvalues will then be observable.
There is also the other option. The singular precision matrix can be used to describe the precision of measurement in the system, in which the number of freedom degrees is bigger then the real number of components, which are measured. Then all measurements may be treated in a coherent way and the measurements from various devices may be taken into account. Thus all measurements from the devices which do not measure all state parameters can be included (i.e. direction finder). In this paper we are focusing on second option.
Each measuring device uses its own dedicated coordinate system. To be able to use the results of measurements performed by different devices, it is necessary to present all measured results in the common coordinate system. By changing coordinate system the measured values are changed as well. The change involves not only the digital values of measured results, but also the precision matrix describing the accuracy of particular measurement. We consider the simplest case namely the linear transformation of variables. Let
In this case the covariance of measurement changes according to formula:
The precision matrix as inverse of covariance matrix changes according to formula:
Consider the case when the transformation
Additional rows of this matrix should be chosen in such way that the obtained precision matrix has block-diagonal form:
it means that additional virtual lines of transformation
In such a way, the additional virtual lines do not have any influence on the computation result of precision matrix in the new coordinate system.
Consider the specific case. Let the measuring device measures target OX and OY positions independently, but with different precision. Measurement in the direction of the axis OX is done with accuracy
We are interested in accuracy of the new variable which is a linear combination of these variables. Let
The precision matrix in the new coordinate system has the following form:
We ignore additional variables which have been added to make
The same result can be obtained using the first method. Then the covariance matrix is of the form:
The value of the covariance of variable
While the second method is more complicated it is nevertheless more general, and permits to find the precision matrix in the new coordinate system even if the precision matrix is singular.
In a case when it is necessary to make nonlinear transformation of variables
In order to find the value of partial derivative, it is necessary to know point
3. Estimation of stationary parameters
In engineering it is often necessary to estimate certain unknown value basing on several measurements. The simplest case is, when all measurements are independent and have the same accuracy. Then, the intuitive approach i.e. using the representation of results as arithmetic average of measurements is correct (it leads to minimal variation unbiased estimator). The problem is more complicated if the measured value is a vector and individual measurements have been performed with different accuracy. Usually, to characterize measurements accuracy the covariance matrix of measurement error distribution is used. Application of precision matrix leads to the formulas which are more general and simple. Further the precision matrix describes the precision of measurements when the number of degrees of freedom is smaller than the dimension of the space in which the measurement has been done. For example, direction finder measures the direction in the three dimensional space. The direction finder can be used as a radar with an unlimited error of measured distance. In this case, the covariance matrix of the measurement does not exist (includes infinite values). But there exists the precision matrix (singular). Information included in the precision matrix permits to build up optimal (minimal variation unbiased) linear estimator.
3.1. Formulation of the problem
Consider the following situation. The series of n measurements
There is no preliminary information concerning the value of the measured quantity
The measured quantity is constant.
All measurements are independent.
All measurements are unbiased (expected value of the measured error is zero).
All measurements can be presented in common coordinate system and the precision matrix Pi which characterizes the measurement error distribution in these coordinates, can be computed.
The problem is to find the optimal linear estimator of unknown quantity
3.2. The solution
The first step consists in presenting all measurement results in the common coordinate system. The precision matrices must be recalculated to characterize the measurement error distribution in the common coordinate system. The method of the recalculation the precision (covariance) matrixes in the new coordinated system have been presented before. If all measurements have been done in the same coordinate system, the first step is omitted.
The unbiased minimal variation estimator of unknown quantity
The presented formula represents the particular case of the general estimator of state
In our case, the formula is radically simplified. All measurement results are presented in the same coordination system, therefore the observation Matrix has the following form:
Since all measurements are independent, the covariance matrix has the block-diagonal form:
After simplification and changing of the multiplication order (symmetrical matrix) we obtain:
As we can see, it is another form of formula (16). Its interpretation is easy to read and easy to remember. The optimal linear estimator represents the weighed average of all measurements, when weighs are the precision matrices of individual measurements.
4. Modified Kalman filter
In many cases it is necessary to estimate the changing state vector of the system. The value of the vector changes between successive measurements. Such situation exists in radiolocation when the position of moving target is tracked. Restricting ourselves to linear systems, the solution of this problem is known as the Kalman filter. In the Kalman filter theory, the model of system is presented as the pair of equations:
The first equation describes dynamics of system in the discrete time domain. Vector
The covariance of internal noise
It is assumed that we have estimate of the value
The equation of the state prediction has the following form:
The covariance matrix of the state prediction has the following form:
The equation of the measurement prediction has the form:
The Kalman gain is defined as:
The new estimate of the state has the following form:
The covariance matrix of new estimate of the state has the following form:
Equations (27-40) constitutes the description of the classic Kalman filter. The Kalman filter can be considered as the system in which the input consists of string of measurements
It is easy to prove that:
We invert both sides:
It can be written as:
Finally, the equation (38) has form:
