1972 kalman filter applications in airborne radar tracking

1. Introduction

Kalman Filter ApplicationsIn AirborneRadar TrackingJOHN B. PEARSON, IIIHughes Aircraft CompanyCanoga Park, Calif. 91304

EDWIN B. STEAR, Member, IEEEUniversity of California at Santa BarbaraSanta Barbara, Calif.

Abstract

This paper studies the application of Kalman filtering to

single-target track systems in airborne radar. An angle channel

Kalman filter is configured which incorporates measures of range,

range rate, and on-board dynamics. Theoretical performance results

are given and a discussion of methods for reducing the complexityof the Kalman gain computation is presented. A suboptimal antenna

controller which operates on the outputs of the angle Kalman filter

is also described. In addition, methodological improvements are

shown to exist in the design of range and range-rate trackers usingthe Kalman filter configuration.

This paper investigates the application of modernestimation theory to the design of single-target tracksystems in airborne radar. The functions of the trackingsystem in airborne radar are to provide estimates ofvariables associated with the relative range vector betweenthe tracking aircraft and the target and to optimize thesystem's ability to maintain lock-on. The estimates are usedfor execution of the intercept mission, for instance, in anairborne interceptor for interceptor guidance and missilelaunch computations, and in an air-to-air missile to formthe missile guidance commands.

The desired estimates are obtained from two basicsystems, called the angle and range tracking systems. Theangle tracking system provides estimates of angularvariables, such as direction and spatial angular rate of thetarget line-of-sight (LOS), while the range tracking systemestimates such variables as scalar range and range rate.Tracking of range rate independent of range measurementsis also available with Doppler radar.

The methodology traditionally used to design thesesystems has been the classical servomechanism theory. Foreach track loop a measure of the control error provided bythe radar receiver is operated on by some form of gain andcompensation network to produce the loop output andfeedback signals. The input characteristics of the system fordesign purposes are the statistics of the input noises and theexpected characteristics of the input geometrical variables.For a fixed-parameter system, the compensator must bechosen so that the error is tolerable for worst-caseconditions [1] . If the resulting accuracy over the total setof conditions is not adequate, the need then arises to adaptthe loop structure as a function of the conditions. Thissituation presents the question of how this adaptation isbest done and, in particular, how to make best use ofmeasurable on-board variables from other subsystems. TheKalman filter provides one approach to answering theseoptimization questions.

The first sections of this paper give a description of oneform of angle channel Kalman filter, along with somediscussion of its performance and means for minimizing itscomputational complexity. A discussion is then given of acontroller structure for an angle track loop using theKalman filter. The paper concludes with a description ofthe design of range and velocity track loops using theKalman approach.

Manuscript received May 18, 1972.

This work was supported in part by the U.S. Air Force Office ofScientific Research under Grant 699-62.

A digital implementation based on the tracking filterstructures described in this paper has been flight tested inan airborne tactical radar system by the Hughes AircraftCompany. The flight tests successfully demonstrated manyof the performance improvements which are predicted intheory and are discussed in the paper.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-10 NO. 3 MAY 1974 319

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WLS = LINE-OF-SIGHT RATE R, R = RANGE AND RANGE RATE

WA = ANTENNA RATE al = INTERCEPTOR ACCELERATION

WAC = COMMANDED ANTENNAARATE AEa = ANGLE POINTING ERROR (aLS = ESTIMATES AT FILTER OUTPUT

= MEASURED POINTING ERROR -aTFI

Fig. 1. Pointing error control system using Kalman estimates.

-' e d) ANTENNA ORTHOGONALCOORDINATE AXES, r ISTHE ELECTRICAL BORESIGHTAX IS

eI,dA) =LINE-OF-SIGHT ORTHOGONALCOORDINATE AXES. r LIESALONG THE LINE-OF-SIGHT

e,e = ANGLE TRACK ERRORS ASSUMEDed SMWALL ENOUGH FOR ALGEBRA

OF INFINITESIMAL ROTATIONSTO BE VALID

Fig. 2. Antenna and line-of-sight coordinate frames.

II. Angle Filter Structure

The general form of the angle track system to be studiedis shown in Fig. 1. The integral of the difference betweenthe line-of-sight rate oLS and the antenna rate cA is seento be the pointing error C, a measure of which is producedby the angle error receiver of the radar. This measurementEm is inputted to the Kalman filter, along with measures ofWA, range and range rate, and the inertial acceleration ofthe interceptor, ai. The outputs of the filter are estimatesof the states C, WLS' and aTn where aTn is the componentof the target acceleration normal to the line-of-sight. Thefilter outputs are seen to be used as the inputs to thecontroller, which forms the commanded antenna rate WA cThe typical antenna stabilization loop functions as aninertial rate servo, with the input and output signals beingthe commanded and actual values of the antenna spatialrate WA In this section we shall consider the structure ofthe Kalman filter portion of the loop. A discussion of thecontroller will be given at a later point in Section V.

In order for the techniques of Kalman filtering to beapplicable for estimating the states of a system, it must bepossible to represent the dynamics and outputs of thesystem sufficiently well by a linear vector differential stateequation of the form

x =Ax + u + w (1)

with linear measurement or output equation

y -Mx + n (2)

where the values of A, u, and M are known functions oftime, and w and n are white noise vectors with knownspectral densities. x is called the system state vector and ythe system output vector, the latter being used as the filterinput. The Kalman filter then computes the bestmean-square (minimum variance [2] ) estimate of x(t) basedon y(T) for to < r < t using the Kalman filter structureparameters determined by the values of A, u, and M, alongwith the statistics of w and n.

