Embed Size (px)
Citation preview
Brigham Young University Brigham Young University
BYU ScholarsArchive BYU ScholarsArchive
Theses and Dissertations
2013-11-04
Simulating the Performance of Tracking a Spinning Missile at C-Simulating the Performance of Tracking a Spinning Missile at C-
Band Band
Darren Robert Kartchner Brigham Young University - Provo
Follow this and additional works at: https://scholarsarchive.byu.edu/etd
Part of the Electrical and Computer Engineering Commons
BYU ScholarsArchive Citation BYU ScholarsArchive Citation Kartchner, Darren Robert, "Simulating the Performance of Tracking a Spinning Missile at C-Band" (2013). Theses and Dissertations. 3877. https://scholarsarchive.byu.edu/etd/3877
This Thesis is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact [email protected], [email protected].
Simulating the Performance of Tracking a Spinning Missile at C-Band
Darren Kartchner
A thesis submitted to the faculty ofBrigham Young University
in partial fulfillment of the requirements for the degree of
Master of Science
Michael D. Rice, ChairBrian D. Jeffs
Randal W. Beard
Department of Electrical and Computer Engineering
Brigham Young University
December 2013
Copyright c© 2013 Darren Kartchner
All Rights Reserved
ABSTRACT
Simulating the Performance of Tracking a Spinning Missile at C-Band
Darren KartchnerDepartment of Electrical and Computer Engineering
Master of Science
The amplitude fluctuation induced by a spinning missile acts as a disturbance ontracking schemes that use sequential lobing (e.g., conscan). In addition, if a tracking systemconverts from S-band to C-band, the beamwidth is narrower and the wrap-around antenna onthe missile requires more patches, and so the margin of error for tracking decreases. Trackingperformance is simulated with a spinning missile with ballistic and fly-by trajectories whilerunning at C-band. The spinning missile causes a periodic component in the pointing error,and when the scan frequency is an integer multiple of the roll rate, several tracking schemeslose track of the target. Remedial techniques are discussed, including increasing the scanfrequency and using simultaneous (monopulse) tracking rather than sequential lobing.
Keywords: aeronautical telemetry, tracking
ACKNOWLEDGMENTS
There are several people whose contributions to the development of this paper are
greatly appreciated: Mr. Steve O’Neill (Tybrin, Edwards AFB), Mr. Bob Selbrede (JT3, Ed-
wards AFB), Mr. Mihail Mateescu (TCS Inc.), Mr. Scott Kujiroaka (NAVAIR–Pt. Mugu),
Mr. Filberto Macias (WSMR), Mr. Juan M. Guadiana (WSMR), Mr. Nathan King (46
RANSS/TSRI, Eglin AFB).
This work was supported in part by the Test Resource Management Center (TRMC)
Test and Evaluation Science and Technology (T&E/S&T) Program through a grant to BYU
from the US Army Program Executive Office for Simulation, Training, and Instrumentation
(PEO STRI) under contract W900KK-09-C-0016. Any opinions, findings and conclusions or
recommendations expressed in this material are those of the author and do not necessarily
reflect the views of the TRMC and T&E/S&T Program and/or PEO STRI. The Executing
Agent and Program Manager work out of the AFFTC.
Table of Contents
List of Tables vi
List of Figures vii
1 Introduction 1
1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Sequential Lobing 5
2.1 Types of Sequential Lobing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Conical Scan (Conscan) . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Analog Conscan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.3 Lissajous Scan and Rosette Scan . . . . . . . . . . . . . . . . . . . . 8
2.2 Impact of Spinning Missile on Sequential Lobing . . . . . . . . . . . . . . . . 10
3 Sequential Lobing Simulation 13
3.1 Antenna Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Ballistic Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Fly-by Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Simultaneous Lobing 40
4.1 Simulating a Monopulse Tracker . . . . . . . . . . . . . . . . . . . . . . . . . 41
iv
5 Conclusion 46
Bibliography 48
v
List of Tables
1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
5.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
vi
List of Figures
1.1 Roll patterns for two conformal wrap-around antennas, operating at S-band(2250 MHz) and at C-band (5135 MHz). . . . . . . . . . . . . . . . . . . . . 2
1.2 Gain patterns for an 8-foot parabolic reflector antenna with η = 0.7, operatingat S-band (2250 MHz) and at C-band (5135 MHz). . . . . . . . . . . . . . . 3
2.1 Normalized S-curves for least squares conscan and DFT conscan. The singledotted line denotes the half–power beamwidth. . . . . . . . . . . . . . . . . . 7
2.2 Normalized S-curves for analog conscan when mixing with a sine wave andwhen mixing with a bandpassed square wave, respectively. The single dottedline denotes the half–power beamwidth. . . . . . . . . . . . . . . . . . . . . . 9
2.3 Feed path for (a) Lissajous scan and (b) Rosette scan. The feed path forconscan is included for reference. . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Normalized S-curves for Lissajous and Rosette scans. The dotted line repre-sents the half-power beamwidth. . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Average azimuth and elevation error estimates of a stationary, rotating targetat boresight, using both methods of conscan. . . . . . . . . . . . . . . . . . . 12
3.1 Block diagram of the antenna controller . . . . . . . . . . . . . . . . . . . . . 14
3.2 Block diagram of the inner loop of the controller . . . . . . . . . . . . . . . . 14
3.3 Block diagram of the outer loop of the controller, with the inner loop treatedas a unity gain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4 Diagram of the ballistic missile simulation . . . . . . . . . . . . . . . . . . . 17
3.5 Azimuth and elevation pointing errors for a non-spinning ballistic missile,using (a) least squares conscan and (b) DFT conscan. . . . . . . . . . . . . . 18
vii
3.6 Azimuth and elevation pointing errors for a non-spinning ballistic missile,using sliding window (a) least squares conscan and (b) DFT conscan, for4T = T/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.7 Azimuth and elevation pointing errors for a non-spinning ballistic missile,using analog conscan, mixed by (a) a sine wave and (b) a bandpassed squarewave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.8 Azimuth and elevation pointing errors for a non-spinning ballistic missile,using (a) Lissajous scan and (b) Rosette scan. . . . . . . . . . . . . . . . . . 21
3.9 Azimuth and elevation pointing errors for a ballistic missile spinning at 2 Hz,using (a) least squares conscan and (b) DFT conscan. . . . . . . . . . . . . . 22
3.10 Azimuth and elevation pointing errors for a ballistic missile spinning at 2Hz, using sliding window (a) least squares conscan and (b) DFT conscan, for4T = T/4. Note that sliding window least squares conscan loses sight of thetarget. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.11 Azimuth and elevation pointing errors for a ballistic missile spinning at 2 Hz,using analog conscan, mixed by (a) a sine wave and (b) a bandpassed squarewave. Note that in the second case the tracker loses sight of the target. . . . 24
3.12 Azimuth and elevation pointing error for a ballistic missile spinning at 2 Hz,using (a) Lissajous scan and (b) Rosette scan. . . . . . . . . . . . . . . . . . 25
3.13 Azimuth and elevation pointing errors for a ballistic missile spinning at 5 Hz,using (a) least squares conscan and (b) DFT conscan. . . . . . . . . . . . . . 26
