Simulating the Performance of Tracking a Spinning Missile at C-Band

Transcription

1 Brigham Young University BYU ScholarsArchive All Theses and Dissertations Simulating the Performance of Tracking a Spinning Missile at C-Band Darren Robert Kartchner Brigham Young University - Provo Follow this and additional works at: Part of the Electrical and Computer Engineering Commons BYU ScholarsArchive Citation Kartchner, Darren Robert, "Simulating the Performance of Tracking a Spinning Missile at C-Band" (213). All Theses and Dissertations This Thesis is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in All Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact scholarsarchive@byu.edu, ellen_amatangelo@byu.edu.

2 Simulating the Performance of Tracking a Spinning Missile at C-Band Darren Kartchner A thesis submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Master of Science Michael D. Rice, Chair Brian D. Jeffs Randal W. Beard Department of Electrical and Computer Engineering Brigham Young University December 213 Copyright c 213 Darren Kartchner All Rights Reserved

3 ABSTRACT Simulating the Performance of Tracking a Spinning Missile at C-Band Darren Kartchner Department of Electrical and Computer Engineering Master of Science The amplitude fluctuation induced by a spinning missile acts as a disturbance on tracking schemes that use sequential lobing (e.g., conscan). In addition, if a tracking system converts from S-band to C-band, the beamwidth is narrower and the wrap-around antenna on the missile requires more patches, and so the margin of error for tracking decreases. Tracking performance is simulated with a spinning missile with ballistic and fly-by trajectories while running 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 schemes lose track of the target. Remedial techniques are discussed, including increasing the scan frequency and using simultaneous (monopulse) tracking rather than sequential lobing. Keywords: aeronautical telemetry, tracking

4 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, Edwards 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 W9KK-9-C-16. 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.

5 Table of Contents List of Tables vi List of Figures vii 1 Introduction Notation Introduction Sequential Lobing Types of Sequential Lobing Conical Scan (Conscan) Analog Conscan Lissajous Scan and Rosette Scan Impact of Spinning Missile on Sequential Lobing Sequential Lobing Simulation Antenna Controllers Ballistic Trajectory Fly-by Trajectory Simultaneous Lobing Simulating a Monopulse Tracker iv

6 5 Conclusion 46 Bibliography 48 v

7 List of Tables 1.1 Notation Summary of Results vi

8 List of Figures 1.1 Roll patterns for two conformal wrap-around antennas, operating at S-band (225 MHz) and at C-band (5135 MHz) Gain patterns for an 8-foot parabolic reflector antenna with η =.7, operating at S-band (225 MHz) and at C-band (5135 MHz) Normalized S-curves for least squares conscan and DFT conscan. The single dotted line denotes the half power beamwidth Normalized S-curves for analog conscan when mixing with a sine wave and when mixing with a bandpassed square wave, respectively. The single dotted line denotes the half power beamwidth Feed path for (a) Lissajous scan and (b) Rosette scan. The feed path for conscan is included for reference Normalized S-curves for Lissajous and Rosette scans. The dotted line represents the half-power beamwidth Average azimuth and elevation error estimates of a stationary, rotating target at boresight, using both methods of conscan Block diagram of the antenna controller Block diagram of the inner loop of the controller Block diagram of the outer loop of the controller, with the inner loop treated as a unity gain Diagram of the ballistic missile simulation Azimuth and elevation pointing errors for a non-spinning ballistic missile, using (a) least squares conscan and (b) DFT conscan vii

9 3.6 Azimuth and elevation pointing errors for a non-spinning ballistic missile, using sliding window (a) least squares conscan and (b) DFT conscan, for T = T/ Azimuth and elevation pointing errors for a non-spinning ballistic missile, using analog conscan, mixed by (a) a sine wave and (b) a bandpassed square wave Azimuth and elevation pointing errors for a non-spinning ballistic missile, using (a) Lissajous scan and (b) Rosette scan Azimuth and elevation pointing errors for a ballistic missile spinning at 2 Hz, using (a) least squares conscan and (b) DFT conscan 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 T = T/4. Note that sliding window least squares conscan loses sight of the target 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 that in the second case the tracker loses sight of the target Azimuth and elevation pointing error for a ballistic missile spinning at 2 Hz, using (a) Lissajous scan and (b) Rosette scan Azimuth and elevation pointing errors for a ballistic missile spinning at 5 Hz, using (a) least squares conscan and (b) DFT conscan 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 T = T/4. Note that sliding window least squares conscan loses sight of the target 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 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 target when using Rosette scan Diagram of the fly-by missile simulation Pointing error for least squares conscan and DFT conscan for a non-spinning fly-by missile viii

10 3.19 Pointing error for sliding least squares conscan and sliding DFT conscan for a non-spinning fly-by missile Pointing error for analog conscan, mixing with a sine wave and a bandpassed square wave, for a non-spinning fly-by missile Pointing error for Lissajous scan and Rosette scan for a non-spinning fly-by missile Pointing error for least squares conscan and DFT conscan for a fly-by missile spinning at 2 Hz Pointing error for sliding least squares conscan and sliding DFT conscan for a fly-by missile spinning at 2 Hz. Note that sliding window least squares conscan loses sight of the target Pointing error for analog conscan, mixing with a sine wave and a bandpassed square wave, for a fly-by missile spinning at 2 Hz. Note that in the second case the tracker loses sight of the target Pointing error for Lissajous scan and Rosette scan for a fly-by missile spinning at 2 Hz Pointing error for least squares conscan and DFT conscan for a fly-by missile spinning at 5 Hz. Note that DFT conscan loses track of the target Pointing error for sliding least squares conscan and sliding DFT conscan for a fly-by missile spinning at 5 Hz. Note that sliding window least squares conscan loses sight of the target Pointing error for analog conscan, mixing with a sine wave and a bandpassed square wave, for a fly-by missile spinning at 5 Hz. Note that in both cases, the tracker loses sight of the target Pointing error for Lissajous scan and Rosette scan for a fly-by missile spinning at 5 Hz Average amplitude variance per scan as a function of scan frequency, for roll rates of 2 Hz and 5 Hz Pointing error for least squares conscan and DFT conscan for a fly-by missile spinning at 5 Hz, for a scan frequency of 5 Hz. The number of samples per cycle remains constant at 4. Compare with Figure Pointing error for sliding least squares conscan and sliding DFT conscan for a fly-by missile spinning at 5 Hz, for a scan frequency of 5 Hz. Compare with Figure ix

11 3.33 Pointing error for analog conscan, mixing with a sine wave and a bandpassed square wave, for a fly-by missile spinning at 5 Hz, for a scan frequency of 5 Hz. Compare with Figure Pointing error for Lissajous scan and Rosette scan for a fly-by missile spinning at 5 Hz, for a scan frequency of 5 Hz. Compare with Figure A block diagram of the monopulse tracking method, reproduced from [1] Normalized S-curves for a monopulse tracker with a carrier frequency of 5135 MHz. The single dotted line is at the half-power beamwidth A comparison of monopulse tracking and the various methods of sequential lobing for a non-spinning fly-by missile. The scan frequency is 5 Hz Windowed mean pointing error for monopulse tracking and the various methods of sequential lobing for a fly-by missile spinning at 5 Hz. The scan frequency is 5 Hz Windowed pointing error variance of monopulse tracking and the various methods of sequential lobing for a fly-by missile spinning at 5 Hz. The scan frequency is 5 Hz x