The particular system model to be studied includesstates which correspond to the components of the angletracking error and the components of the spatial rate of theline-of-sight. A Kalman filter whose mechanization is basedon this system model will thus estimate the angular positionand spatial angular rate of the line-of-sight, which are thegeometrical variables about which information is alsoprovided by a classically designed angle tracking system.Turning to a definition of coordinate systems, ' will beused throughout to designate a right-hand orthogonalcoordinate frame. Let OA be the antenna coordinate framewith axes r, e, and d. The antenna electrical boresight, orforward-pointing centerline axis, will be labeled the r axis.The e and d axes are defined as being the axes about whichthe two components of the angle tracking error aremeasured by the two channels of the angle error receiver(see Fig. 2). The components of the angle tracking errorabout the e and d axes will be labeled Ce and C . Define asecond frame, to be called the line-of-sight frame OLS, asthat orthogonal coordinate frame which is obtained byperturbing the antenna frame OA through the small anglesce about the e axis and Cd about the d axis. Let thecorresponding axes of the OLS coordinate frame be calledthe r', e', and d' axes. The r' axis is then, by definition,coincident with the target line-of-sight.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS MAY 1974320


The nomenclature to be used to describe the inertial (orspatial) angular rate of a coordinate frame will be co. LetCOA and coLS be the inertial angular rates of the frames tAand OLS . The components of WjA along the axes of OA arethen CAr' COAe' and WA d, respectively, while thecomponents of co LS along the axes of kLS are WLS,',LLSe', and COLSd-

With the above definitions in mind, let us next examinethe following set of equations:

Ye =keee + ne (3)

Yd kded + nd (4)

ee =LSe' -Ae +± ArEd (5)

6=ULS'-XA -WA 6 (6)ed = OLSd CJAd Cor e(6

CoLSet = - (2R/R)WLSe' + (1IR)(aT_d' + aId)+COA rCOL Sd' (7)

COLSd = (2R/R)WLSd' +±(lIR)(aTe'-Ie)

- ArCLSe (8)

aT_d' = (I1rTzT d' + COAraTe, +Wd (9)

aT = 1IT)aTe'- AraTa d' +We (10)e

Equations (3) and (4) are models for the angle errorreceiver outputs Ye and Yd' where ke and kd are the angleerror receiver slopes and ne and nd are additive noises. Thederivation of (5) through (10) is indicated in the Appendix,and is based on the assumption that the track errors aresmall enough for the operations of small angletransformations to be valid. Equations (5) and (6) definethe dynamical interrelations between the tracking errorsand the components of coWA and cLSI while (7) and (8)represent the dynamical interrelations between thecomponents of coLS and the inertial accelerations of thetwo vehicles. The quantities aTe. and aT-d' designate thecomponents of the target acceleration along the e' and -d'axes, while ale and aId are components of interceptor (orownship) acceleration along the e and d axes. (Writing theequations in terms of aT-d' instead of aTd, is a notationalconvenience which simplifies the discussion at a laterpoint.) The quantities R and R denote range to the targetand its derivative, while T represents the time constant ofthe assumed target acceleration model and we, and Wd, areindependent white noises with equal spectral densities.

As discussed in the Appendix, the target accelerationstate equations (9) and (10) are especially appropriate forthe situation in which the magnitudes of the LOS spatialrates COLSe, and CoLSd, do not exceed 15 or 20 degrees persecond. Although the use of these models could yieldadequate results for higher LOS rate cases, a more nearlyoptimum approach in this situation involves modeling thecomponents of the target acceleration vector alongspace-stabilized axes as states [3]. A significantdisadvantage of the latter approach is that it iscomputationally more complex and is less amenable to themethod of reducing computation discussed in the nextsection. It is also noted that the low angular rate situation isthe primary one of interest in airborne tracking, since atlong ranges the LOS rates are perforce small, while at shortranges the geometric condition of interest for gun androcket firing is the lead collision course, for which thespatial rates are again relatively small.

The above state and measurement equations, (3) through(10), form the system model on which the angle channelKalman filter is based, with the filter thus producingestimates of the components of e, Cobs, and' T Adiscussion is next given of how the parameters required forthe mechanization may be obtained on-line. In the contextof (1) and (2), these parameters are those associated withthe matrixes A, u, and M, along with the statistics of thesystem driving and measurement noises. The receiver gainand noise statistics associated with the measurementequations (3) and (4) may often be determined from apriori analysis of the specific receiver being implemented.In relation to the state equations (5) through (10), it will beseen that the filter parameter requirements are wellmatched to the outputs of on-board sensors which are oftenavailable. Considering first the measurement of the antennarates coAe and CoAd, the antenna stabilization loopstypically provide space stabilization of the antenna aboutthe e and d axes using gyros mounted on these axes. Thesegyros are usually either of the rate or rate integrating type.If rate gyros are used, their outputs provide direct measuresof the rates CoA e and CoAd. Alternately, if rate integratinggyros are employed, the input signals to the stabilizationloops can be used as the measures of COA e and CAd in thefiltering process, since these inputs are the commandedvalues of the rates CoAe and WA d. Although a measure ofCAr is often not used in design, it is seen to be easilyobtainable. The simplest method is to also mount a rategyro on the r axis of the antenna. As to the remainingmeasureable parameters, measures of R and R are oftenavailable from range channel processing. Also, it is oftenpossible to obtain measures of the ownship accelerationcomponents a,e and aId. These would be obtainable fromthe outputs of on-board accelerometers by first eliminatingthe gravity component from their outputs, and thenresolving the resulting vector into antenna coordinates.Finally, the parameters which specify the statisticalcharacteristics of the target acceleration models, namely thetime constant tan and the spectral densities of we' and wd',

PEARSON & STEAR KALMAN FILTERS IN AIRBORNE RADAR TRACKING 321


are chosen by the designer in accordance with the expectedbehavior of the actual target acceleration.