3.14 Azimuth and elevation pointing errors for a ballistic missile spinning at 5Hz, using sliding window (a) least squares conscan and (b) DFT conscan, for4T = T/4. Note that sliding window least squares conscan loses sight of thetarget. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.15 Azimuth and elevation pointing errors for a ballistic missile spinning at 5 Hz,using analog conscan, mixed by (a) a sine wave and (b) a bandpassed squarewave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.16 Azimuth and elevation pointing error for a ballistic missile spinning at 5 Hz,using (a) Lissajous scan and (b) Rosette scan. Note that the tracker losessight of the target when using Rosette scan. . . . . . . . . . . . . . . . . . . 29
3.17 Diagram of the fly-by missile simulation . . . . . . . . . . . . . . . . . . . . 30
3.18 Pointing error for least squares conscan and DFT conscan for a non-spinningfly-by missile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
viii
3.19 Pointing error for sliding least squares conscan and sliding DFT conscan fora non-spinning fly-by missile. . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.20 Pointing error for analog conscan, mixing with a sine wave and a bandpassedsquare wave, for a non-spinning fly-by missile. . . . . . . . . . . . . . . . . . 32
3.21 Pointing error for Lissajous scan and Rosette scan for a non-spinning fly-bymissile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.22 Pointing error for least squares conscan and DFT conscan for a fly-by missilespinning at 2 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.23 Pointing error for sliding least squares conscan and sliding DFT conscan fora fly-by missile spinning at 2 Hz. Note that sliding window least squaresconscan loses sight of the target. . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.24 Pointing error for analog conscan, mixing with a sine wave and a bandpassedsquare wave, for a fly-by missile spinning at 2 Hz. Note that in the secondcase the tracker loses sight of the target. . . . . . . . . . . . . . . . . . . . . 34
3.25 Pointing error for Lissajous scan and Rosette scan for a fly-by missile spinningat 2 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.26 Pointing error for least squares conscan and DFT conscan for a fly-by missilespinning at 5 Hz. Note that DFT conscan loses track of the target. . . . . . 35
3.27 Pointing error for sliding least squares conscan and sliding DFT conscan fora fly-by missile spinning at 5 Hz. Note that sliding window least squaresconscan loses sight of the target. . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.28 Pointing error for analog conscan, mixing with a sine wave and a bandpassedsquare wave, for a fly-by missile spinning at 5 Hz. Note that in both cases,the tracker loses sight of the target. . . . . . . . . . . . . . . . . . . . . . . . 36
3.29 Pointing error for Lissajous scan and Rosette scan for a fly-by missile spinningat 5 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.30 Average amplitude variance per scan as a function of scan frequency, for rollrates of 2 Hz and 5 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.31 Pointing error for least squares conscan and DFT conscan for a fly-by missilespinning at 5 Hz, for a scan frequency of 50 Hz. The number of samples percycle remains constant at 40. Compare with Figure 3.26. . . . . . . . . . . . 38
3.32 Pointing error for sliding least squares conscan and sliding DFT conscan for afly-by missile spinning at 5 Hz, for a scan frequency of 50 Hz. Compare withFigure 3.27. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
ix
3.33 Pointing error for analog conscan, mixing with a sine wave and a bandpassedsquare wave, for a fly-by missile spinning at 5 Hz, for a scan frequency of 50Hz. Compare with Figure 3.28. . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.34 Pointing error for Lissajous scan and Rosette scan for a fly-by missile spinningat 5 Hz, for a scan frequency of 50 Hz. Compare with Figure 3.29. . . . . . . 39
4.1 A block diagram of the monopulse tracking method, reproduced from [1]. . . 41
4.2 Normalized S-curves for a monopulse tracker with a carrier frequency of 5135MHz. The single dotted line is at the half-power beamwidth. . . . . . . . . . 42
4.3 A comparison of monopulse tracking and the various methods of sequentiallobing for a non-spinning fly-by missile. The scan frequency is 50 Hz. . . . . 42
4.4 Windowed mean pointing error for monopulse tracking and the various meth-ods of sequential lobing for a fly-by missile spinning at 5 Hz. The scan fre-quency is 50 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.5 Windowed pointing error variance of monopulse tracking and the variousmethods of sequential lobing for a fly-by missile spinning at 5 Hz. The scanfrequency is 50 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
x
Chapter 1
Introduction
1.1 Notation
Table 1.1: Notation of the Paper
Variable DefinitionG0 Boresight gainJ1(·) Modified Bessel function of the first orderk Wavenumber of carrier frequencyD Diameter of parabolic reflectorλ Wavelength of carrier frequencyη Antenna efficiencya(t) Received signal amplitudeN Number of samples per scan cycleT Time spacing between samples of signal amplitude
a(nT ) Sampled signal amplitudef0 Scan frequency (in rotations per second)ε(t) Angular displacement between target and boresightr Squint angleεaz Azimuth component of pointing errorεel Elevation component of pointing errorA0 Average amplitude over one scan cycleA(m) m-th component of the inverse FFT of a(nT )xi, yi Cartesian position of tracking feed at sample i during a scanti Time at sample i during a scan
αL, βL Harmonics used to determine feed path during a Lissajous scanαR, βR Harmonics used to determine feed path during a Rosette scanω3dB Controller 3 dB bandwidth (in rad/s)ζ Controller damping constant
1
1.2 Introduction
As frequency bands are auctioned, allocated, and reallocated, equipment designed
to operate at specific frequencies will experience changes in performance. In particular,
the transition from lower S-band (2000-2300 MHz) to lower C-band (4000-5500 MHz) is
of interest in the field of missile range testing. If the carrier frequency of a transmitter-
receiver system changes, the gain patterns of the antennas change as well. A missile is
usually equipped with a wrap-around antenna comprising patches spaced approximately
half a wavelength apart [2]. At higher frequencies, more patches are needed to maintain
proper spacing (assuming the radius of the missile remains constant). Generally speaking,
the gain pattern of a wrap-around transmit antenna exhibits more lobing at a higher carrier
frequency. Figure 1.1 compares two actual roll patterns of conformal wrap-around antennas
designed for a missile with a 5-inch diameter.
−15
−5
5
30
210
60
240
90
270
120
300
150
330
180 0
2250 MHz5135 MHz
Figure 1.1: Roll patterns for two conformal wrap-around antennas, operating at S-band (2250MHz) and at C-band (5135 MHz).
2
The radiation pattern of the transmit antenna is non-isotropic. Therefore, if the
missile spins, the amplitude of the received signal will fluctuate [3]. For tracking systems
that estimate the target’s position based on signal amplitude, these fluctuations induced by
the spinning of the missile act as a disturbance on the tracker.
In addition to the changes on the transmitting end, the receiver’s performance differs
as the carrier frequency changes. For an ideal, uniformly illuminated parabolic reflector, the
gain pattern is described in [4] as
G(φ) = G0 × 2J1 (0.5kD sin (φ))
0.5kD sin (φ), (1.1)
where G0 is
G0 =
(πD
λ
)2
η. (1.2)
Figure 1.2 illustrates the gain patterns for an 8-foot parabolic reflector operating at 2250
MHz and 5135 MHz. The well-known tradeoff between boresight gain and beamwidth is
apparent upon first glance. For the purposes of this paper, beamwidth is of greater concern.
