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In the case of the normal Gaussian distribution, max p(qˆn | Yn ) corresponds to the mathematical expectation of the vector of estimated target track parameters. Consequently, the problem of target track parameter estimation by a posteriori probability maximum is reduced, in our case, to definition of the parameters θˆ n and Ψn in (5.115). Using (5.111) through (5.114) for pdfs included in (5.115), we obtain after the logarithmic transformations the following formula:

Taking into consideration (5.106) and (5.110) for qˆexn and Yexn, the main relationships of the optimal sequential filtering algorithm can be presented in the following form:

The system of equation (5.118) represents the optimal recurrent linear filtering algorithm and is called the Kalman filtering equations [9–16]. These equations can be transformed into a more convenient form for realization:

A general filter flowchart realizing (5.119) is shown in Figure 5.6. A discrete optimal recurrent filter possesses the following properties:

•Filtering equations have recurrence relations and can be realized well by a computer system.

•Filtering equations represent simultaneously a description of procedure to realize this filter; in doing so, a part of the filter is similar to the model of target track (compare Figures 5.2 and 5.6).

FIGURE 5.6  General Kalman filter flowchart.

•The correlation matrix of errors of target track parameter estimation Ψn is computed independently of measuring Yn; consequently, if statistical characteristics of measuring errors are given, then the correlation matrix Ψn can be computed in advance and stored in a memory device; this essentially reduces the time of realization of target track parameter filtering.

5.5.2  Filtering of Linear Target Track Parameters

Formulas of the sequential filtering algorithm of linear target track parameters are obtained directly from (5.119). The target track coordinate and velocity of its changes at the instant of last nth measuring are considered as the filtered parameters. We assume that all measurements are equally discrete with the period Teq.

1. Assume the vector of filtered target track parameters

and the error correlation matrix of target track parameter estimations

are obtained by the (n − 1) previous measurements of the coordinate x.

6. Taking into consideration the relations obtained, we are able to compute the estimations of linear target track parameters in the following form:

7.Under equally discrete and uniformly precise measurements of the target track coordinates, we obtain

Substituting (5.132) into (5.128) and (5.129), we obtain finally

Dependences of the coefficients An and Bn versus the number of observations n are presented in Figure 5.7. As we can see from Figure 5.7, with an increase in n the filter gains on the coordinate and velocity are approximated asymptotically to zero. Consequently, with an increase in n the results of last measurements at filtering the coordinate and velocity are taken into consideration with the less weight, and the filtering algorithm ceases to respond to changes in the input signal. Moreover, essential problems in the realization of the filter arise in computer systems with limited capacity of number representation. At high n, the computational errors are accumulated and are commensurable with a value of the lower order of the computer system, which leads to losses in conditionality and positive determinacy of the correlation matrices of extrapolation errors and filtering of the target track parameters. The “filter divergence” phenomenon appears

FIGURE 5.7  Coefficients An and Bn versus the number of observations n.

when the filtering errors increase sharply and the filter stops operating. Thus, if we do not take specific measures in correction, the optimal linear recurrent filter cannot be employed in complex automatic radar systems. Overcoming this problem is a great advance in automation of CRSs.

5.5.3  Stabilization Methods for Linear Recurrent Filters

In a general form, the problem of recurrent filter stabilization is the problem of ill-condi- tioned problem solution, namely, the problem in which small deviations in initial data cause arbitrarily large but finite deviations in solution. The method of stable (approximated) solution was designed for ill-conditioned problems. This method is called the regularization or smoothing method [17]. In accordance with this method, there is a need to add the matrix αI to the matrix of measuring errors Rn given by (5.127), where I is the identity matrix, under a synthesis of the regularizing algorithm of optimal filtering of the unperturbed dynamic system parameters:

where

δ is the accuracy of matrix Rn assignment

ε(δ) and α0(δ) are the arbitrary decreasing functions tending to approach zero as δ → 0

Thus, in the given case, a general approach to obtain the stable solutions by the regularization or smoothing method is the artificial rough rounding of measuring results. However, the employment of this method in a pure form is impossible since a way to choose the regularization parameter α is generally unknown. In practice, a cancellation of divergence of the recurrent filter can be ensured by effective limitation of memory device capacity including in the recurrent filter. Consider various ways to limit a memory device capacity in the recurrent filter.