The foregoing discussion indicates that implementing aKalman filter based on the state and measurementequations (3) through (10) is viable from the point of viewof obtaining the required parameters needed in the filterstructure. The estimates ce and ed produced by the filterwill be seen in a later section to be useful for antennacontrol. As measures of the LOS rate components arestandard quantities needed for the intercept mission, thefilter's estimates CJLSe, and WLSd' can be used for thispurpose. The estimates aTe, and aT-d' of the targetacceleration components are of potential use, for instance,in optimal air-to-air missile navigation [5] .

It is of interest to note that several of the on-boarddynamical measurements, specifically those of coAe) WAd,ajeiR, and a,dIR, are treated as elements of the knowndeterministic driving vector u in (1). As the Kalman filteraccuracies are theoretically independent of thedeterministic input u [4], assuming that u is used as aninput to the filter, this implies, for the system at hand, thatthe filter accuracies will be rglatively independent ofinterceptor acceleration and also the e and d components ofantenna rate. This latter fact implies that if the Kalmanfilter were operating on the outputs of a track loop notclosed through the filter, then the filter estimationaccuracies could be expected to be close to independent ofthe loop compensation and bandwidth, provided that thereceiver linearity assumptions embodied by (3) and (4)remained in force.

111. Simplified Gain Computation

A major factor contributing to the complexity of theKalman filter discussed in the previous section is theKalman gain computation, which, for the filter underdiscussion, would, in general, require the solution of asix-by-six matrix Ricatti error covariance equation. It willbe seen in this section that in some cases of interest it ispossible to compute the required filter gains usingconsiderably less computation, while at the same timeincurring little or no reduction in estimation accuracy. Twonot atypical sets of conditions will be cited under which itwould be permissible to compute the gains from thesolution of a three-by-three instead of a six-by-six errorcovariance equation.A first situation under which a simplified gain

calculation is legitimate is specified by the following"6symmetric" conditions: 1) receiver gains ke and kd areequal, and receiver noises ne and nd are statisticallyindependent white noises with equal spectral densities; 2)the initial error covariance matrix of the states, P(O), isdiagonal with the diagonal elements being such that P1 I (0)= P22(0), P33(0) = P44(0), and P 5(0) = P66(0); and 3)the target acceleration state equations have the propertiesdiscussed in the Appendix, namely, that the states aTe andaTd , are initially statistically independent and have equalinitial variance, with the white noises we and wd in (9)

and (10) being statistically independent but having equalspectral densities. Condition 2) states that the initialuncertainties on the two components of e are equal, as alsoare those on the two components of COLS and aT.

Under the above conditions, the complexity of the gaincomputation can be greatly reduced since the gains can beobtained from those associated with the analogoustwo-dimensional problem. The equivalent two-dimensionalproblem is specified by the state and measurementequations of either the elevation or azimuth channel whenWAr is set to zero in (3) through (10). Define the 3 X 1Kalman gain matrix associated with the planar state andmeasurement equations as having the elements K1 1, K2 1,and K3 1. Next assume that the parameters R, R, k, and r,along with the statistics of n and w, are identical for thetwo- and three-dimensional cases, and, further, let thesymmetrical conditions given in 1) through 3) hold for thethree-dimensional case. Then, an examination of the errorcovariance and gain equations for the six-state filter of thethree-dimensional case shows that the 6 X 2 Kalman gainmatrix K can be written in the particularly simple form

(I1)K I 0 K22 0 K31 0 T

K0=O K1 l 0 K2IO K3I

where the Ki1 shown are the two-dimensional gains. Thatis, the three-dimensional gain matrix can, in this case, becomputed from its two-dimensional equivalent and isindependent of the antenna roll rate WA r. The zeroelements in the gain matrix (11) indicate that there is nomeasurement residual cross coupling between the channels,so that the filter differential equations for the elevationchannel states are not driven by the measured azimuthtrack error, and vice versa. It is also noted that, in this case,the computed estimation error variances for each of thetwo components of e, WLS' and AT in thethree-dimensional case are identical and equal to theirequivalents in the two-dimensional solution.A somewhat more general set of conditions can also be

cited for which the computational reduction discussedabove is possible. Suppose that the conditions met in theprevious case hold, except that ke does not equal kd, and,also, the receiver noise spectral densities are not necessarilythe same. To examine the legitimacy of computing thegains for both angle channels from one two-dimensionalgain computation in this case, it is first noted that thereceiver inputs Ye and Yd of (3) and (4) can bemultiplicatively scaled so that the two resulting receivergains are equal. As a result of this scaling, the present casediffers from the previous one only in that the spectraldensities of the e and d channel receiver noises are notnecessarily equal. However, if the Kalman gains arecomputed for the two-dimensional situation using the largerof the receiver noise spectral densities, and these gains areused in the three-dimensional filter as shown in (11), then itis easily demonstrated that the estimation error variancesare upper bounded by their corresponding values associated



with the two-dimensional gain solution. Consequently, ifthese upper bounds give sufficient accuracy, the suggestedprocedure can be used and provides the desiredsimplification in computing the three-dimensional gains.