The beamwidth narrows upon transitioning from S-band to C-band; this imposes a smaller
margin of error in terms of tracking.
−10 −5 0 5 10
−10
0
10
20
30
40
Azimuth Angle (deg)
Gai
n (d
B)
2250 MHz5135 MHz
Figure 1.2: Gain patterns for an 8-foot parabolic reflector antenna with η = 0.7, operatingat S-band (2250 MHz) and at C-band (5135 MHz).
3
This paper will address two questions: what effect does a spinning target have on
tracking at C-band, and how can it affect trade-off decisions? In Chapter 2, several imple-
mentations of sequential lobing tracking methods are detailed, including conscan, Lissajous
scan, and Rosette scan. In addition, the impact of the spinning target on scanning trackers is
reviewed, particularly when the scan frequency is an integer multiple of the target roll rate.
In Chapter 3, simulations are outlined for tracking a ballistic target and a fly-by target, and
the results are plotted. The controller that directs boresight pointing is derived. A section is
dedicated to the effects of increasing scan frequency and the impact on simulations. Chapter
4 compares the results of Chapter 3 with monopulse, a simultaneous lobing tracking method
which is unaffected by the signal fluctuation from a spinning missile. The paper concludes
by comparing each tracking method’s performance with its complexity of implementation in
order to see where trade-offs occur in the scenario of a spinning target.
4
Chapter 2
Sequential Lobing
Sequential lobing entails tracking techniques where the pointing error is estimated by
received signal amplitude over a period of time. Typically, the receiver feed is attached to a
motor which steers the feed over a periodic path off boresight. Each time the feed completes
a cycle, the received signal amplitude a(t) is then used to estimate the pointing error. There
are a variety of methods in which sequential lobing is used.
2.1 Types of Sequential Lobing
2.1.1 Conical Scan (Conscan)
In conical scan (or conscan), the receiver feed deviates slightly off boresight, at an
angle called the squint angle [5]. The squint angle is selected so that the loss from pointing
away from boresight is about 0.1 dB [6]. The antenna feed is simply rotated about the
boresight axis, and so the gain pattern follows a conical trajectory (hence the name). The
component of a(t) at the scan frequency f0 is used to estimate the azimuth and elevation
pointing errors. The in-phase and quadrature elements of the conscan frequency component
are used to determine azimuth and elevation pointing error, respectively. These signals are
used to drive an automatic gain control (AGC), which drives the motors of the antenna [7].
Conscan can either be implemented in discrete time or in analog. In discrete-time versions
of conscan, the signal amplitude is sampled:
a(t)→ a(nT ), n = 0, 1, · · · , N − 1. (2.1)
In this paper, two discrete-time methods of executing conscan are used: the least squares
method, and the DFT method.
5
Least Squares Method
The angular displacement between boresight and the target as a function of time is
given by [8]:
ε(t)2 = r2 + ε2az + ε2el − 2rεaz cos(f0t)− 2rεel sin(f0t). (2.2)
The angular displacement ε(t) determines the amplitude of the received signal according
to the gain pattern, as seen in (1.1). Under the assumption that the target is relatively
stationary during the scan cycle, the only time-varying elements of (2.2) are the sine and
cosine. After some approximation, [8] writes the following vector equation:
Pc(t) =[1 cos(f0t) sin(f0t)
]P0
P0((ks/h)εaz)
P0((ks/h)εel)
, (2.3)
where ks/h is the conscan slope divided by the half-power beamwidth, and P0 is the average
power over the scan period. When applied to a(nT ), (2.3) becomes
a(T )
a(2T )...
a(NT )
=
1 cos
(2π
N
)sin
(2π
N
)1 cos
(2
2π
N
)sin
(2
2π
N
)...
......
1 cos(2π) sin(2π)
A0
A0((ks/h)εaz)
A0((ks/h)εel)
, (2.4)
or
A = Y C. (2.5)
From (2.5), C is estimated as
C = (Y >Y )−1Y >A, (2.6)
where (Y >Y )−1Y > can be pre-computed. At this point, εaz is estimated by dividing the
second element of C by the first element of C, then dividing by the slope of the resulting
S-curve. The elevation error is estimated the same way, substituting the second element of
C with the third element.
6
DFT Method
In this method, pointing error is estimated from the frequency component of a(nT )
at the conscan frequency. This can be found by using the inverse FFT [9]:
A(m) = IFFT {a(nT )} =1
N
N−1∑n=0
a(nT ) exp
(j
2πmn
N
), for m = 0, 1, · · · , N − 1 (2.7)
The azimuth and elevation error estimates, then, are simply the real and imaginary parts of
A(1), respectively, divided by the slope of the resultant S-curves [10].
Figure 2.1 compares examples of S-curves using both least squares and DFT methods
for a carrier frequency of 5135 MHz and a conscan frequency of 25 Hz (note that the slopes
of the S-curves at the origin are normalized to 1).
−1 −0.5 0 0.5 1−5
0
5
Leas
t squ
ares
err
or e
stim
ate
(deg
)
−1 −0.5 0 0.5 1−2
−1
0
1
2
Azimuth angle error (deg)
DFT
err
or e
stim
ate
(deg
)
Figure 2.1: Normalized S-curves for least squares conscan and DFT conscan. The singledotted line denotes the half–power beamwidth.
7
Sliding Window Conscan
Using sliding window conscan, the error estimate updates every N4T seconds, where
4T < T . The first update occurs once the feed completes a full cycle; then at every update,
the estimate is formed using the most recent N samples [11]. This technique can be used with
the least squares method by permuting the rows of the pseudo-inverse matrix (Y >Y )−1Y >
appropriately; or with the DFT method by appropriately shifting the phase of a(nT ). From
[11], the primary benefit of sliding window conscan over regular conscan is that sliding
window resolves pointing error faster when there are abrupt changes in target position.
2.1.2 Analog Conscan
Analog conscan functions similarly to DFT conscan, in that the pointing error is
estimated by isolating the frequency component of the amplitude at the scan frequency.
However, analog conscan uses a(t) to estimate pointing error instead of a(nT ). The amplitude
is multiplied by a cosine wave with frequency f0, which mixes the frequency component to
baseband. The cosine wave can be produced using a local oscillator or by constructing a
square wave and applying a passband filter tuned to f0 [12]. A lowpass filter is then applied
to isolate the DC component of the signal, which then corresponds to the azimuth error
signal. The elevation error signal is found by multiplying a(t) by a sine wave, then following
the same process. The primary benefit of analog conscan is its low complexity and cost,
opposed to the additional signal processing that least squares or DFT conscan requires. The
corresponding S-curves for analog conscan can be seen in Figure 2.2.
2.1.3 Lissajous Scan and Rosette Scan
Lissajous scan is similar to least squares conscan, but the tracking feed follows a path
defined by
xiyi
=
r sinαLωti
r sin βLωti
. (2.8)
8
−1 −0.5 0 0.5 1−2
−1
0
1
2
Azimuth angle error (deg)
E−s
can
outp
ut (d
eg)
−1 −0.5 0 0.5 1−2
−1
0
1
2
Azimuth angle error (deg)
E−s
can
outp
ut (d
eg)
Figure 2.2: Normalized S-curves for analog conscan when mixing with a sine wave and whenmixing with a bandpassed square wave, respectively. The single dotted line denotes the half–power beamwidth.
Rosette scan follows a path defined by
xiyi
=
r sinαRωti − r sin βRωti
r cosαRωti + r cos βRωti
. (2.9)
Using the values αL = 4, βL = 3, αR = 1, and βR = 3 (values taken from [11]), the
path of the feed for Lissajous and Rosette scans can be seen in Figure 2.3 (with a circle of
radius r for comparison). According to literature, both tracking methods are used more often
for deep space tracking, where a scan period can be 60–120 seconds, much too long to track
a missile. However, let us make the assumption that equipment is available to construct
a ground antenna capable of maneuvering the feed along the paths described in (2.8) and
(2.9). Figure 2.4 depicts the S-curves of the Lissajous and Rosette scans when using the
same parameters for the previous S-curves.
9
−0.4 −0.2 0 0.2 0.4−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Azimuth (deg)
Ele
vatio
n (d
eg)
(a)
−0.4 −0.2 0 0.2 0.4−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Azimuth (deg)
Ele
vatio
n (d
eg)
(b)
Figure 2.3: Feed path for (a) Lissajous scan and (b) Rosette scan. The feed path for conscanis included for reference.
2.2 Impact of Spinning Missile on Sequential Lobing
The main feature of sequential lobing is the use of variation in received signal am-
plitude over one scan period to estimate pointing error. Therefore, when the transmitted
signal amplitude changes due to the missile spinning, the result is a disturbance on sequential
lobing. The missile’s roll rate has a large role in dictating how much disturbance occurs. To
10
−1 −0.5 0 0.5 1−4
−2
0
2
4
Azimuth angle error (deg)
Liss
ajou
s sc
an o
utpu
t (de
g)
−1 −0.5 0 0.5 1−4
−2
0
2
4
Azimuth angle error (deg)
Ros
ette
sca
n ou
tput
(deg
)
Figure 2.4: Normalized S-curves for Lissajous and Rosette scans. The dotted line representsthe half-power beamwidth.
illustrate this point, Figure 2.5 plots the pointing error estimates produced by least squares
and DFT conscan when a rotating target is at boresight (note that when using either method
of conscan, a target at boresight should output zero error). The gain pattern of the missile
is the C-band gain pattern from Figure 1.1. The abscissa is the roll rate of the missile, and
the ordinate is the average output of conscan after the missile spins several times. Upon
studying this figure, we note there are large peaks whenever the conscan frequency is an
integer multiple of the roll rate. This is because conscan picks up the harmonic generated
by the spinning missile, resulting in an especially bad disturbance.
11
0 5 10 15 20 25
−0.2
0
0.2
Azi
mut
h po
intin
g er
ror (
deg)
Least SquaresDFT
0 5 10 15 20 25
−0.2
0
0.2
Roll Rate (Hz)
Ele
vatio
n po
intin
g er
ror (
deg)
Least SquaresDFT
Figure 2.5: Average azimuth and elevation error estimates of a stationary, rotating target atboresight, using both methods of conscan.
12
Chapter 3
Sequential Lobing Simulation
In this chapter, each of the tracking methods described in Chapter 2 is tested for
performance in a number of scenarios. Section 3.2 deals with tracking a ballistic missile
fired in an arc, and Section 3.3 contains the data from tracking a fly-by missile. For each
trajectory, the target spins at 0, 2, and 5 Hz. In all scenarios, noise and attenuation due to
range are neglected, under the assumption that any tracking error in such a setting will occur
in a more realistic environment. However, the dynamics of the antenna pointing controller
will be modeled.
3.1 Antenna Controllers
The tracking algorithm outputs a current used to drive the electric motors which
steer the antenna. Two motors drive the antenna: one for azimuth, and one for elevation.
For these simulations, the azimuth and elevation controllers are independent and identical.
The driving current is proportional to the torque which the motor exerts. The torque affects
the pointing angle of the antenna in the following way:
τ = Jθ, (3.1)
where τ is torque, J is the mass moment of inertia of the antenna, and θ is the angular
acceleration. The angular velocity, then, is the integration of θ, and the angular position,
the integration of the angular velocity. A PI controller is used to control the torque applied
to the system, and another is used to control the velocity. Figure 3.1 is a block diagram of
the controller. Note that J does not appear in the diagram because the PI controller gains
can be adjusted proportionally to J .
13
PI Controller
Tracking Algorithm _
+ 1s
PI Controller
velocity loop
position loop
θθ
1s
Figure 3.1: Block diagram of the antenna controller
Successive loop closure is used to determine the gains to tune the controllers [13].
Under successive loop closure, the innermost loop (see Figure 3.2) is tuned first. The transfer
_+ 1
sPI
Controller
velocity loop
Figure 3.2: Block diagram of the inner loop of the controller
function of a PI controller is kp + kis
. And so, the transfer function of the inner loop is
H(s) =kps+ ki
s2 + kps+ ki. (3.2)
This is a second-order system, whose canonical form is
H(s) =2ζωns+ ω2
n
s2 + 2ζωns+ ω2n
. (3.3)
14
From (3.3), values for the PI gains can be derived from ζ and ωn:
kp = 2ζωn, (3.4)
ki = ω2n. (3.5)
To find the 3 dB bandwidth, consider the magnitude squared of the transfer function:
|H(jω)|2 =4ζ2ω2
nω2 + ω4
n
(ω2n − ω2)2 + 4ζ2ω2
nω2. (3.6)
The 3 dB bandwidth of the inner loop is the frequency ω3dB where (3.6) is 1/2:
1
2=
4ζ2ω2nω
23dB + ω4
n
(ω2n − ω2
3dB)2 + 4ζ2ω2nω
23dB
. (3.7)
Solving for ω3dB results in
ω3dB = ωn
√2ζ2 + 1 +
√(2ζ2 + 1)2 + 1. (3.8)
Therefore, for a desired 3 dB bandwidth ω3dB and a damping constant ζ, the PI gains are
kp =2ζω3dB√
2ζ2 + 1 +√
(2ζ2 + 1)2 + 1, (3.9)
ki =ω23dB
2ζ2 + 1 +√
(2ζ2 + 1)2 + 1. (3.10)
Once the inner loop has been tuned, the next step is to tune the outer loop. Assuming
that the 3 dB bandwidth of the outer loop is entirely within the 3 dB bandwidth of the inner
loop, the inner loop can be treated as a unity gain from the perspective of the outer loop. In
addition, the S-curves of all tracking methods from Chapter 2 support the argument that for
small pointing errors, the tracking algorithm block can be modeled as a negative feedback
sum block (see Figure 3.3). The resulting block model for the outer loop is mathematically
equivalent to the block model of the inner loop, and so (3.9) and (3.10) can also be used to
determine the PI controller gains for the outer loop. The only difference is that the 3 dB
15
bandwidth for the outer loop should be selected so that the inner loop can be approximated
as unity gain over all of the outer loop bandwidth. A good rule of thumb is for the outer
loop bandwidth to be one-tenth of the inner loop bandwidth. For example, if the desired
loop bandwidth for the overall system is 3 Hz, the outer loop 3 dB bandwidth should be 6π
rad/s, and the inner loop 3 dB bandwidth should be 60π rad/s.
PI Controller
θ1s1θ
+_
Figure 3.3: Block diagram of the outer loop of the controller, with the inner loop treated asa unity gain.
3.2 Ballistic Trajectory
To simulate performance while tracking a spinning ballistic missile, consider the sce-
nario illustrated in Figure 3.4. A missile fired the ground, from south to north, 15◦ from
the ground. The missile has a velocity of 1012 m/s (about Mach 3), and the missile is fired
15 km east and 27 km south of the tracking antenna. The antenna controllers are tuned to
a loop bandwidth of 3 Hz, with all damping constants set to ζ = 0.7071. Figure 3.4 is a
diagram of the simulation. Figures 3.5–3.16 plot the azimuth and elevation pointing errors
of the tracking algorithms mentioned in Chapter 2, for three cases: when the missile is not
spinning, when the missile spins at a rate of 2 Hz (a rate coprime with the scan frequency),
and when the missile spins at a rate of 5 Hz (a harmonic of the scan frequency). The gain
pattern of the target is the C-band pattern from Figure 1.1. Based on the results, when
the target is not spinning, all methods can track with relatively small error, with the largest
error occuring at the time of largest angular velocity. When the missile spins at 2 Hz, a
periodic component is introduced to the error, and some tracking methods lose sight of the
16
target. When the missile spins at 5 Hz, the RMS pointing error increases, and even more
methods lose track.
15 km
54 km
v = 1012 m/s
θ = 15°
N
Figure 3.4: Diagram of the ballistic missile simulation
17
0 10 20 30 40 500
0.1
0.2
0.3
0.4
Time (s)
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50−0.1
−0.05
0
0.05
0.1
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(a)
0 10 20 30 40 500
0.1
0.2
0.3
0.4
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50−0.1
−0.05
0
0.05
0.1
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(b)
Figure 3.5: Azimuth and elevation pointing errors for a non-spinning ballistic missile, using(a) least squares conscan and (b) DFT conscan.
18
0 10 20 30 40 500
0.1
0.2
0.3
0.4
Time (s)
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50−0.04
−0.02
0
0.02
0.04
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(a)
0 10 20 30 40 500
0.1
0.2
0.3
0.4
Time (s)
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50−0.04
−0.02
0
0.02
0.04
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(b)
Figure 3.6: Azimuth and elevation pointing errors for a non-spinning ballistic missile, usingsliding window (a) least squares conscan and (b) DFT conscan, for 4T = T/4.
19
0 10 20 30 40 500.05
0.1
0.15
0.2
0.25
Time (s)
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50−0.1
−0.05
0
0.05
0.1
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(a)
0 10 20 30 40 500
0.1
0.2
0.3
0.4
Time (s)
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50−0.1
−0.05
0
0.05
0.1
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(b)
Figure 3.7: Azimuth and elevation pointing errors for a non-spinning ballistic missile, usinganalog conscan, mixed by (a) a sine wave and (b) a bandpassed square wave.
20
0 10 20 30 40 500
0.05
0.1
0.15
0.2
Time (s)
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50−0.04
−0.02
0
0.02
0.04
Ele
vatio
n po
intin
g er
ror (
deg)
Time (s)
(a)
0 10 20 30 40 500
0.05
0.1
0.15
0.2
Time (s)
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50−0.04
−0.02
0
0.02
0.04
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(b)
Figure 3.8: Azimuth and elevation pointing errors for a non-spinning ballistic missile, using(a) Lissajous scan and (b) Rosette scan.
21
0 10 20 30 40 50−1
−0.5
0
0.5
1
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50
−0.5
0
0.5
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(a)
0 10 20 30 40 50−1
−0.5
0
0.5
1
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50
−0.5
0
0.5
Ele
vatio
n po
intin
g er
ror (
deg)
Time (s)
(b)
Figure 3.9: Azimuth and elevation pointing errors for a ballistic missile spinning at 2 Hz,using (a) least squares conscan and (b) DFT conscan.
22
0 10 20 30 40 500
10
20
30
40
50
Time (s)
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50
−40
−20
0
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(a)
0 10 20 30 40 50−0.5
0
0.5
1
Time (s)
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50−0.5
0
0.5
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(b)
Figure 3.10: Azimuth and elevation pointing errors for a ballistic missile spinning at 2 Hz,using sliding window (a) least squares conscan and (b) DFT conscan, for 4T = T/4. Notethat sliding window least squares conscan loses sight of the target.
23
0 10 20 30 40 50−0.5
0
0.5
1
Time (s)
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50−0.5
0
0.5
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(a)
0 10 20 30 40 50−0.5
0
0.5
1
1.5
2
Time (s)
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50−2
−1.5
−1
−0.5
0
0.5
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(b)
Figure 3.11: Azimuth and elevation pointing errors for a ballistic missile spinning at 2 Hz,using analog conscan, mixed by (a) a sine wave and (b) a bandpassed square wave. Note thatin the second case the tracker loses sight of the target.
24
0 10 20 30 40 50−0.5
0
0.5
1
Time (s)
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50−0.5
0
0.5
1
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(a)
0 10 20 30 40 50−1
−0.5
0
0.5
1
Time (s)
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50−1
−0.5
0
0.5
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(b)
Figure 3.12: Azimuth and elevation pointing error for a ballistic missile spinning at 2 Hz,using (a) Lissajous scan and (b) Rosette scan.
25
0 10 20 30 40 50−1.5
−1
−0.5
0
0.5
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50−1.5
−1
−0.5
0
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(a)
0 10 20 30 40 50−1.5
−1
−0.5
0
0.5
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50−1.5
−1
−0.5
0
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(b)
Figure 3.13: Azimuth and elevation pointing errors for a ballistic missile spinning at 5 Hz,using (a) least squares conscan and (b) DFT conscan.
26
0 10 20 30 40 50−300
−200
−100
0
Time (s)
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50−100
0
100
200
300
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(a)
0 10 20 30 40 50−1
−0.5
0
0.5
Time (s)
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 500
0.5
1
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(b)
Figure 3.14: Azimuth and elevation pointing errors for a ballistic missile spinning at 5 Hz,using sliding window (a) least squares conscan and (b) DFT conscan, for 4T = T/4. Notethat sliding window least squares conscan loses sight of the target.
27
0 10 20 30 40 50−1
−0.5
0
0.5
Time (s)
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 500
0.5
1
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(a)
0 10 20 30 40 50−1
−0.5
0
0.5
Time (s)
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 500
0.5
1
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(b)
Figure 3.15: Azimuth and elevation pointing errors for a ballistic missile spinning at 5 Hz,using analog conscan, mixed by (a) a sine wave and (b) a bandpassed square wave.
28
0 10 20 30 40 50−0.2
0
0.2
0.4
0.6
Time (s)
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50−0.4
−0.2
0
0.2
0.4
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(a)
0 10 20 30 40 50−1
0
1
2
3
Time (s)
Azi
mut
h po
intin
g er
ror (
deg)
0 10 20 30 40 50−3
−2
−1
0
1
Time (s)
Ele
vatio
n po
intin
g er
ror (
deg)
(b)
Figure 3.16: Azimuth and elevation pointing error for a ballistic missile spinning at 5 Hz,using (a) Lissajous scan and (b) Rosette scan. Note that the tracker loses sight of the targetwhen using Rosette scan.
29
3.3 Fly-by Trajectory
To simulate a fly-by, assume the missile flies from north to south at a constant altitude
of 11000 feet. The velocity and gain pattern remain the same, and the flight path is 15 km
east of the tracking antenna. The total flight time is 20 seconds. Figure 3.17 illustrates the
simulation. Figures 3.18–3.29 plot the pointing error of each tracking algorithm when the
missile spins at 0, 2, and 5 Hz. Since the elevation angle is nearly constant for this scenario,
only the azimuth pointing error will be considered.
15 km
20 km
v = 1012 m/s
N
Figure 3.17: Diagram of the fly-by missile simulation
Changing Scan Frequency As seen in previous figures, the tracking error is highest when
the angular velocity of the targest is highest. In such moments, the angular displacement of
the target per scan cycle is at its peak. By increasing the scan frequency, the target does not
move as far per cycle; thus, the displacement per cycle is reduced, and so the assumption
that the target is stationary during a conscan cycle is closer to the truth. In addition to a
30
0 5 10 15 20−0.4
−0.3
−0.2
−0.1
0
Time (s)
LS p
oint
ing
erro
r (de
g)
0 5 10 15 20−0.4
−0.3
−0.2
−0.1
0
Time (s)
DFT
poi
ntin
g er
ror (
deg)
Figure 3.18: Pointing error for least squares conscan and DFT conscan for a non-spinningfly-by missile.
0 5 10 15 20−0.4
−0.3
−0.2
−0.1
0
Time (s)
LS p
oint
ing
erro
r (de
g)
0 5 10 15 20−0.4
−0.3
−0.2
−0.1
0
Time (s)
DFT
poi
ntin
g er
ror (
deg)
Figure 3.19: Pointing error for sliding least squares conscan and sliding DFT conscan for anon-spinning fly-by missile.
31
0 5 10 15 20
−0.25−0.2
−0.15−0.1
−0.05
Time (s)
Sin
e−m
ixed
poi
ntin
g er
ror (
deg)
0 5 10 15 20−0.8
−0.6
−0.4
−0.2
0
Time (s)
Squ
are−
mix
ed p
oint
ing
erro
r (de
g)
Figure 3.20: Pointing error for analog conscan, mixing with a sine wave and a bandpassedsquare wave, for a non-spinning fly-by missile.
0 5 10 15 20−0.2
−0.15
−0.1
−0.05
0
Time (s)
Liss
ajou
s po
intin
g er
ror (
deg)
0 5 10 15 20−0.2
−0.15
−0.1
−0.05
0
Time (s)
Ros
ette
poi
ntin
g er
ror (
deg)
Figure 3.21: Pointing error for Lissajous scan and Rosette scan for a non-spinning fly-bymissile.
32
0 5 10 15 20−1.5
−1
−0.5
0
0.5
Azi
mut
h P
oint
ing
Err
or (d
eg)
0 5 10 15 20−1.5
−1
−0.5
0
0.5
Azi
mut
h P
oint
ing
Err
or (d
eg)
Time (s)
Figure 3.22: Pointing error for least squares conscan and DFT conscan for a fly-by missilespinning at 2 Hz.
0 5 10 15 20−10
−5
0
5
10
Time (s)
LS p
oint
ing
erro
r (de
g)
0 5 10 15 20−1
−0.5
0
0.5
Time (s)
DFT
poi
ntin
g er
ror (
deg)
Figure 3.23: Pointing error for sliding least squares conscan and sliding DFT conscan for afly-by missile spinning at 2 Hz. Note that sliding window least squares conscan loses sight ofthe target.
33
0 5 10 15 20−0.8
−0.6
−0.4
−0.2
0
Time (s)
Sin
e−m
ixed
poi
ntin
g er
ror (
deg)
0 5 10 15 20−3
−2
−1
0
1
Time (s)
Squ
are−
mix
ed p
oint
ing
erro
r (de
g)
Figure 3.24: Pointing error for analog conscan, mixing with a sine wave and a bandpassedsquare wave, for a fly-by missile spinning at 2 Hz. Note that in the second case the trackerloses sight of the target.
0 5 10 15 20−0.5
0
0.5
Time (s)
Liss
ajou
s po
intin
g er
ror (
deg)
0 5 10 15 20−1
−0.5
0
0.5
1
Time (s)
Ros
ette
poi
ntin
g er
ror (
deg)
Figure 3.25: Pointing error for Lissajous scan and Rosette scan for a fly-by missile spinningat 2 Hz.
34
0 5 10 15 20−1.5
−1
−0.5
0
Azi
mut
h P
oint
ing
Err
or (d
eg)
0 5 10 15 20−1.5
−1
−0.5
0
Azi
mut
h P
oint
ing
Err
or (d
eg)
Time (s)
Figure 3.26: Pointing error for least squares conscan and DFT conscan for a fly-by missilespinning at 5 Hz. Note that DFT conscan loses track of the target.
0 5 10 15 20−50
0
50
Time (s)
LS p
oint
ing
erro
r (de
g)
0 5 10 15 20−2
−1.5
−1
−0.5
0
Time (s)
DFT
poi
ntin
g er
ror (
deg)
Figure 3.27: Pointing error for sliding least squares conscan and sliding DFT conscan for afly-by missile spinning at 5 Hz. Note that sliding window least squares conscan loses sight ofthe target.
35
0 5 10 15 20−3
−2
−1
0
1
Time (s)
Sin
e−m
ixed
poi
ntin
g er
ror (
deg)
0 5 10 15 20−3
−2
−1
0
1
Time (s)
Squ
are−
mix
ed p
oint
ing
erro
r (de
g)
Figure 3.28: Pointing error for analog conscan, mixing with a sine wave and a bandpassedsquare wave, for a fly-by missile spinning at 5 Hz. Note that in both cases, the tracker losessight of the target.
0 5 10 15 20−0.6
−0.4
−0.2
0
0.2
Time (s)
Liss
ajou
s po
intin
g er
ror (
deg)
0 5 10 15 20
−1
−0.5
0
0.5
1
Time (s)
Ros
ette
poi
ntin
g er
ror (
deg)
Figure 3.29: Pointing error for Lissajous scan and Rosette scan for a fly-by missile spinningat 5 Hz.
36
lower angular displacement per cycle, the target experiences less roll per cycle, so there is
less disturbance on the amplitude of the received signal. In general, the disturbance from
the target’s rotation can be reduced by decreasing the scan period. Figure 3.30 illustrates
the average amplitude variance per scan as a function of scan frequency. The gain pattern
used is the C-band pattern from Figure 1.1, and plots for a missile roll rate of 2 Hz and 5
Hz are included.
0 10 20 30 40 500
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Scan frequency (Hz)
Am
plitu
de V
aria
nce
2 Hz5 Hz
Figure 3.30: Average amplitude variance per scan as a function of scan frequency, for rollrates of 2 Hz and 5 Hz.
Figures 3.31–3.34 illustrate the difference in tracking in the fly-by scenario when the
scan frequency is 50 Hz instead of 25 Hz, while the missile roll rate is 5 Hz (note that the
scan frequency is still an integer multiple of the roll rate). There are a few positive changes
when the scan frequency increases to 50 Hz. First, none of the tracking methods lose sight
of the target; second, the RMS pointing error lower in general; and third, the amplitude of
the periodic component of the error is lower in general.
37
0 5 10 15 20−0.4
−0.2
0
0.2
0.4
Time (s)
LS p
oint
ing
erro
r (de
g)
0 5 10 15 20−0.4
−0.2
0
0.2
0.4
Time (s)
DFT
poi
ntin
g er
ror (
deg)
Figure 3.31: Pointing error for least squares conscan and DFT conscan for a fly-by missilespinning at 5 Hz, for a scan frequency of 50 Hz. The number of samples per cycle remainsconstant at 40. Compare with Figure 3.26.
0 5 10 15 20−0.5
0
0.5
1
Time (s)
LS p
oint
ing
erro
r (de
g)
0 5 10 15 20−0.4
−0.2
0
0.2
0.4
Time (s)
DFT
poi
ntin
g er
ror (
deg)
Figure 3.32: Pointing error for sliding least squares conscan and sliding DFT conscan for afly-by missile spinning at 5 Hz, for a scan frequency of 50 Hz. Compare with Figure 3.27.
38
0 5 10 15 20−0.4
−0.2
0
0.2
0.4
Time (s)
Sin
e−m
ixed
poi
ntin
g er
ror (
deg)
0 5 10 15 20−0.3
−0.2
−0.1
0
0.1
Time (s)
Squ
are−
mix
ed p
oint
ing
erro
r (de
g)
Figure 3.33: Pointing error for analog conscan, mixing with a sine wave and a bandpassedsquare wave, for a fly-by missile spinning at 5 Hz, for a scan frequency of 50 Hz. Compare withFigure 3.28.
0 5 10 15 20−0.4
−0.3
−0.2
−0.1
0
Time (s)
Liss
ajou
s po
intin
g er
ror (
deg)
0 5 10 15 20−0.8
−0.6
−0.4
−0.2
0
Time (s)
Ros
ette
poi
ntin
g er
ror (
deg)
Figure 3.34: Pointing error for Lissajous scan and Rosette scan for a fly-by missile spinningat 5 Hz, for a scan frequency of 50 Hz. Compare with Figure 3.29.
39
Chapter 4
Simultaneous Lobing
Simultaneous lobing differs from sequential lobing in that rather than estimating
pointing error given amplitude over time, the tracker estimates pointing error using signals
from multiple feeds at once. Monopulse tracking is the most well-known example of simul-
taneous lobing; it uses four stationary feeds pointed away from boresight at the same squint
angle used in conscan [14]. The feeds are positioned like four corners of a square [15]. Az-
imuth and elevation differences are produced using sums and differences of the feed output
amplitudes, as summarized in Figure 4.1. The azimuth and elevation pointing error esti-
mates are the azimuth and elevation differences divided by the sum signal [16], then divided
by the slope of the S-curve. The normalized S-curves are shown in Figure 4.2. Sometimes
a “scan frequency” is reported with the monopulse method, but this is not a “scan” in the
same sense as sequential lobing, but rather the rate at which the feed output amplitudes are
sampled for other purposes.
There are multiple receive feeds in monopulse, and so there is a potential for mutual
coupling between the feeds. Mutual coupling can be minimized by introducing additional
coupling to cancel out the inherent coupling of the system [17]. In addition, variations in
system temperature can lead to gain drift in the system. Since the focus of this paper is on
the effect of a spinning target on tracking, it is assumed that there is no coupling between
feeds and that the gain patterns of the feeds are identical and time-invariant.
With respect to the issue of a spinning target, simultaneous lobing has a couple
advantages over sequential lobing. For one, the temporal delay between signals goes away,
since the four feeds are receiving simultaneously. In addition, simultaneous lobing divides out
the fluctuating amplitude of the received signal. Therefore, tracking methods like monopulse
40
1 2
3 4+ +
__ +
_
_
++ +
+ +
Σ
Σ
Σ
Σ∆
∆
∆
FrontElevation Difference
Azimuth Difference Sum
1
2
34
Figure 4.1: A block diagram of the monopulse tracking method, reproduced from [1].
are unaffected by the disturbance induced by a spinning target. If there were a figure simular
to Figure 2.5, but for the monopulse method, the output would be zero for all roll rates.
4.1 Simulating a Monopulse Tracker
Under the same scenario of a spinning fly-by missile, simultaneous lobing can be com-
pared to sequential lobing. Figure 4.3 is a plot of the azimuth pointing error for monopulse
and all previously described forms of sequential lobing. To better comprehend the data for
tracking a fly-by missile spinning at 5 Hz, the information is divided into two plots: mean
pointing error (Figure 4.4) and pointing error variance (Figure 4.5). The mean and variance
are calculated from non-overlapping windows of 4000 data points.
41
−1 −0.5 0 0.5 1−5
0
5
Azimuth angle error (deg)
Mon
opul
se e
rror
est
imat
e (d
eg)
Figure 4.2: Normalized S-curves for a monopulse tracker with a carrier frequency of 5135MHz. The single dotted line is at the half-power beamwidth.
0 2 4 6 8 10 12 14 16 18 20−0.3
−0.28
−0.26
−0.24
−0.22
−0.2
−0.18
−0.16
−0.14
−0.12
−0.1
Time (s)
Azi
mut
h P
oint
ing
Err
or (d
eg)
Conscan (Least Squares)Conscan (DFT)Sliding Window Conscan (LS)Sliding Window Conscan (DFT)Analog Conscan (Sine−mixed)Analog Conscan (Square−mixed)Lissajous ScanRosette ScanMonopulse
Figure 4.3: A comparison of monopulse tracking and the various methods of sequential lobingfor a non-spinning fly-by missile. The scan frequency is 50 Hz.
42
0 2 4 6 8 10 12 14 16 18 20−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Time (s)
Win
dow
ed A
zim
uth
Poi
ntin
g E
rror
(deg
)
Conscan (Least Squares)Conscan (DFT)Sliding Window Conscan (LS)Sliding Window Conscan (DFT)Analog Conscan (Sine−mixed)Analog Conscan (Square−mixed)Lissajous ScanRosette ScanMonopulse
Figure 4.4: Windowed mean pointing error for monopulse tracking and the various methodsof sequential lobing for a fly-by missile spinning at 5 Hz. The scan frequency is 50 Hz.