12 Chapter 1 Introduction 1.1 Notation Table 1.1: Notation of the Paper Variable Definition G Boresight gain J 1 ( ) Modified Bessel function of the first order k Wavenumber of carrier frequency D Diameter of parabolic reflector λ Wavelength of carrier frequency η Antenna efficiency a(t) Received signal amplitude N Number of samples per scan cycle T Time spacing between samples of signal amplitude a(nt ) Sampled signal amplitude f Scan frequency (in rotations per second) ɛ(t) Angular displacement between target and boresight r Squint angle ɛ az Azimuth component of pointing error ɛ el Elevation component of pointing error A Average amplitude over one scan cycle A(m) m-th component of the inverse FFT of a(nt ) x i, y i Cartesian position of tracking feed at sample i during a scan t i Time at sample i during a scan α L, β L Harmonics used to determine feed path during a Lissajous scan α R, β R Harmonics used to determine feed path during a Rosette scan ω 3dB Controller 3 db bandwidth (in rad/s) ζ Controller damping constant 1

13 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 (2-23 MHz) to lower C-band (4-55 MHz) is of interest in the field of missile range testing. If the carrier frequency of a transmitterreceiver 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 MHz 5135 MHz Figure 1.1: Roll patterns for two conformal wrap-around antennas, operating at S-band (225 MHz) and at C-band (5135 MHz). 2

14 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) = G 2 J 1 (.5kD sin (φ)), (1.1).5kD sin (φ) where G is G = ( ) 2 πd η. (1.2) λ Figure 1.2 illustrates the gain patterns for an 8-foot parabolic reflector operating at 225 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. Gain (db) MHz 5135 MHz Azimuth Angle (deg) Figure 1.2: Gain patterns for an 8-foot parabolic reflector antenna with η =.7, operating at S-band (225 MHz) and at C-band (5135 MHz). 3

15 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 implementations 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

16 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 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.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 f 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 =, 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

17 Least Squares Method given by [8]: The angular displacement between boresight and the target as a function of time is ɛ(t) 2 = r 2 + ɛ 2 az + ɛ 2 el 2rɛ az cos(f t) 2rɛ el sin(f t). (2.2) The angular displacement ɛ(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: [ ] P c (t) = 1 cos(f t) sin(f t) P ((k s /h)ɛ az ), (2.3) P ((k s /h)ɛ el ) P where k s /h is the conscan slope divided by the half-power beamwidth, and P is the average power over the scan period. When applied to a(nt ), (2.3) becomes ( ) ( ) 2π 2π a(t ) 1 cos sin N N ( a(2t ) = 1 cos 2 2π ) ( sin 2 2π ) A N N A. ((k s /h)ɛ az ), (2.4)... A ((k s /h)ɛ el ) a(nt ) 1 cos(2π) sin(2π) or From (2.5), C is estimated as A = Y C. (2.5) Ĉ = (Y Y ) 1 Y A, (2.6) where (Y Y ) 1 Y can be pre-computed. At this point, ɛ az is estimated by dividing the second element of Ĉ by the first element of Ĉ, then dividing by the slope of the resulting S-curve. The elevation error is estimated the same way, substituting the second element of Ĉ with the third element. 6

18 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= ( a(nt ) exp j 2πmn ), for m =, 1,, N 1 (2.7) N 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 [1]. 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). Least squares error estimate (deg) DFT error estimate (deg) Azimuth angle error (deg) Figure 2.1: Normalized S-curves for least squares conscan and DFT conscan. The single dotted line denotes the half power beamwidth. 7

19 Sliding Window Conscan Using sliding window conscan, the error estimate updates every N T seconds, where T < 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 ) 1 Y 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 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 f, 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 f [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 Lissajous Scan and Rosette Scan defined by Lissajous scan is similar to least squares conscan, but the tracking feed follows a path x i = r sin α Lωt i. (2.8) y i r sin β L ωt i 8