IV. Line-of-Sight Rate Estimation Accuracies

This section presents some theoretical accuracies in theestimates of the LOS spatial rate components produced bythe filter being studied and discusses their implications. Theaccuracies to be given are those associated with the diagonalelements of the error covariance matrix used in the filtergain computation. The symmetrical conditions discussed inthe previous section have been assumed for the calculations.As indicated previously, these conditions imply that thethree-dimensional filtering accuracies are identical withthose arising in the corresponding two-dimensional errorcovariance calculation, and the latter computation has beenused here for convenience.A delineation is first made between three types of target

accelerations which represent altemative target phenomenaencountered in airborne tracking. The first type assumesthat the target acceleration is relatively small, asexemplified by the situation in which a stationary groundtarget is being tracked. The second case is one in which thetarget acceleration is sizable, but constant or slowlyvarying, such as when tracking a ballistic vehicle. The thirdsituation is that of tracking a maneuvering airborne vehicle.The general procedure for designing the filter under studyin each of these cases involves the choice of an accelerationmodel which best represents the ensemble of expectedtarget accelerations. The theoretical accuracies which areobtained from the error covariance matrix solution in thegain computation then represent an averaging of the filter'sperformance over the statistics of the target accelerationmodel. Since the filter parameters associated with the Aand M matrixes, as well as the deterministic input u and thenoise statistics, are assumed known in the computationused here, the actual filter performance will be somewhatdegraded relative to the curves, with the amount ofdegradation depending on the degree of modeling error,that is, on the accuracy with which the parameters such asR and R are known and the extent to which the targetacceleration violates the assumed acceleration model.

Fig. 3 indicates the computed accuracies in estimatingeach of the line-of-sight rate components, assumingrepresentative models for the three target accelerationclasses given above. The parameters specifying the assumedtarget acceleration models are given in the figure. Typicalvalues of range, range rate, receiver characteristics, andinitial conditions are assumed for the computations and arealso given in the figure. Curves 1, 2, and 3 in the figurerepresent, respectively, the situations in which the targetacceleration is relatively small, large but slowly varying, andlarge and rapidly varying. A trend which would be expecteda priori is seen in comparing the curves, namely, that moreuncertainty in the estimate of target acceleration impliesdecreased accuracy in the line-of-sight rate estimates.

ERROR 0.5 DE~Gl

cn

30

102 64 1

b

FILTERING TIME IN SECONDS

TARGET ACCELERATION MODELS ASSUMED:

CURVE (FT /SEC2) arT(SEC)1 10 32 64 103 64 1

Fig. 3. Comparison of LOS estimation accuracies.

Fig. 3 may be used as a reference point for the followingbrief summary of some of the main advantages of using theKalman filter approach in place of the classical technique.It is first seen that a priori knowledge about thecharacteristics of the target acceleration may be included inthe design. Secondly, it is recalled that the accuracy ofthese estimates can be made independent of theaccelerations of the interceptor to the extent that thequantities aIe and aId can be measured and used to drivethe filter. In the same vein, the accuracies can be maderelatively independent of the antenna motion orclosed-loop tracking behavior, since the antenna ratecomponents WAe and COAd enter in as deterministic drivingterms, and, for the symmetrical conditions discussed in theprevious section and assumed for the above computations,the accuracies are also theoretically independent of theantenna roll rate coAr about the boresight axis. This lattereffect is a decided departure from the noticeable effect ofroll rates on the performance of the classical design. A finaltype of advantage is that the filtering computations adaptthemselves to R and R, as well as the receiver gains andnoise statistics, to the extent that these quantities can bemeasured and inputted to the calculations. The aboveeffects taken together imply that the angle channel Kalmanfilter which has been described offers the flexibility forproviding increased accuracies in estimating the angularvariables needed in airborne radar tracking.

PEARSON & STEAR: KALMAN FILTERS IN AIRBORNE RADAR TRACKING 323


TIME IN SECONDS

PARAMETERS SPECIFYING INPUT GEOMETRY: E(WLS It>)) = 0

rwLS (to) = 0.133 RAD/SEC, INITIAL RANGE, 15,000 FT

RANGE RATE = -1000 FPS

RECEIVER PARAMETERS: GAIN = 1, NOISE SPECTRALDENSITY = 5 x 10-4 iDEG)2/CPS

STABILIZATION LOOP MODEL: GSL(S) = Wm2/ S2+2K,wmS +Wm2)

t m = 125, 5 = 0.5

Fig. 4. Tracking error performance using the suboptimal controller.

V. Simplified Angle Loop Controller Design

This section demonstrates a simplified angle loopcontroller structure with close to optimal properties undersome conditions. To simplify the discussion, thedevelopment will primarily be concerned with thetwo-dimensional version of the control problem.

Assuming that the control loop is of the form shown inFig. 1, the particular suboptimal controller of interest issuggested by the following arguments. A heuristic approachfor reducing the angle tracking error to a minimum wouldbe to drive the antenna boresight axis at a spatial rate equalto the target line-of-sight rate WLS' and also to null out anyangular differences between the two axes. Animplementation of this appraoch based on the filter outputswould be to let the antenna rate command be

A A (12)()A LS 12

C

where the quantities with a caret are the filter outputs andX is a constant. The first term 2LS drives the antenna atthe Kalman estimated spatial rate of the line-of-sight, andthe second term has the effect of driving the antenna in the

direction needed to null the Kalman estimated trackingerror. It is noted that the use of the LOS rate estimate LSin the controller equation is akin to a technique called"aided tracking" in classical loop design, in which anexternally provided measure of coLS' if available, is addedto the antenna rate command signal. With the Kalman filterin-the-loop approach, aided tracking is seen to be implicitlyavailable in the form of the estimate LLS

Fig. 4 indicates the performance of the suboptimalcontroller introduced above under some typical operatingconditions. A second-order model is used to approximatethe antenna stabilization loop dynamics. The figure givesthe rms value of the tracking error e averaged over thereceiver noise and over an ensemble of input LOS ratesstatistically generated as shown in the figure. Theparameters used in the state equations which generate theinput w LS process are also given in the figure and assumedto be known to the filter. Two additional curves are givenin the figure for comparison purposes. One of the curvesmakes it possible to compare the performance of thespecified suboptimal controller with that of an equivalentclassical configuration, as it indicates the rms pointing errorunder identical conditions, except that the antenna ratecommand SAC is formed as Xem where em is the angleerror receiver output and X is the same controller gain valueused in the suboptimal controller. The third curve in thefigure shows a lower bound on the pointing error accuracywhich could be obtained if the controller were designedusing the principles of optimal control theory, and indicatesthat the suboptimal accuracy lies close to the lower boundafter an initial transient period. The lower bound is basedon [61, where it is shown that, in the feedback solution tothe optimal linear stochastic control problem obtainedusing the "separation principle" [7], [8], the mean-squarevalues of the system states being controlled arelower-bounded by the mean-square errors in the Kalmanestimates of these states. In the present context, the systembeing controlled would be the planar version of the stateequations given by (5) through (10), plus a set of linearstate equations representing the antenna stabilization loop(see Fig. 1), which relates the control variable or antennarate command WAC to the actual antenna rate 0A' Anoptimal feedback control minimizing a quadratic functionalin e would be a linear function of the estimated states, andapplication of the discussion given in [6] shows that theresulting variance on the track error would belower-bounded by the variance on the Kalman estimate ofe. Further, the Kalman error variance, assuming the systemdescribed above, is itself lower-bounded by the errorvariance of the Kalman estimate when the antenna rate coAis assumed known, for which case the states associated withthe stabilization loop are eliminated in the filter structure.The square root of the latter Kalman error variance isplotted in Fig. 4.

The suboptimal controller described above is of thesimplest generic aided tracking form. However, additionalsimulation has also shown that its accuracies fall close to


I

324


the lower bound under a fairly wide set of encounterconditions specified by R, R, and the receivercharacteristics, assuming that the stabilization loop modelshown in the figure is used. For general purposes, a majorfactor in the question of whether the simplified suboptimalcontroller assumed above is adequate for a given situation isthe bandwidth of the stabilization loop. If the stabilizationloop bandwidth is sufficiently high, as in the case of themodel shown in Fig. 4, then the performance of the simplesuboptimal controller is close to the lower bound and alsonotable constant over a sizable range of values of thecontroller gain X. The upper bound on this range of valuesis specified by loop stability considerations, while the lowerbound occurs when the tracking loop begins to be sluggish.If the bandwidth of the stabilization loop to be used isreduced, the performance of the loop with the baselinesuboptimal controller becomes increasingly degradedrelative to the high bandwidth case. In these situations it isthus desirable to consider modifying the controllerstructure to account for the stabilization loop dynamics.Although the design of such a controller would appear tobe most optimally approached using the principles ofoptimal control, an alternate suboptimal technique is tomodify (12) by first passing e through a lead lagcompensator. For an excellent study of additional aidedtracking concepts from the modern viewpoint, see [9].

The extension of the foregoing discussion to thethree-dimensional situation implies the use of thesuboptimal controller type given in (12) for each of theangle control channels. The primary modification in theform of each of the channels relative to Fig. 1 becomes thecross-coupling terms between the channels caused byantenna roll motion about the boresight axis [see (5) and(6)]. Roll cross-coupling terms have, however, close tonegligible effect for systems with low antenna roll rates, sothat in these cases the planar analysis given above remainsmeaningful for each of the angle control channels.Examples of systems with low antenna roll rates areroll-stabilized missiles and interceptor radars withroll-stabilized antennas.

A very practical situation in which the controller typediscussed above has additional appeal is that in which it isnecessary to optimally extrapolate through blind zones, orperiods in which no valid receiver data is available. Theestimator/controller system described in this section posesan optimal type solution for this environment. First, itmakes best use of the available receiver and on-boarddynamical information to determine the Kalman estimatesof e(t) and WLS(t) which are to be used to control theantenna. During the blind zones the filter is informed thatthe angle error receiver slopes are zero, and it automaticallygoes into an extrapolation which forms the best predictionof the angular motion of the line-of-sight. The controllerthen ensures that the antenna boresight remains closelyaligned with the best predicted direction of theline-of-sight. Such a system provides significantmethodological improvements over the antenna

extrapolation schemes which have been used in conjunctionwith classical designs.

VI. Range and Range-Rate Estimation and Control

This section discusses the application of Kalman filteringto range and range-rate tracking in airborne radar. Thetypes of airborne radar systems which will be of interest arethose whose fundamental measures are R, R, or, perhaps,both R and R. Examples of these are, respectively, pulse,pulsed-Doppler, and range-gated pulsed-Doppler type radars[11. In a range type radar, the estimate of R must beobtained by taking the derivative of a filtered measure of R.In a pulsed-Doppler radar, whose primary measurement isR, it is sometimes possible to obtain a direct measure of Rusing range gating. When this is not possible, a measure ofRmay be obtained by some less direct means, such as FMranging [ 1] .

The form of the Kalman filter which can be used for theR and R tracking operation is next described. Consider thefollowing state space description, in which the states arexl = R, X2 =R, and x3= aTr', where aTr' is thecomponent of the target acceleration along the LOS:

X1 =X2 (13)

(14)X2 =&Ls Xl + X3 ai

X3 =-(l 1*/3 + W3 (15)

(16)Yi =klxl + ni

Y2 - k2x2 ±n2 - (17)

Equation (14) is derived in the Appendix. The parameterLS designates the sum of the squares of the LOS angular

rate components which are measured traditionally by theangle tracking system, and which may be estimated usingKalman filtering, as described in Section II. Either of thesemeasures may be used to compute an estimate of thequantity coLS2, which may then be considered for use inthe role of a known system parameter. The quantity aIr, isthe component of the interceptor acceleration along theLOS. A measure of its value can be obtained by resolvinginterceptor acceleration measurements into the antennaaxes, and may be treated as a deterministic input. Equation(15) is a first-order Markov state equation model for thetarget acceleration component aTr" with w3 being whitenoise. Finally, (16) and (17) form the observation structureassociated with the foregoing state equations. Theparameters k, and k2 will be seen to correspond to thediscriminator slopes, and the white noises n, and n2represent the additive noises in the discriminator outputs.



R

ccb

Fig. 5. Range/range-rate tracking using Kalman filtering.

FILTERING TIME, SECONDS

Fig. 5 indicates how a Kalman filter based on the stateand measurement equations (13) through (17) would beused to provide the R and R tracking functions. It is seenthat the filter's estimates of R and R are used not only asthe tracker outputs, but also to position the range andvelocity gates; in other words, to center the discriminators.Assuming that the track errors are small enough for thediscriminators to be operating in their linear region, thediscriminator outputs are of the form

Y1 = kl (R-R) + nj (18)

Y2 =k2'(R-R) +n2 (19)

where kl' and k2j are the discriminator slopes, n1 and n2are the additive receiver noises, and R and A are therange-rate estimates produced by the filter. Equations (16)and (17) show that the discriminator outputs Y1 and Y2may be used to exactly mechanize the residuals required forthe Kalman filter, and that the discriminator gains kl' andk2', together with the discriminator noise statistics, wouldthen be those associated with the measurement structure,(16) and (17), in computing the Kalman gains.

The general form of the system indicated in Fig. 5applies to situations in which range and/or range-rateR and R estimates produced by the filter. Equations (16)cases when only range or only range-rate are available maybe obtained respectively by setting k2 and k5 to zero in(17) and (16). In the case when only R measurements areto be available, a particular question of interest is whetheran accurate estimate of range is produced by the filter. Thiswill depend on the magnitude of the centripetalacceleration LS2R relative to the uncertainty in the radialtarget acceleration aTr. For instance, if a stationary groundtarget is being tracked by an airborne vehicle flyingoverhead, then aTrr is small and close to being known, and,

RECEIVER PARAMETERS: RANGE DISCRIMINATOR GAIN = 1

RANGE RATE DISCRIMINATOR GAIN = 1 OR 0

RANGE NOISE SPECTRAL DENSITY = 25 (FT)2/CPSR NOISE SPECTRAL DENSITY = 1 lFPS)2/CPS

GEOMETRY PARAMETERS: CiLS2 = 0.018

TARGET ACCELERATION MODEL: caT 64 FT/SEC2

TaT' 3 SEC

Fig. 6. Typical accuracy of the range tracking Kalman filter.

for sufficiently low altitudes, coS 2R is sizable. Underthese conditions a meaningful estimate of range will beprovided by the filter. At the other extreme, if the range is

large, so that LS2R is small, and the uncertainties in aTr,are relatively large, as, for instance, when tracking a

maneuvering airborne vehicle, then Co 2R will be smallrelative to the uncertainties in aTr and will beindistinguishable from aTr . In such Cases, an accurateestimate of range cannot be obtained and there will beessentially no loss of accuracy in estimating R using a filterwhich is configured with two states instead of three, thetwo states being R and the sum of CoLS2R and aTr.

The benefits of using the Kalman approach to R and R

tracking are similar to those discussed in connection withthe angle channel. The approach makes systematic use ofparameters such as Co 2 and aIr which may be availablefrom other on-board su%systems, and optimally adapts toon-line changes in the discriminator gains and noisestatistics if these changes can be measured and provided tothe filter. An important example in the latter category isthat the system provides optimal capability forextrapolating through radar blind zones, thus maximizingthe probability that the target will reappear in the rangeand velocity gates when the signal returns.

A numerical example is given next which demonstratesthe advantage of processing range and velocitydiscriminator outputs simultaneously, when both areavailable, as opposed to separately, which is typical inclassical servo design. Fig. 6 shows the rms tracking error in



range using the above described filter for a typicalgeometry, and compares the case of processing only rangemeasurements with that of simultaneously processing rangeand range-rate measurements. The curves were obtained bysolving the Kalman error covariance equation associatedwith the two filters, and thus represent theoretical orcomputed accuracies which assume no mismatch in theparameters used. The initial covariance matrix was assumedto be diagonal with the values on the diagonal as given inthe figure. The relative difference between the curvesindicates that a significant improvement in range trackingaccuracy is possible for the case in which range and velocitydata are processed simultaneously, as opposed toseparately.

effects, radome error, and receiver nonlinearities whichoccur if the tracking errors are large [1] .

Implementation of the filters described would generallyrequire the use of a digital computer due to the complexityand time-varying nature of the solution. The filter design inthese cases would proceed by forming a discretized versionof the state and measurement equations with respect towhich the discrete version of the Kalman filter would beimplemented in the computer.