178 Signal Processing in Radar Systems

5.5.3.1  Introduction of Additional Term into Correlation Matrix of Extrapolation Errors

In this case, we obtain

where Ψ0 is the arbitrary positive definite matrix. Under separate filtering of the polynomial target track parameters using the results of equally discrete and uniformly precise coordinating measurements, we obtain

where

σ2xm is the variance of coordinating measuring errors ci is the constant coefficients, i = 0, …, s

In order for the recurrent filter to have a limited memory capacity, there is a need that the components of the vector Gn should be converged to the constant values 0 < γ 0 < 1,…, 0 < γ s < 1 and be in the safe operating area of the recurrent filter.

Using filtering equations, we are able to establish a relation for each specific case between the gains γ1n ,…, γ sn and the values c0 , c1,…, cs. Passing to limit as n → ∞, we are able to obtain a func-

where nef is the fixed effective finite memory capacity of the recurrent filter. At the same time, the variances of random errors of target track parameter estimations in steady filtering coincide with analogous parameters for nonrecursive filters, namely,

Thus, the recurrent filter with additional term introduced into the error correlation matrix of extrapolation approximates the filter with finite memory capacity at the corresponding choice of the coefficients c0, c1,…, cs.

5.5.3.2  Introduction of Artificial Aging of Measuring Errors

This operation is equivalent to a replacement of the correlation matrix Rn−i of measuring errors at the instant tn−i by the matrix

where s = exp(cT0 ) > 1. At the same time, the correlation matrix of extrapolation errors is determined in the following form:

In this case, under the equally discrete and uniformly precise coordinating measurements, the smoothing filter coefficients are converged to positive constants in the filter safe operating area. However, it is impossible to find the parameter s for this filter so that the variance and dynamic errors of these filters and the filter with finite memory capacity are matched.

5.5.3.3  Gain Lower Bound

In the simplest case of the equally discrete and uniformly precise coordinating measurements, the gain bound is defined directly by the formulas for Gn at the given effective memory capacity of the filter. Computation and simulation show that the last procedure is the best way by the criterion of realization cost and rate to define the variances of errors under the equally discrete and uniformly precise coordinating measurements from the considered procedures to limit the recurrent filter memory capacity. The first procedure, that is, the introduction of an additional term into the correlation matrix of extrapolation errors, is a little worse. The second procedure, that is, an introduction of the multiplicative term into the correlation matrix of extrapolation errors, is worse in comparison with the first and third ones by realization costs and rate of convergence of the error variance to the constant magnitude.

5.6  ADAPTIVE FILTERING ALGORITHMS OF MANEUVERING TARGET TRACK PARAMETERS

5.6.1  Principles of Designing the Filtering Algorithms

of Maneuvering Target Track Parameters

Until now, in considering filtering methods and algorithms of target track parameters we assume that a model equation of target track corresponds to the true target moving. In practice, this correspondence is absent, as a rule, owing to target maneuvering. One of the requirements of successful solution of problems concerning the real target track parameter filtering is to take into consideration a possible target maneuver.

The state equation of maneuvering target takes the following form:

where

Fnqn−1 is the equation of undisturbed target track, that is, the polynomial of the first order

gmn is the l-dimensional vector of the disturbed target track parameters caused by the willful target maneuver

ηn is the p-dimensional vector of disturbances caused by stimulus of environment and uncertainty in control (control noise)

Γn and Kn are the known matrices

According to precision of CRS characteristics and estimated target maneuver, three approaches are possible to design the filtering algorithm of real target track parameters.

5.6.1.1  First Approach

We assume that the target has limited possibilities for maneuver; for example, there are only random unpremeditated disturbances in the target track. In this case, the second term in (5.145) is equal to zero and sampled values of the vector ηn are the sequence subjected to the normal Gaussian distribution with zero mean and the correlation matrix:

Nonzero elements of this matrix represent a set of a priori data about an intensity of target maneuver by each coordinate (r, β, ε). In this case, a record of target track disturbance in the filtering algorithm is reduced to filter bandwidth increase. For this purpose, a computation of the error correlation matrix of target track parameter estimations into an extrapolated point is carried out by the formula

where the s × l matrix Kn can be presented in the following form:

= tn − tn −1 is the interval of target track parameter extrapolation. Other formulas of the recurrent filtering algorithm are the same as in the case of a nonmaneuvering target.