The only problem which can appear here is the possible singularity of the matrix
5.1. Estimation of the static system.
In order to demonstrate the proposed technique, the following example is used. The position of tracking target is determined using three measurements. The first measurement comes from the radar. The second one comes from the onboard GPS device. The third one comes from the direction finder. The radar is a device which measures three coordinates: slant range (
Basing on these results we want to obtain the estimator of
Transforming to new coordinate system, we obtain following results:
In order to determine the precision matrix we invert the corresponding covariance matrices:
The accuracy of the GPS device is 100 m. Therefore we consider 100 as the standard deviation of error distribution.
The standard deviation of the direction finder is 0.001 rad. The variance and precision are accordingly:
After inserting zeros to the rows and columns corresponding to missing variables we obtain the following precision matrices:
When applying the linear transformation of variables:
The Jacobian matrix of such transformation has the following form:
Transforming variables from the local coordinate system of the measuring device to global system, we obtain as follows:
For the radar:
In case of the GPS device we do not have any change of variables. We are only adding the zero values corresponding to additional variable to the precision matrix:
For the direction finder:
Finally, we obtain the position estimator:
Note that utilization of the measurements from GPS device and direction finder, which do not measure the height, actually results in some improvement of altitude estimation. This is due to the fact that the variables which provide the radar measurements in XYZ coordinates are not independent, but they are strongly correlated. The linear combination of them is measured with high accuracy. Thus, by improving the estimation of XY variables we also improve the estimation of the variable in Z direction.
5.2. Estimation in the dynamic system
The next example presents the application of the proposed technique to the estimation of the parameters of a dynamic system. It is the most typical situation when the racking target is moving. Our purpose is the actualization of motion parameters of tracking target based on three measurements. We assume that the tracking is two dimensional. The following parameters are tracked:
Based on previous measurements, we obtain estimation of the initial parameters of tracked target motion:
The covariance matrix of this estimation has the form:
The tracked target has been detected by the radar R1 which is located at the beginning of our coordinate system
The parameters of the detection are:
We assume that the radar measurement is unbiased. The covariance matrix of the measurement error has the following form:
The next position measurements of the tracking target took place at the time:
We obtain the bearing of tracked target. The direction finder was located at the point:
The bearings value is:
The covariance measurement error is:
At the near time:
We obtain bearing from another direction finder which was located an point:
The bearing value is:
The covariance measurement error is:
Traditionally, to solve such a problem, the theory of Kalman filter is used. In this theory two assumptions are usually accepted. First that all measurements are carried out by the same device. Second, that the measurements are done in regular time intervals. None of the above assumptions is satisfied in our example. To address the issue of different measuring device, we assume that the measurements are always presented in the common global coordinate system, and their dispersion is described by the precision matrix which can be singular. Using the modified formulas of Kalman filters (46), the estimation can be done, even if the precision matrix is singular. In order to address the issue of the irregular time intervals of the measurements, the Kalman filter has been modified to consider the changes of time interval between individual measurements. It means that the matrix
We introduce the following state vector of Kalman filter:
Therefore, the dynamic matrix
We also assume that the internal noise of the process has additive character and it is the source of change of motion velocity. In such a case, the velocity variation is increasing linearly with time:
The internal noise related to velocity affects the position through the dynamic process described by matrix
Additionally we assume that the observation vector has the following form:
Therefore, the observation matrix
The direction finder, as well as the radar, make measurements at spherical coordinate system. To present measurements results at the global reference system we use formulas (63-65) described in the first example. The necessary elements have been already discussed. The algorithm of the estimation of the motion parameters can be described on the basis of measurements coming from different sources. Utilizing previous measurements we know the estimation
Calculate the prediction of the state vector
using formula (35)
Calculate the covariance matrix
of the prediction of the state using formula (36)
Calculate the prediction of detection
at the global coordinate system
Calculate the prediction of detection
at the local radar (direction finder) coordinate system
Expand the measurements vector
. The values which has not been measured are replaced by predicted values from the previous measurements.
Create the precision matrix
of the measurements at the local radar (direction finder) coordinate system. The values which have not been measured have zero precision.
Transform the expanded vector of measurements
and corresponding precision matrix to global coordinate system and .
Note that tracking is done in two dimensional space. The altitude Z is not tracked and is fixed as Z=2000. If the altitude is required in computation like for example in formula (97) we take Z=2000m.
For the first measurements we have accordingly:
The covariance matrix of the prediction:
The prediction of the detection at the global coordinate system XYZ:
In the last formula we use information on tracking target altitude 2000m. Now, we consider the system of the radar R1. The prediction of the detection at the radar local coordinate system:
The radar measures all coordinates. It is not necessary to increase measuring vector and to use the prediction.
The dispersion of the radar measurements is described by the precision matrix of the form:
After transformation to the global coordinate system XYZ we obtain:
The precision matrix is transformed according to formula:
Because we are tracking only the components XY, we ignore the last row in the measurement vector
Now, we can determine Kalman gain
It is the end of the first step of the determination of the estimate of tracking target motion parameters.
The next step is more interesting because it demonstrates how one can use the measurements which were done by the direction finder. Accordingly:
For the first measurement we have:
The covariance matrix of prediction is:
The prediction of the detection at the global coordinate system XYZ
We are taking into consideration the direction finder N2. The prediction of the detection at the direction finder local coordinate system:
The direction finder measures only azimuth
The precision matrix variables calculated using the prediction have zero precision:
After the transformation to global XYZ coordinate system we obtain:
As before, we determine the precision matrix
It is obvious that in the case of the second bearing we act similarly:
The covariance matrix of prediction:
The prediction of the detection in global XYZ coordinate system:
Now, we consider the local coordinate system of the direction finder N3. The prediction of the detection at the direction finder local coordinate system:
The direction finder measures only azimuth
The precision matrix variables defined using the prediction have zero precision:
After transformation to global XYZ coordinate system we obtain:
As before, we determine the precision matrix
The presented example demonstrates the power of the described technique. Even a single bearing can improve the estimation of motion parameters of the tracking target. If several bearings are utilized, they need not to be simultaneous.
In the proposed technique the measurements carried out by different devices are treated in a simple and uniform manner. The key idea comprises in expansion the vector of the measured values and insertion of missing variables based on the prediction. The precision matrix is used to describe their dispersion. In order to obtain the estimate of the tracked target state we have to present all measurements and corresponding precision matrixes in a common coordinate system. The formulas (16-17, 18-21) should be used for static systems or the modified Kalman filters equations (27-40, 46) should be used for dynamic systems.
For linear objects, the Kalman filters are the optimal estimators. In the presented example dynamics of the tracked target has been described by linear differential equation The presented methodology can be applied also in the case when the dynamics of tracked target is nonlinear. In this case, instead of formula (27) we use linearization. Let the dynamics of the tracked process be described by a nonlinear function:
In order to determine the covariance of the state prognosis we use linearization
In order to increase accuracy, we can decrease time interval. Calculations (131) could be done several times. Let:
In the past, this approach was rejected due to computer high power requirements. Presently it creates no problem at all.
An interesting case of proposed methodology application is the use of measurements from Doppler radar. The Doppler radar measures frequency shifting of received signal permitting to detect approaching speed of tracking targets. More precisely, it permits to determine speed radial component
The missing values are calculated utilizing the prediction. The corresponding precision matrix has the following form:
After transformation to the global coordinate system XYZ we can use measurements from the Doppler radar to improve estimation of the tracking target motion parameters.
In all so-far considered examples, we assumed that time error does not exist. In practice, the time is always measured with certain error. If for example the measuring time is presented in seconds, then during one second the tracked target can move several hundreds meters. Fortunately, in our case the error of time measuring can be easily taken into consideration. Time error
Assuming that the time measuring error is independent from the position measurement error, the variances of both errors are added:
The error of the time measurement can be easily taken into consideration by the proper increase of the error covariance value
When tracking the state of dynamic system by means of the modified Kalman filter the assumption that the measurements are done in regular time intervals has been abandoned. Nevertheless, it has been assumed that the received measurements are sequential:
The presented method is in 100% correspondence with the well-known least squares method (LSM). The careful analysis of the presented examples with taking into consideration the different weights of particular errors, existing correlations between them as well as the flow of time will provide the same results for the estimated values as the LSM method. Therefore the proposed methodology do note give better estimators then the classical theory. However the presented methodology avoids the very complicated and arduous computations and presents all measurements in a uniform way. Such approach is very useful for the automatic data processing by an automatic tracking system.
Since the presented technique is in 100% correspondence with the LSM, it also it discloses the weak points of LSM. In particular, it is not robust methodology. It is not robust for the values drastically odd (different) and for large errors. Even the single erroneous measurement not fitting the assumed model of errors distribution results in the drastic increase of the final error of the estimator. Because of this, it is recommended that the preliminary preselection of tracking measurements should be done. Drastically different values should be rejected.
The Kalman filter represents an optimal estimator for linear system only if the real model of the process corresponds to the one accepted during Kalman filter (27-34) design. It is particularly important when considering the error values. Consider for example the ladar (laser radar) which measures the position of an object from the distance of a few tens of kilometers with accuracy up to 5 m. Naturally, the error