VIII. Conclusions

VIl. Additional Commen ts

In Section II it was seen that the system dynamicalequations associated with the angular motion of the LOSwere linear in the components of C, cLS, and aT,. Thismade it possible to consider the application of linearestimation to this problem by choosing these componentsas states, and using measurements ofR and R in the role ofknown system parameters. Also, in the previous section itwas seen that the system dynamics associated with R, R,and aTr. were likewise linear in these variables, so that theKalman structure could again be applied if a measure ofcULS obtained from the angle channel were used as aknown system parameter. If these two channels areconsidered as one system, and the states of the system arechosen to be R, R, the components of e and oLS' and thestates associated with the target acceleration vector, theresulting system model is seen to be nonlinear, so thatnonlinear filtering techniques could as well be applied tothe estimation of the states with a potential improvementin performance. An important disadvantage of thisapproach is, of course, the increased computationalrequirement, as the gain computation for the nine-statefilter would be considerably more complex than when usingthe linearization in the foregoing sections.

In a more general vein, in considering the application ofthe above discussed filters to a particular radar trackingsystem, simulation studies must be made to ascertain theeffect on performance of the fact that the parameters usedin the filtering computations are not known exactly. Forthe angle filter these include range, range-rate, and theantenna rate components, while for the range channel filter,the measured LOS rate is used as an input. Additionalparameters for both filters are the interceptor accelerationcomponents along the antenna axes and the parametersassociated with the receiver characteristics. If both rangeand angle channel filters are used, a simulationencompassing both is appropriate for specifying the systemperformance. Effects of general interest in modeling thereceiver gains and additive noises include thermal receivernoise, clutter, angle, range, and amplitude scintillation

It has been shown that the Kalman filter provides anatural framework for configuring the estimation processesrequired in airborne tracking and that the resulting systemhas features not as readily available with the classical servoapproach., These features, obtained by espousing themethods of optimal estimation, include adaptivity of theprocessing to the specific encounter conditions, minimizingthe effect of ownship acceleration and antenna dynamicson the estimation accuracies, and accounting for variationsin receiver characteristics when these are measurable andincluded as parameters in the filter. The states estimated bythe filters are important quantities needed for solution ofthe intercept problem and are natural variables foroptimizing the feedback signals which drive the antennaand range and velocity gates.

AppendixSummary of Derivation of State Equations

The derivation of the state equations (3) through (10)and (12) is indicated in this Appendix. The state equationsfor the track errors Ce and Ed are considered first.

Recall that the antenna and line-of-sight frames, OA andOLS' have angular rate vectors coA and coWLS) respectively,and that the direction cosine matrix between OA and epLS is-a function of the track errors ee and Ed. Let E be thisdirection cosine matrix expressed as a skew symmetricmatrix function of ee and Ed [9], and let WA and WLS bethe skew symmetric angular rate matrixes involving thecomponents of cA and c LS' Then a well knowndifferential property of the direction cosine matrix specifiesthat F = WLsE-EWA. Examination of the appropriateelements of this matrix equation provides the desired stateequations, (3) and (4), for Ce and ed.

The state equations for the variables ALse and CALSd'are next considered. Let 0I be an inertial frome, OLSX again,be the line-of-sight frame, and let R be the relative positionvector. Then the twice application of the Coriolis equation[9] gives



(d2 R/dt2 )(I) = (d2 R/dt2 )(OL S)+ 20LS(dR/dt)QPLS)

+ (d dt)( R ) LS(LS X R).(20)

If each vector in (20) is expressed in terms of itscomponents in the 4LS frame, and the vector products areevaluated, (20) will then be expressed in terms of itscomponents along the r', e', and d' axes of the 0LScoordinate system. These three component equations maybe arranged as

R-(&jLSe + cLSd')R + aTr- a1 (21)

CLSe' (2R/R)C'LS- (l/R)(aTd -aId)

+WLSr'CLSd' (22)

L Sd' (2R/R)CL Sd' ± (1IR)(aTe'- a,e)

C LSr'(-LSe' (23)

where R, R, and R are scalar range and its derivatives, andwhere the aT and ai variables are the components of thetarget and interceptor inertial acceleration vectors along the

OLS axes. Equation (21) is used as a state equation in thediscussion of range tracking, and (22) and (23) are thedynamical relationships from which the state equations forWLSe) and L.Sd t are derived, as discussed in the following.

Equations (22) and (23) may be written, to closeapproximation [3 ], as

coLSe= - (2RR)ALSe-(1/R)(aTd,-aId)+ A LStr d

+ [(leaI )/R + (edwAr e

ceA )dLS ,

cLS (2R/R) LS (/R)(aT-aI)-WA WLSe,

+ [(EdaIr)R - eC A d)LSe (25)

The WLSee and WLSd' state equations used in the maintext are (24) and (25) with the bracketed terms neglected.Although the bracketed expressions can be included in thestate equations, giving rise to nonlinear terms in filteringprocess, their inclusion is not expected to be necessary [3]in typical airborne tracking cases. The reason for this stemsfrom the fact that neglecting these terms is equivalent tointroducing errors in the measured interceptor accelerationcomponents ale and aId. Typically, the tracking errors ceand Ed are sufficiently close to zero for the magnitude ofthese equivalent acceleration measurement errors to beinappreciable. Also, a slight change in nomenclature is made

in (24) when used in the main text; namely, (24) is, forlater convenience, written in terms of aT-d' instead ofaTd, where aT d' =-aTd.

The remainder of this appendix indicates the derivationfor the target acceleration state equations, (9) and (10). Itis assumed that an adequate general approach for modelingthe target acceleration vector aT is to model its componentaTx along any fixed or slowly varying direction x in spaceby a first-order Markov process, that is, by aTx- (lIr)aTx + w, where w is white noise. Models for thetarget acceleration components along any two orthogonalaxes in space will further be assumed statisticallyindependent.