There is a need to note that the methods discussed in the previous section to limit the memory capacity of the recurrent filter by their sense and consequences are equivalent to the considered method to record the target maneuver since a limitation in memory capacity of the recurrent filter, increasing a stability, can simultaneously decrease the sensitivity of the filter to short target maneuvering.

5.6.1.2  Second Approach

We assume that the target performs only a single premeditated maneuver of high intensity within the observation time interval. In this case, the target track can be divided into three parts: before the start of, in the course of, and after finishing the target maneuver. In accordance with this target track division, the intensity vector of a premeditated target maneuver can be presented in the following form:

where tstart and tfinish are the instants to start and finish the target maneuver. In this case, the intensity vector of target maneuver as well as the instants to start and finish the target maneuvering are sub-

jected to statistical estimation by totality of input signals (measuring coordinates). Consequently, in the given case, the filtering problem is reduced to designing a switching filtering algorithm or switching filter with switching control based on the analysis of input signals. This algorithm concerns the class of simplest adaptive algorithms with a self-training system.

5.6.1.3  Third Approach

It is assumed that the targets subjected to tracking have good maneuvering abilities and are able to perform a set of maneuvers within the limits of observation time. These maneuvers are related to air miss relative to other targets or flight in the space given earlier. In this case, to design the formula flowchart of filtering algorithms of target track parameters, there is a need to have data about the mathematical expectation E(gmn ) and the variance σ2g of target maneuver intensity for each target and within the limits of each interval of information updating. These data (estimations) can be obtained only based on an input information analysis, and the filtering is realized by the adaptive recurrent filter.

Henceforth, we consider the principles of designing the adaptive recurrent filter based on the Bayes approach to define the target maneuver probability [19].

5.6.2  Implementation of Mixed Coordinate Systems under Adaptive Filtering

The problems of target maneuver detection or definition of the probability to perform a maneuver by target are solved in one form or another by adaptive filtering algorithms. Detection of target maneuver is possible by deviation of target track from a straight-line trajectory by each filtered target track coordinate. However, in a spherical coordinate system, when the coordinates are measured by a CRS, the trajectory of any target, even in the event that the target is moving uniformly and in a straight line, is defined by nonlinear functions. For this reason, a detection and definition of target maneuver characteristics under filtering of target track parameters using the spherical coordinate system are impossible.

To solve the problem of target maneuver detection and based on other considerations, it is worthwhile to carry out filtering of the target track parameters using the Cartesian coordinate system, the origin of which is matched with the CRS location. This coordinate system is called the local Cartesian coordinate system. The formulas of coordinate transformation from the spherical coordinate system to the local Cartesian one are given by (see Figure 5.8)

x = r cos ε cosβ;

y = r cos ε sin β; (5.149)

z = r sin ε.

Z

y

Y

FIGURE 5.8  Transformation from the spherical coordinate system into the local Cartesian coordinate system.

Transformation to the local Cartesian coordinate system leads to an appearance of nonuniform precision and correlation between the coordinates at the filter input, which in turn leads to complexity of filter structure and additional computer cost in realization. Also, there is a need to take into consideration that other operations of signal reprocessing by a CRS, for example, target pip gating, target pip identification and so on, are realized in the simplest form in the spherical coordinate system. For this reason, the filtered target track parameters must be transformed from the local Cartesian coordinate system to the spherical one during each step of target information update.

Thus, to solve the problem of adaptive filtering of the track parameters of maneuvering targets, it is worthwhile to employ the recurrent filters, where the filtered target track parameters are represented in the Cartesian coordinate system and comparison between the measured and extrapolated coordinates is carried out in the spherical coordinate system. In this case, a detection of target maneuver or definition of the probability performance of target maneuver can be organized based on the analysis of deviation of the target track parameter estimations from magnitudes corresponding to the hypotheses of straight-line and uniform target moving.

Consider the main relationships of the recurrent filter, in which the filtered target track parameters are represented in the local Cartesian coordinate system and a comparison between the extrapolated and measured target track parameters coordinates is carried out in the spherical coordinate system. Naturally, the linear filter is considered as the basic filter, and equations defining the operation abilities of the considered linear filter are represented by the equation system given by (5.119). For simplicity, we can be restricted by the case of a two-dimensional radar measured CRS that defines the target range rtg and azimuth βtg. In this case, the transposed vector of target track moving parameters at the previous (n − 1) step takes the following form:

and the error correlation matrix of the target track parameter estimations contains 4 × 4 nonzero elements: