Tài liệu Tracking and Kalman filtering made easy P1 doc

Then we could predict the distance target 1 would move during the scan-to-scan period and as a result have an estimate of the target’s future position.Assume this prediction was done for

a radar the fan beam rotates continually through 360
, typically with a period of

10 sec Such a radar provides two-dimensional information about a target Thefirst dimension is the target range (i.e., the time it takes for a transmitted pulse

to go from the transmitter to the target and back); the second dimension is theazimuth of the target, which is determined from the azimuth angle (see Figure1.1-1) the fan beam is pointing at when the target is detected [1] Figures 1.1-2through 1.1-6 show examples of fan-beam radars

Assume that at time t¼ t1 the radar is pointing at scan angle
and twotargets are detected at ranges R1 and R2; see Figure 1.1-7 Assume that on thenext scan at time t¼ t1þ T (i.e., t1þ 10 see), again two targets are detected;see Figure 1.1-7 The question arises as to whether these two targets detected onthe second scan are the same two targets or two new targets The answer to thisquestion is important for civilian air traffic control radars and for militaryradars In the case of the air traffic control radar, correct knowledge of thenumber of targets present is important in preventing target collisions In thecase of the military radar it is important for properly assessing the number oftargets in a threat and for target interception

Assume two echoes are detected on the second scan Let us assume wecorrectly determine these two echoes are from the same two targets as observed

on the first scan The question then arises as to how to achieve the properassociation of the echo from target 1 on the second scan with the echo from

3Copyright # 1998 John Wiley & Sons, Inc ISBNs: 0-471-18407-1 (Hardback); 0-471-22419-7 (Electronic)

Figure 1.1-1 Example of fan-beam surveillance radar.

Figure 1.1-2 New combined Department of Defense (DOD) and Federal AviationAdministration (FAA) S Band fan-beam track-while-scan Digital Airport SurveillanceRadar (DASR) ASR-11 This primary system uses a 17-kW peak-power solid-state

‘‘bottle’’ transmitter Mounted on top of ASR-11 primary radar antenna is L-band array rectangular antenna of colocated Monopulse Secondary Surveillance Radar(MSSR) Up to 200 of these systems to be emplaced around the United States (Photocourtesy of Raytheon Company.)

open-target 1 on the first scan and correspondingly the echo of open-target 2 on the secondscan with that of target 2 on the first scan.

If an incorrect association is made, then an incorrect velocity is attached to agiven target For example, if the echo from target 1 on the second scan isassociated with the echo from target 2 of the first scan, then target 2 isconcluded to have a much faster velocity than it actually has For the air trafficcontrol radar this error in the target’s speed could possibly lead to an aircraftcollision; for a military radar, a missed target interception could occur.The chances of incorrect association could be greatly reduced if we couldaccurately predict ahead of time where the echoes of targets 1 and 2 are to beexpected on the second scan Such a prediction is easily made if we had anestimate of the velocity and position of targets 1 and 2 at the time of the firstscan Then we could predict the distance target 1 would move during the scan-to-scan period and as a result have an estimate of the target’s future position.Assume this prediction was done for target 1 and the position at which target 1

is expected at scan 2 is indicated by the vertical dashed line in Figure 1.1-7.Because the exact velocity and position of the target are not known at the time

of the first scan, this prediction is not exact If the inaccuracy of this prediction

is known, we can set up a 3 (or  2) window about the expected value,where  is the root-mean-square (rms), or equivalently, the standard deviation

of the sum of the prediction plus the rms of the range measurement Thiswindow is defined by the pair of vertical solid lines straddling the expectedposition If an echo is detected in this window for target 1 on the second scan,

Figure 1.1-3 Fan-beam track-while-scan S-band and X-band radar antennas emplaced

on tower at Prince William Sound Alaska (S-band antenna on left) These radars are part

of the Valdez shore-based Vessel Traffic System (VTS) (Photo courtesy of RaytheonCompany.)

then with high probability it will be the echo from target 1 Similarly, a 3window is set for target 2 at the time of the second scan; see Figure 1.1-7.For simplicity assume we have a one-dimensional world In contrast to aterm you may have already heard, ‘‘flatland’’, this is called ‘‘linland’’ Weassume a target moving radially away or toward the radar, with xnrepresentingthe slant range to the target at time n In addition, for further simplicity weassume the target’s velocity is constant; then the prediction of the targetposition (range) and velocity at the second scan can be made using thefollowing simple target equations of motion:

where xn is the target range at scan n; _xn is the target velocity at scan n, and Tthe scan-to-scan period These equations of motion are called the systemdynamic model We shall see later, once we understand the above simple case,Figure 1.1-4 Fan-beam track-while-scan shipboard AN=SPS-49 radar [3] Twohundred ten radars have been manufactured (Photo courtesy of Raytheon Company.)