Define a coordinate frame (r', el, d1) such that r' isalong the line-of-sight, and such that the frame is inertiallynonrotating about the r' axis. Since the el and d1 axes havezero inertial rate about the r' axis, their only angularmotion is that about the d, and e1 axes, respectively.Assume next that the angular rates about these axes are lessthan 15 or 20 degrees per second. In this case, the el andd1 axes can be taken as slowly varying directions in space,and aTe and aTd modeled as statistically independentand identical first-order Markov processes. Assuming this tobe the case, the desired acceleration components whosestate equations are next to be found are those along the e'and d' axes. But, the derivatives of aTef and aTd may berelated to those of aTe and aTd using the Conolisequation. Applying this property results in the followingdifferential equations for aTe' and aTd:'

(26)aT =-(l /T)a±T LS4 aT ,+ WLee e r d

(27)aTd - (lIr)aTd )cLSr,aTe, +wd±

where we' and Wd' are again statistically independent whitenoises with equal spectral density, and COLSr, is the rate ofthe ¢ LS frame about the r' axis. The state equations for thetarget acceleration components used in the main text areobtained from (26) and (27) by approximating WLSr bywA and by rewriting the equations given here in terms ofaT d' instead of aTd.

References

[11 D.J. Povejsil, R.S. Raven, and P. Waterman, Airborne Radar.Princeton, N.J.: Van Nostrand, 1961.

[21 R.E. Kalman, "New methods in Wiener filtering theory,"Proc. 1st Symp. of Engrg. Application of Random FunctionTheory and Probability. New York: Wiley, 1963, pp.270-388.

[31 J.B. Pearson, I1I, "Basic studies in airborne radar trackingsystems," Ph.D. Dissertation, University of California at LosAngeles, September 1970.



[41 H.W. Sorenson, "Kalman filtering techniques," in Advances in [71 T.L. Gunkel and G.F. Franklin, "A general solution for linearControl Systems, vol. 3, C.T. Leondes, Ed. New York: sampled-data control," J. Basic Engrg.. vol. 85D, pp. 197-201,Academic Press, 1966. 1963.

181 P. Joseph and J.T. Tou, "On linear control theory," AIES[51 A.E. Bryson, Jr., "The 1966 Martin Minta Lecture, Trans.,vol. 80, pp. 193-196, 1961.

'Applications of optimal control theory in aerospaceTrns, o. '0 pp 19-16 1961

[91 J.M. Fitts, "Aided tracking applied to high accuracy pointingengineering'," J. Spacecraft and Rockets, vol. 4, pp. ' systems," IEEE Trans. Aerospace and Electronic Systems, vol.May 1968. AES-9, pp. 350-368, May 1973.

[61 A.E. Bryson, Jr., and Y.C. Ho, Applied Optimal Control. 1101 H. Goldstein, Classical Mechanics. Cambridge, Mass.:Waltham, Mass.: Blaisdale, 1969, p. 417. Adison-Wesley, 1950.

John B. Pearson, III was born in Washington, D.C., on November 14, 1935. He receivedi:iXthe B.S.E.E. and M.S.E.E. degrees from Stanford University, Stanford, Calif., in 1956 and

1957, respectively, and the Ph.D. degree in engineering from the University of Californiaat Los Angeles in 1970.

Since 1958 he has been employed by the Hughes Aircraft Company, and has workedprincipally in the areas of inertial navigation, tracking systems, and missile dynamics. Heis presently a Senior Staff Engineer with Hughes Radar Avionics in Canoga Park, Calif. Hehas held the Hughes Master of Science and Doctoral Fellowships and has two patentspending in the area of digital tracking. He has published in the field of nuclear engineeringand has recently been concentrating on applications of optimal filter and control theoryto radar avionics and missile system design.

Edwin B. Stear (A'57 M'70) was born in Peoria, Ill., on December 8, 1932. He receivedthe M.S. degree in mechanical engineering from the University of Southern California,Los Angeles, in 1956, and the Ph.D. degree in engineering from the University ofCalifornia, Los Angeles, in 1961.

From 1954 to 1959 he was employed as a member of the Technical Staff of HughesAircraft Company, Culver City, Calif., where he worked on various aspects of air-to-airmissile design and analysis. He held the Hughes Master of Science Fellowship from 1954to 1956 and the Hughes Doctoral Fellowship from 1956 to 1958. From 1959 to 1961 hewas an Assistant Research Engineer at the University of California at Los Angeles, wherehe did research on control systems theory. From 1961 to 1963 he was a Project Officer inthe U.S. Air Force in charge of several applied research programs in aerospace vehicleflight control. From 1963 to 1964 he was manager of the Control and CommunicationsLaboratory at Lear Siegler, Inc., where he directed research programs in control andcommunications theory and practice. He joined the faculty of the University of Californiaat Los Angeles in 1964 as an Assistant Professor, where he taught and supervised graduatestudents in the control and systems area. In 1969 he transferred to the University of

:.......:- California at Santa Barbara, where he is currently Professor of Electrical Engineering, incharge of the academic program in control and systems science. He has published over

twenty papers in various aspects of control and systems theory and practice, includingseveral papers on hormonal control. He is also co-editor of the book "Hormonal ControlSystems" (American Elsevier). He has consulted- widely for various government andindustrial organizations, and is currently Senior Technical Consultant to the Air ForceSpace and Missile Test Center.

Dr. Stear is a member of the American Institute of Aeronautics and Astronautics andof Sigma Xi. He is currently a member of the United States Air Force Scientific AdvisoryBoard. He is also Associate Editor of the American Institute of Aeronautics andAstronautics Journal ofAircraft.

PEARSON & STEAR: KALMAN FILTERS IN AIRBORNE RADAR TRACKING 329


1972 kalman filter applications in airborne radar tracking

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