43
0 2 4 6 8 10 12 14 16 18 200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Time (s)
Win
dow
ed A
zim
uth
Poi
ntin
g E
rror
Var
ianc
e (d
eg2 )
Conscan (Least Squares)Conscan (DFT)Sliding Window Conscan (LS)Sliding Window Conscan (DFT)Analog Conscan (Sine−mixed)Analog Conscan (Square−mixed)Lissajous ScanRosette ScanMonopulse
Figure 4.5: Windowed pointing error variance of monopulse tracking and the various methodsof sequential lobing for a fly-by missile spinning at 5 Hz. The scan frequency is 50 Hz.
44
Upon observation of this information, it can be seen that in the scenario of a non-
spinning missile, monopulse performs roughly as well as least squares or DFT conscan.
However, when the missile spins, the performance of monopulse is unaffected. Although
some implentations of sequential lobing exhibit lower RMS error (e.g., DFT conscan), the
error variance of monopulse is negligent, while all forms of sequential lobing feature some
error variance due to the periodic component of the error induced by the spinning missile.
45
Chapter 5
Conclusion
The simulations demonstrate how tracking performance ties into the trade-offs re-
garding which tracking algorithm to use. For the sequential lobing techniques discussed in
this paper, all forms of conscan drive the feed along the simpler trajectory of a circle, while
Rosette and Lissajous scan require more complex controllers to drive the feed [18]. Simul-
taneous lobing, on the other hand, uses several non-moving feeds. Of the circuitry needed
to estimate the pointing error given a(t) or a(nT ), the methods that use least squares need
memory to store the pseudo-inverse matrix, plus 3N multiplications per update. DFT con-
scan, however, needs only the circuitry to calculate an N -point FFT of a(nT ). Analog
conscan simplifies the process by mixing a(t) and then applying a simple lowpass filter. The
mixing signal can be produced using either a local oscillator at the scan frequency, but the
bandpassed square wave is even simpler to make [1]. The results from Chapters 3 and 4
can lead to a general statement that increasing the scan frequency improves tracking perfor-
mance. In addition, comparing the results of all tracking algorithms in Chapter 4, along with
the information of implementation complexity, permits us to make more informed decisions
and trade-offs when choosing a tracking method and the associated parameters. Table 5.1
summarizes the findings of this paper.
46
Table
5.1:
Su
mm
ary
ofR
esu
lts
Tra
ckin
gA
lgor
ithm
Har
dw
are
Com
ple
xit
yC
omputa
tion
alC
omple
xit
y
RM
Ser
ror
for
non
-spin
nin
gta
rget
(deg
)
RM
Ser
ror
for
targ
etsp
innin
gat
har
mon
icra
te(d
eg)
Err
orva
rian
cefo
rta
rget
spin
nin
gat
5H
z(d
eg2)
Con
scan
(LS)
Low
Med
ium
0.19
0.02
0.02
6C
onsc
an(D
FT
)L
owM
ediu
m0.
195
0.06
0.00
75Slidin
gW
indow
Con
scan
(LS)
Low
Med
ium
0.19
0.14
0.06
4
Slidin
gW
indow
Con
scan
(DF
T)
Low
Med
ium
0.19
50.
058
0.01
Anal
ogC
onsc
an(s
ine-
mix
ed)
Low
Low
0.24
0.14
0.00
75
Anal
ogC
onsc
an(s
quar
e-m
ixed
)L
owV
ery
Low
0.26
0.17
0.00
3
Lis
sajo
us
Sca
nH
igh
Med
ium
0.16
0.17
0.00
3
Ros
ette
Sca
nH
igh
Med
ium
0.18
50.
220.
015
Mon
opuls
eM
ediu
mM
ediu
m0.
190.
20N
one
47
Bibliography
[1] M. Skolnik, Ed., Radar Handbook, 2nd ed. New York, NY: McGraw-Hill, 1990.
[2] R. Hall and D. Wu, “Modeling and design of circularly-polarized cylindrical wraparoundmicrostrip antennas,” in Antennas and Propagation Society International Symposium,vol. 1, 1996, pp. 672–675.
[3] C. Brockner, “Angular jitter in conventional conical-scanning, automatic-tracking radarsystems,” in Proceedings of the IRE, 1951, pp. 51–55.
[4] J. Damonte and D. Stoddard, “An analysis of conical scan antennas for tracking,” inIRE International Convention Record, vol. 4, 1956, pp. 39–47.
[5] N. Levanon, “Upgrading conical scan with off-boresight measurements,” IEEE Trans-actions on Aerospace and Electronic Systems, vol. 33, no. 4, pp. 1350–1357, 1997.
[6] D. Eldred, “An improved conscan algorithm based on a Kalman filter,” TDA ProgressReport 42-116, pp. 223–231, February 1994.
[7] A. Guesalaga and S. Tepper, “Synthesis of automatic gain controllers for conical scantracking radar,” IEEE Transactions on Aerospace and Electronic Systems, vol. 36, no. 1,pp. 302–309, 2000.
[8] L. Alvarez, “Analysis of open-loop conical scan pointing error and variance estimators,”TDA Progress Report 42-115, pp. 81–90, November 1993.
[9] M. Rice, Digital Communications: A Discrete-Time Approach. Upper Saddle River,NJ: Pearson Prentice Hall, 2009.
[10] A. Mileant and T. Peng, “Pointing a ground antenna at a spinning spacecraft using coni-cal scan-simulation results,” in IEEE International Conference on Systems Engineering,1989, 1989, pp. 541–547.
[11] W. Gawronski, Modeling and Control of Antennas and Telescopes. New York, NY:Springer, 2008.
[12] G. Brooker, “Conical-scan antennas for W-band radar systems,” in Proceedings of theInternational Radar Conference, 2003, 2003, pp. 406–411.
[13] R. Beard and T. McLain, Small Unmanned Aircraft: Theory and Practice. Princeton,NJ: Princeton University Press, 2012.
48
[14] U. Nickel, “Overview of generalized monopulse estimation,” Aerospace and ElectronicSystems Magazine, IEEE, vol. 21, no. 6, pp. 27–56, 2006.
[15] P. Willett, W. Blair, and Y. Bar-Shalom, “Correlation between horizontal and verticalmonopulse measurements,” IEEE Transactions on Aerospace and Electronic Systems,vol. 39, no. 2, pp. 533–549, 2003.
[16] F. Ulaby, E. Michielssen, and U. Ravaioli, Fundamentals of Applied Electromagnetics,6th ed. Upper Saddle River, NJ: Prentice Hall, 2010.
[17] W. Bridge, “Cross coupling in a five horn monopulse tracking system,” Antennas andPropagation, IEEE Transactions on, vol. 20, no. 4, pp. 436–442, 1972.
[18] H. Lee and D. Schmidt, “Robust two-degree-of-freedom H∞ control of a seeker scanloop system,” Control Theory and Applications, IEE Proceedings -, vol. 149, no. 2, pp.149–156, 2002.
49