Figure 1.1-5 L-band fan-beam track-while-scan Pulse Acquisition Radar of HAWKsystem, which is used by 17 U.S allied countries and was successfully used duringDesert Storm Over 300 Hawk systems have been manufactured (Photo courtesy ofRaytheon Company.)

Figure 1.1-6 New fan-beam track-while-scan L-band airport surveillance radar 23SS consisting of dual-beam cosecant squared antenna shown being enclosed inside50-ft radome in Salahah, Oman This primary radar uses a 25-kW peak-power solid-state ‘‘bottle’’ transmitter Mounted on top of primary radar antenna is open-arrayrectangular antenna of colocated MSSR This system is also being deployed in HongKong, India, The People’s Republic of China, Brazil, Taiwan, and Australia

ASR-that we can easily extend our results to the real, multidimensional world where

we have changing velocity targets

The –, ––, and Kalman tracking algorithms described in this book areused to obtain running estimates of xn and _xn, which in turn allows us to do theassociation described above In addition, the prediction capabilities of thesefilters are used to prevent collisions in commercial and military air trafficcontrol applications Such filter predictions also aid in intercepting targets indefensive military situations

The fan-beam ASR-11 Airport Surveillance Radar (ASR) in Figure 1.1-2 is

an example of a commercial air traffic control radar The fan-beam marine radar

of Figure 1.1-3 is used for tracking ships and for collision avoidance These twofan-beam radars and those of the AN/SPS-49, HAWK Pulse Acquisition Radar(PAR), and ASR-23SS radars of Figures 1.1-4 to 1.1-6 are all examples ofradars that do target tracking while the radar antenna rotates at a constant ratedoing target search [1] These are called track-while-scan (TWS) radars Thetracking algorithms are also used for precision guidance of aircraft onto therunway during final approach (such guidance especially needed during badweather) An example of such a radar is the GPS-22 High PerformancePrecision Approach Radar (HiPAR) of Figure 1.1-8 [1–4] This radar useselectronic scanning of the radar beam over a limited angle (20
in azimuth,

Figure 1.1-7 Tracking problem

Figure 1.1-8 Limited-scan, electronically scanned phased-array AN/GPS-22 HiPAR.Used for guiding aircraft during landing under conditions of poor visibility [1–3] Sixtysystems deployed around the world [137] (Photo courtesy of Raytheon Company.)

Figure 1.1-9 Multifunction PATRIOT electronically scanned phased-array radar used

to do dedicated track on many targets while doing search on time-shared basis [1–3].One hundred seventy-three systems built each with about 5000 radiating elements forfront and back faces for a total of about 1.7 million elements [137] (Photo courtesy ofRaytheon Company.)

8
in elevation) instead of mechanical scanning [1–4] An example of a angle electronically scanned beam radar used for air defense and enemy targetintercept is the PATRIOT radar of Figure 1.1-9 used successfully during DesertStorm for the intercept of SCUD missiles Another example of such a radar isthe AEGIS wide-angle electronically scanned radar of Figure 1.1-10.

wide-The Kalman tracking algorithms discussed in this book are used toaccurately predict where ballistic targets such as intercontinental ballisticmissiles (ICBMs) will impact and also for determining their launch sites (whatcountry and silo field) Examples of such radars are the upgraded wide-angleelectronically steered Ballistic Missile Early Warning System (BMEWS) andthe Cobra Dane radars of Figures 1.1-11 and 1.1-12 [1–3] Another such wide-angle electronically steered radar is the tactical ground based 25, 000-elementX-band solid state active array radar system called Theater High Altitude Area

Figure 1.1-10 Multifunction shipboard AEGIS electronically scanned phased-arrayradar used to track many targets while also doing search on a time-shared basis [1, 3].Two hundred thirty-four array faces built each with about 4000 radiating elements andphase shifters [137] (Photo courtesy of Raytheon Company.)

Figure 1.1-11 Upgrade electronically steered phased-array BMEWS in Thule,Greenland [1] (Photo courtesy of Raytheon Company.)

Figure 1.1-12 Multifunction electronically steered Cobra Dane phased-array radar (inShemya, Alaska) Used to track many targets while doing search on a time-shared basis[1, 3] (Photo by Eli Brookner.)

Defense (THAAD; formerly called GBR) system used to detect, track, andintercept, at longer ranges than the PATRIOT, missiles like the SCUD; seeFigure 1.1-13 [136, 137] Still another is the Pave Paws radar used to tracksatellites and to warn of an attack by submarine-launched ballistic missiles; seeFigure 1.1-14 [1–3].

Figure 1.1-13 A 25,000-element X-band MMIC (monolithic microwave integratedcircuit) array for Theater High Altitude Area Defense (THAAD; formerly GBR) [136,137] (Photo courtesy of Raytheon Company.)

Figure 1.1-14 Multifunction electronically steered two-faced Pave Paws solid-state,phase-steered, phased-array radar [1–3] (Photo by Eli Brookner.)

Two limited-scan electronically steered arrays that use the algorithmsdiscussed in this book for the determination of artillery and mortar launch sitesare the Firefinder AN=TPQ-36 and AN=TPQ-37 radars of Figures 1.1-15 and1.1-16 [1] An air and surface–ship surveillance radar that scans its beamelectronically in only the azimuth direction to locate and track targets is theRelocatable Over-the-Horizon Radar (ROTHR) of Figure 1.1-17 [1].

All of the above radars do target search while doing target track Someradars do dedicated target track An example of such a radar is the TARTARAN=SPG-51 dish antenna of Figure 1.1-18, which mechanically slews apencil beam dedicated to tracking one enemy target at a time for missileinterception Two other examples of dedicated pencil beam trackers are theHAWK and NATO SEASPARROW tracker-illuminators; see Figures 1.1-19and 1.1-20

Figure 1.1-15 Long-range limited electronically scanned (phase–phase), phased-arrayartillery locating Firefinder AN=TPQ-37 radar [1] One hundred two have been built and

it is still in production [137] (Photo courtesy of Hughes Aircraft.)

Assume the target is estimated to have a velocity at time n 1 of 200 ft=sec.Let the scan-to-scan period T for the radar be 10 sec Using (1.1-a) we estimatethe target to be (200 ft=sec) (10 sec)¼ 2000 ft further away at time n than it was

at time n 1 This is the position xn indicated in Figure 1.2-1 Here we areassuming the aircraft target is flying away from the radar, corresponding to thesituation where perhaps enemy aircraft have attacked us and are now leaving

Figure 1.1-16 Short-range, limited-scan, electronically scanned (phase-frequency)phased-array artillery locating Firefinder AN=TPQ-36 radar [1] Two hundred forty-three have been built (Photo courtesy of Hughes Co.)

(b)

Figure 1.1-17 Very long range (over 1000 nmi) one-dimensional electronicallyscanned (in azimuth direction) phased-array ROTHR: (a) transmit antenna; (b) receiveantenna [1] (Photos courtesy of Raytheon Company.)

Figure 1.1-18 AN=SPG-51 TARTAR dedicated shipboard tracking C-band radarusing offset parabolic reflector antenna [3] Eighty-two have been manufactured (Photocourtesy of Raytheon Company.)

Figure 1.1-19 Hawk tracker-illuminator incorporating phase 3 product improvementkit, which consisted of improved digital computer and the Low Altitude SimultaneousHAWK Engagement (LASHE) antenna (small vertically oriented antenna to the left ofmain transmit antenna, which in turn is to the left of main receive antenna) Plans areunderway to use this system with the AMRAAM missile (Photo courtesy of RaytheonCompany.)

and we want to make sure they are still leaving Assume, however, at time n theradar observes the target to be at position yn instead, a distance 60 ft furtheraway; see Figure 1.2-1 What can we conclude as to where the target really is?

Is it at xn, at yn, or somewhere in between? Let us initially assume the radarrange measurements at time n 1 and n are very accurate; they having beenmade with a precise laser radar that can have a much more accurate rangemeasurement than a microwave radar Assume the laser radar has a rangemeasurement accuracy of 0.1 ft In this case the observation of the target at adistance 60 ft further out than predicted implies the target is going faster than

we originally estimated at time n 1, traveling 60 ft further in 10 sec, or(60 ft)=(10 sec)¼6 ft=sec faster than we thought Thus the updated targetvelocity should be

Updated velocity ¼ 200 ft=sec þ 60 ft

Figure 1.1-20 NATO SEASPARROW shipborne dedicated tracker-illuminatorantenna [3] One hundred twenty-three have been built (Photo courtesy of RaytheonCompany.)

6-ft=sec apparent velocity increase (How we choose the fraction 101 will beindicated later.) The updated velocity now becomes

if the target velocity were really 200 ft=sec, then on successive observations themeasured position of the target would be equally likely to be in front or behindthe predicted position so that on average the target velocity would not bechanged from its initial estimated value of 200 ft=sec

Putting (1.2-2) in parametric form yields

_xn ¼ _xnþ hn

yn xnT

ð1:2-3ÞThe fraction 1

10is here represented by the parameter hn The subscript n is used

to indicate that in general the parameter h will depend on time The aboveequation has a problem: The symbol for the updated velocity estimate after theFigure 1.2-1 Target predicted and measured position, xn and yn, respectively, on nthscan

measurement at time n is the same as the symbol for the velocity estimate attime n just before the measurement was made, both using the variable _xn Todistinguish these two estimates, a second subscript is added This secondsubscript indicates the time at which the last measurement was made for use inestimating the target velocity Thus (1.2-3) becomes

_x n;n ¼ _xn;n1þ hn

yn xn;n1T

!

ð1:2-4Þ

The second subscript, n 1, for the velocity estimate _xn;n1 indicates anestimate of the velocity of the target at time n based on measurement made attime n 1 and before.y The second subscript n for the velocity estimate _x n;ngiven before the equal sign above indicates that this velocity estimate uses therange measurement made at time n, that is, the range measurement yn Thesuperscript asterisk is used to indicate that the parameter is an estimate Withoutthe asterisk the parameters represent the true values of the velocity and position

of the target Figure 1.2-2 gives Figure 1.2-1 with the new notation

We now have the desired equation for updating the target velocity, (1.2-4).Next we desire the equation for updating the target position As before assumethat at time n 1 the target is at a range of 10 nautical miles (nmi) and at time

n, T¼ 10 sec later, the target with a radial velocity of 200 ft=sec is at a range

2000 ft further out As before, assume that at time n the target is actually

y This is the notation of reference 5 Often, as shall be discussed shortly, in the literature [6, 7] a caret over the variable is used to indicate an estimate.

Figure 1.2-2 Target predicted, filtered, and measured positions using new notation

observed to be 60 ft further downrange from where predicted; see Figure 1.2-2.Again we ask where the target actually is At x n;n1, at yn, or somewhere inbetween? As before initially assume a very accurate laser radar is being used forthe measurements at time n 1 and n It can then be concluded that the target is

at the range it is observed to be at time n by the laser radar, that is, 60 ft furtherdownrange than predicted Thus

Updated position¼ 10 nmi þ 2000 ft þ 60 ft ð1:2-5Þ

If, however, we assume that we have an ordinary microwave radar with a 1

of 50 ft, then the target could appear to be 60 ft further downrange than expectedjust due to the measurement error of the radar In this case we cannot reasonablyassume the target is actually at the measured range yn, at time n On the otherhand, to assume the target is at the predicted position is equally unreasonable

To allow for the possibility that the target could actually be a little downrangefrom the predicted position, we put the target at a range further down thanpredicted by a fraction of the 60 ft Specifically, we will assume the target is16of

60 ft, or 10 ft, further down in range (How the fraction 16 is chosen will beindicated later.) Then the updated range position after the measurement at time

n is given by

Updated position¼ 10 nmi þ 2000 ft þ1

6ð60 ftÞ ð1:2-6ÞPutting (1.2-6) in parametric form yields

x n;n ¼ xn;n1þ gnðyn xn;n1Þ ð1:2-7Þwhere the fraction16is represented by the parameter gn, which can be dependent

on n Equation (1.2-7) represents the desired equation for updating the targetposition

Equations (1.2-4) and (1.2-7) together give us the equations for updating thetarget velocity and position at time n after the measurement of the target range

yn has been made It is convenient to write these equations together here as thepresent position and velocity g–h track update (filtering) equations:

_x n;n ¼ _xn;n1þ hn

yn xn;n1T

!

ð1:2-8aÞ

x n;n ¼ xn;n1þ gnðyn xn;n1Þ ð1:2-8bÞThese equations provide an updated estimate of the present target velocity andposition based on the present measurement of target range ynas well as on priormeasurements These equations are called the filtering equations The estimate

x n;nis called the filtered estimate an estimate of xn at the present time based onthe use of the present measurement y as well as the past measurements This

estimate is in contrast to the prediction estimate xn;n1, which is an estimate of

xn based on past measurements The term smoothed is used sometimes in place

of the term filtered [8] ‘‘Smoothed’’ is also used [7] to indicate an estimate ofthe position or velocity of the target at some past time between the first and lastmeasurement, for example, the estimate x h;n, where n0 < h < n, n0 being thetime of the first measurement and n the time of the last measurement In thisbook, we will use the latter definition for smoothed

Often in the literature [6, 7] a caret is used over the variable x to indicate that

x is the predicted estimate x n;n1while a bar over the x is used to indicate that

x is the filtered estimate x n;n Then g–h track update equations of (1.2-8a) and(1.2-8b) become respectively

_xn ¼ ^_xnþ hn

yn ^xnT

x nþ1;n ¼ xn;nþ T _xnþ1;n ð1:2-10bÞThese equations allow us to transition from the velocity and position at time n

to the velocity and position at time nþ 1 and they are called the transitionequations We note in (1.2-10a) the estimated velocity at time nþ 1, _xnþ1;n, isequal to the value _x n;n at time n, because a constant-velocity target model isassumed

Equations (1.2-8) together with (1.2-10) allow us to keep track of a target In

a tracking radar, generally one is not interested in the present target position xnbut rather in the predicted target position xnþ1 to set up the range predictionwindows In this case (1.2-8) and (1.2-10) can be combined to give us just twoequations for doing the track update We do this by substituting (1.2-8) into(1.2-10) to yield the following prediction update equations:

_x nþ1;n ¼ _xn;n1þhn

x nþ1;n ¼ xn;n1þ T _xnþ1;nþ gnðyn xn;n1Þ ð1:2-11b)Equations (1.2-11) represent the well-known g–h tracking-filter equations.These g–h tracking-filter equations are used extensively in radar systems [5, 6,8–10] In contrast to the filtering equation of (1.2-8), those of (1.2-11) are called

prediction equations because they predict the target position and velocity at thenext scan time An important class of g–h filters are those for which g and h arefixed For this case the computations required by the radar tracker are verysimple Specifically, for each target update only four adds and three multipliesare required for each target update The memory requirements are very small.Specifically, for each target only two storage bins are required, one for the latestpredicted target velocity and one for the latest predicted target position, pastmeasurements and past predicted values not being needed for futurepredictions.

We have developed the filtering and prediction equations (1.2-8) and (1.2-10)above through a simple heuristic development Later (Section 1.2.6, Chapter 2and the second part of the book) we shall provide more rigorous developments

In the meantime we will give further commonsense insight into why theequations are optimal

In Figure 1.2-2 we have two estimates for the position of the target at time n,

x n;n1and yn The estimate yn is actually the radar measurement at time n Theestimate x n;n1 is based on the measurement made at time n 1 and allpreceding times What we want to do is somehow combine these two estimates

to obtain a new best estimate of the present target position This is the filteringproblem We have the estimates yn and x n;n1 and we would like to find acombined estimate x n;n, as illustrated in Figure 1.2-3 The problem we face ishow to combine these two estimates to obtain the combined estimate x n;n If ynand x n;n1 were equally accurate, then we would place x n;n1 exactly in themiddle between ynand x n;n1 For example, assume you weigh yourself on twoscales that are equally accurate with the weight on one scale being 110 lb andthat on the other scale being 120 lb Then you would estimate your weight based

on these two measurements to be 115 lb If on the other hand one scale weremore accurate than the other, then we would want the combined estimate of theweight to be closer to that of the more accurate scale The more accurate thescale, the closer we would place our combined estimate to it This is just whatthe filtering equation (1.2-8b) does To see this, rewrite (1.2-8b) as a positionfiltering equation:

x n;n ¼ xn;n1ð1  gnÞ þ yngn ð1:2-12Þ

Figure 1.2-3 Filtering problem mate of x n;n based on measurement yn

Esti-and prediction x 

The above equation gives the updated estimate as a weighted sum of the twoestimates The selection of the fraction gn determines whether we put thecombined estimate closer to yn or to x n;n1 For example, if yn and x n;n1 areequally accurate, then we will set gn equal to 12 In this case the combinedestimate x n;nis exactly in the middle between ynand x n;n1 If on the other hand

x n;n1 is much more accurate than yn (perhaps because the former is based onmany more measurements), then we will want to have the coefficient associatedwith yn much smaller than that associated with x n;n1 For example, in this case

we might want to pick gn¼1

5, in which case the combined estimate is muchcloser to x n;n1 How we select gn shall be shown later

1.2.2 a–b Filter

Now that we have developed the g–h filter, we are in a position to develop the

– filter To obtain the – filter, we just take (1.2-11) and replace g with and h with —we now have the – filter Now you know twice as much,knowing the – filter as well as the g–h filter

1.2.3 Other Special Types of Filters

In this section we will increase our knowledge 22-fold because we will cover 11new tracking filters Table 1.2-1 gives a list of 11 new tracking filters Theequations for all of these filters are given are given by (1.2-11) Consequently,all 11 are g–h filters Hence all 11 are – filters Thus we have increased ourtracking-filter knowledge 22 fold! You are a fast learner! How do these filtersdiffer? They differ in the selection of the weighting coefficients g and h as shall

be seen later (Some actually are identical) For some of these filters g and hdepend on n This is the case for the Kalman filter It is worthwhile emphasizingthat (1.2-11a) and (1.2-11b) are indeed the Kalman filter prediction equations,albeit for the special case where only the target velocity and position are beingtracked in one dimension Later we will give the Kalman filter for the multi-

TABLE 1.2-1 Special Types of Filters

1 Wiener filter

2 Fading-memory polynomial filter

3 Expanding-memory (or growing-memory) polynomial filter

9 Discounted least-squares g–h filter

10 Critically damped g–h filter

11 Growing-memory filter

dimensional situation involving multiple states For many of the g–h trackingfilters of Table 1.2-1, g and h are related.

1.2.4 Important Properties of g–h Tracking Filters

1.2.4.1 Steady-State Performance for Constant-Velocity Target

Assume a target with a constant-velocity trajectory and an errorless radar rangemeasurement Then in steady state the g–h will perfectly track the targetwithout any errors; see Figure 1.2-4 This is not surprising since the equationswere developed to track a constant-velocity target In the steady state theestimate of the target velocity obtained with the tracking equations of (1.2-11)will provide a perfect estimate of the target velocity if there is no range errorpresent in the radar For this case in steady state gn and hn become zero Theprediction equations given by (1.2-11) then become the transition equations ortarget equation of motion as given by (1.2-10)

1.2.4.2 For What Conditions is the Constant-Velocity

Assumption Reasonable

Assume a target having a general arbitrary one-dimensional trajectory as afunction of time t given by x(t) Expressing xðtÞ in terms of its Taylor expansionyields

xðtÞ ¼ xðtnÞ þ t _xðtnÞ þðtÞ

22! xðtnÞ

þ t3

3! _xðtn ð1:2-13ÞFor

ðtÞ22! xðtnÞ

Figure 1.2-4 The g–h filter predicts position of constant-velocity target perfectly insteady state if there are no measurement errors

xðtnþ1Þ ¼ xðtnÞ þ T _xðtnÞ ð1:2-14Þwhere we replaced t by T Equation (1.2-14) is the equation for a target having

a constant velocity Thus the assumption of a constant-velocity target is areasonable one as long as the time between observations T is small or the targetacceleration x is small or their combination is small

1.2.4.3 Steady-State Response for Target with Constant AccelerationAssume we are using the prediction equations given by (1.2-11) that weredeveloped for a target having a constant velocity but in fact the target has aconstant acceleration given by x We ask: How well does the tracking filter do?

It turns out that in steady state the constant g–h tracking filter will have aconstant prediction error given by b  for the target position that is expressed interms of the target acceleration and scan-to-scan period by [5, 12]

Equation (1.2-15) is not surprising The acceleration error is proportional to1

2(x T2) and we see that correspondingly b  is proportional to1

2ðx T2Þ Equation(1.2-15) also indicates that the steady-state error b , called the lag error, is alsoinversely proportional to the parameter h of (1.2-11a) This is not unreasonable.Assume as in Figure 1.2-2 that yn is 60 ft further downrange than predicted andinterpret the additional distance as due to the target having an increased speed

of 6 ft=sec by making hn¼ 1 In this case our tracking filter respondsimmediately to a possible increase in the target velocity, increasing the velocityimmediately by 6 ft=sec If on the other hand we thought that the location of thetarget 60 ft further downrange could be primarily due to inaccuracies in theradar range measurement, then, by only allotting a fraction of this 6 ft/sec to thetarget velocity update by setting hn¼ 1

10, the tracking filter will not respond asquickly to a change in velocity if one actually did occur In this latter caseseveral scans will be necessary before the target velocity will have increased by

6 ft=sec, if indeed the target actually did increase by 6 ft=sec Thus the larger

is h, the faster the tracking filter responds to a change in target velocity.Alternatively, the smaller is h, the more sluggish is the filter Thus quitereasonably the lag error for the filter is inversely proportional to h

When tracking a constant-accelerating target with a g–h filter, there will also

be in steady state constant lag errors for the filtered target position x n;nand thevelocity _x n;n given respectively by [12]

TABLE 1.2-2 Common Names for Steady-State

Prediction Error b  Due to Constant Acceleration

Lag errorDynamic errorSystematic errorBias errorTruncation error

since it is a systematic error rather than a random error, again caused by thetarget motion Appropriately lag error is called a bias error since the error is afixed deviation from the true value in steady state Finally, lag error is called atruncation error, since the error results from the truncation of the accelerationterm in the Taylor expansion given by (1.2-13).

1.2.4.4 Tracking Errors due to Random Range Measurement ErrorThe radar range measurement yn can be expressed as

n is a random zeromean variable with an rms of  that is the same for all n Because represents the rms of the range x measurement error, we shall replace it by xfrom here on The variance of the prediction x nþ1;n is defined as

VARðx nþ1;nÞ ¼ E½fxnþ1;n Eðxnþ1;nÞg2 ð1:2-18Þwhere E

VARðx nþ1;n) in terms of x and the tracking-filter parameters In the literatureexpressions are given for a normalized VARðxnþ1;nÞ Specifically it is givennormalized with respect to 2

x, that is, it is given for ½VARðxnþ1;nÞ=2

x Thisnormalized variance is called the variance reduction factor (VRF) Usingthe tracking prediction equations of (1.2-11), in steady state the VRF for

x nþ1;n for a constant g–h filter is given by [12; see also problems 1.2.4.4-1 and1.2.6-2]

VRFðxnþ1;nÞ ¼VARðxnþ1;nÞ

2 ¼2g

2þ 2h þ ghgð4  2g  hÞ ð1:2-19ÞThe corresponding VRFs for x n;n and _x nþ1;n are given respectively by [12]

VRFðxn;nÞ ¼VARðx n;nÞ

2 ¼2g

2þ 2h  3ghgð4  2g  hÞ ð1:2-20ÞVRFð_xnþ1;nÞ ¼VARð_x nþ1;nÞ

T2

2h2gð4  2g  hÞ ð1:2-21ÞThus the steady-state normalized prediction error is given simply in terms of gand h Other names for the VRF are given in Table 1.2-3

1.2.4.5 Balancing of Lag Error and rms Prediction Error

Equation (1.2-19) allows us to specify the filter prediction error VARðxnþ1;nÞ interms of g and h The non random lag prediction error b  is given in turn by

(1.2-15) in terms of x, T, and h For convenience let the rms prediction error(i.e., the rms of x nþ1;n) be disignated as nþ1;n In designing a tracking filterthere is a degree of freedom in choice of the magnitude of nþ1;nrelative to b .They could be made about equal or one could be made much smaller than theother Generally, making one much smaller than the other does not pay becausethe total prediction error is the sum b þ nþ1;n Fixing one of these errors andmaking the other error much smaller does not appreciably reduce the totalprediction error We would be paying a high price to reduce one of theprediction errors without significantly reducing the total prediction error.Intuitively a good approach would be a balanced system where the randomand non random error components are about the same order of magnitude Oneway to do this is to make b  equal to three times the 1 rms prediction error,that is,

3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVARðxnþ1;nÞ

q

ð1:2-23aÞ

where b  is the lag error obtained for the maximum expected targetacceleration The choice of the factor 3 above is somewhat arbitrary Onecould just as well have picked a factor 2, 1.5, or 1, as will be done later

in Section 5.10 Using (1.2-22) determines b  if nþ1;n is known Equation(1.2-15) can then in turn be used to determine the track filter update period Tfor a given maximum target acceleration x This design procedure will shortly

Variance reduction factor [5]

Variance reduction ratio [12]

Noise ratio [72]

Noise amplification factor [12]

is minimized [11] What is done is ET is plotted versus g for a given T and xand the minimum found Using this procedure yielded that the total error could

be reduced as much as 7% for 1 g  0:9 and by as much as 15% for

0 g  0:9 [11] The improvement tended to be smaller for larger g becausethe minimum ETN point tended to be broader for larger g These results wereobtained in reference 11 for the critically damped and Benedict–Bordner filtersand will be detailed in Sections 1.2.6 and 1.2.5, where these filters will beintroduced

It is worth recalling that the larger is h, the smaller is the dynamic error,however; in turn, the larger VARðxnþ1;nÞ will be and vice versa; see (1.2-15)and (1.2-19) Typically, the smaller are g and h, the larger is the dynamic errorand the smaller are the errors of (1.2-19) to (1.2-21) due to the measurementnoise In fact for the critically damped and Benedict–Bordner g–h filtersintroduced in Sections 1.2.5 and 1.2.6, nþ1;n decreases monotonically withdecreasing g while the prediction bias b nþ1;n increases monotinically withdecreasing g [11] Similarly, for the critically damped filter and steady-stateKalman g–h–k filter (often called optimum g–h–k filter) introduced in Sections1.3 and 2.4, nþ1;n decreases monotonically with decreasing h while theprediction bias b nþ1;n increases monotonically with decreasing h [11] Here thebias is calculated for a constant jerk, that is, a constant third derivative of x, ; seeSection 1.3 The subject of balancing the errors nþ1;n and b  as well asminimization of their total is revisited in Sections 5.10 and 5.11

1.2.5 Minimization of Transient Error (Benedict–Bordner Filter)Assume the g–h tracking filter is tracking a constant-velocity target and at timezero the target velocity takes a step function jump to a new constant-velocityvalue This is illustrated in Figure 1.2-6 where the target position versus time isplotted as the g–h filter input Initially the target is at zero velocity, jumping attime zero to a nonzero constant-velocity value Initially the g–h tracking filterwill not perfectly follow the target, the filter’s sluggishness to a change in target

Figure 1.2-6 Transient error resulting from step jump in velocity for target beingtracked with g–h filter

velocity being the culprit The filter output in Figure 1.2-6 illustrates this As aresult, the tracking filter has an initial error in its prediction, the differencebetween the true trajectory and the predicted trajectory, as indicated in thefigure A measure of the total transient error is the sum of the squares of thesedifferences, that is,

Dx  nþ1;n¼X1 n¼0

One would like to minimize this total transient error In the literature such aminimization was carried out for the weighted sum of the total transient errorplus the variance of the prediction error due to measurement noise errors, that

is, the minimization of

E¼ VARðxn;nþ1Þ þ Dxnþ1;n ð1:2-26ÞThis minimization was done for a step jump in velocity Rather thanconcentrating merely on the g–h tracking filters, which minimize the totalerror given in (1.2-26), what was done in the literature was to find a filter among

a much wider class of tracking filters minimizing the total error of the (1.2-26).From this much wider class of filters the filter minimizing the total error of(1.2-26) is (surprise) the constant g–h tracking filter of (1.2-11) with g and hrelated by

h¼ g2

Table 1.2-4 gives a list of values for nþ1;n=x; g, and h for the Benedict–Bordner filter The parameter
in the g–h filter will be explained shortly, at theend of Section 1.2.6 Basically, for the present think of it as just a convenientindex for the tabulation in Table 1.2-4 In the literature the Benedict–Bordnerfilter has also been referred to as an optimum g–h filter [11, 12]

The transient error in x nþ1;n defined by (1.2-25) for any constant g–h filter isgiven by [6, 12]

Dx  n;nþ1 ¼v

2T2ð2  gÞ

where v is the step change in velocity that produces the filter transient error.The corresponding transient errors for x n;n and _x n;n are given respectively by[12]

Dx  n; n¼v

2T2ð2  gÞð1  gÞ2

D_x  nþ1;n ¼v

2½g2ð2  gÞ þ 2hð1  gÞ

Figure 1.2-7 gives convenient design curves of three times the normalizedprediction error 3nþ1;n=x and normalized b  for the Benedict–Bordner filterversus g Using these curves the design point at which 3nþ1;n ¼ b can easily

be found Figure 1.2-8 plots the sum of these normalized errors

versus g for different values of the normalized maximum acceleration given by

AN ¼ T2xmax=x These curves allow us to obtain the Benedict–Bordner filters,which minimizes the total error given by (1.2-31) These minimum total errordesigns are plotted in Figure 12-9 Problems 1.2.5-1 and 1.2.5-2 compareBenedict–Bordner filter designs obtained using (1.2-23) and obtained for when(1.2-31) is minimized

TABLE 1.2-4 Smoothing Constant for Benedict–

1.2.6 New Approach to Finding Optimum g–h Tracking Filters

(The Critically Damped Filter)

To obtain further insight into the development of tracking equations, let us startafresh with a new point of view We take an approach that is optimal in someother sense We shall use a very simple commonsense approach

Let y0; y1; ; y6be the first seven range measurements made of the target to

be tracked These observations could be plotted as shown in Figure 1.2-10 Withthese measurements we would like to now predict where the target will mostFigure 1.2-7 Normalized bias error b N¼ b =x and prediction error ¼ nþ1;n=x

versus weight g for Benedict–Bordner g–h filter (After Asquith and Woods [11].)

versus weight g for BenedictBordner gh filter (After Asquith and Woods [11].)