20 2 E scan output (deg) Azimuth angle error (deg) 2 E scan output (deg) Azimuth angle error (deg) Figure 2.2: Normalized S-curves for analog conscan when mixing with a sine wave and when mixing with a bandpassed square wave, respectively. The single dotted line denotes the half power beamwidth. Rosette scan follows a path defined by x i = r sin α Rωt i r sin β R ωt i. (2.9) r cos α R ωt i + r cos β R ωt i y i Using the values α L = 4, β L = 3, α R = 1, and β R = 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 6 12 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

21 Elevation (deg) Azimuth (deg) (a) Elevation (deg) Azimuth (deg) (b) Figure 2.3: Feed path for (a) Lissajous scan and (b) Rosette scan. The feed path for conscan is 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 amplitude 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 1

22 Rosette scan output (deg) Lissajous scan output (deg) Azimuth angle error (deg) Azimuth angle error (deg) Figure 2.4: Normalized S-curves for Lissajous and Rosette scans. The dotted line represents the 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

23 Azimuth pointing error (deg).2.2 Least Squares DFT Elevation pointing error (deg).2.2 Least Squares DFT Roll Rate (Hz) Figure 2.5: Average azimuth and elevation error estimates of a stationary, rotating target at boresight, using both methods of conscan. 12

24 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, 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 = J θ, (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

25 θ Tracking Algorithm PI Controller + PI 1 Controller s _ 1 s θ^ velocity loop position loop 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 + PI 1 Controller s _ velocity loop Figure 3.2: Block diagram of the inner loop of the controller function of a PI controller is k p + k i s. And so, the transfer function of the inner loop is H(s) = This is a second-order system, whose canonical form is k ps + k i s 2 + k p s + k i. (3.2) H(s) = 2ζω ns + ωn 2. (3.3) s 2 + 2ζω n s + ωn 2 14

26 From (3.3), values for the PI gains can be derived from ζ and ω n : k p = 2ζω n, (3.4) k i = ω 2 n. (3.5) To find the 3 db bandwidth, consider the magnitude squared of the transfer function: H(jω) 2 = 4ζ 2 ω 2 nω 2 + ω 4 n (ω 2 n ω 2 ) 2 + 4ζ 2 ω 2 nω 2. (3.6) The 3 db bandwidth of the inner loop is the frequency ω 3dB where (3.6) is 1/2: Solving for ω 3dB results in 1 2 = 4ζ 2 ωnω 2 3dB 2 + ω4 n. (3.7) (ωn 2 ω3db 2 )2 + 4ζ 2 ωnω 2 3dB 2 ω 3dB = ω n 2ζ (2ζ 2 + 1) (3.8) Therefore, for a desired 3 db bandwidth ω 3dB and a damping constant ζ, the PI gains are k p = 2ζω 3dB 2ζ , (3.9) (2ζ 2 + 1) k i = ω 2 3dB 2ζ (2ζ 2 + 1) (3.1) 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 (3.1) can also be used to determine the PI controller gains for the outer loop. The only difference is that the 3 db 15

27 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 6π rad/s. θ + _ PI Controller 1 1 s θ^ Figure 3.3: Block diagram of the outer loop of the controller, with the inner loop treated as a unity gain. 3.2 Ballistic Trajectory To simulate performance while tracking a spinning ballistic missile, consider the scenario illustrated in Figure 3.4. A missile fired the ground, from south to north, 15 from the ground. The missile has a velocity of 112 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 =.771. Figure 3.4 is a diagram of the simulation. Figures 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

28 target. When the missile spins at 5 Hz, the RMS pointing error increases, and even more methods lose track. 54 km v = 112 m/s θ = 15 N 15 km Figure 3.4: Diagram of the ballistic missile simulation 17

29 Azimuth pointing error (deg) Elevation pointing error (deg) (a) Azimuth pointing error (deg) Elevation pointing error (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

30 Elevation pointing error (deg) Azimuth pointing error (deg) Azimuth pointing error (deg) (a) Elevation pointing error (deg) (b) Figure 3.6: Azimuth and elevation pointing errors for a non-spinning ballistic missile, using sliding window (a) least squares conscan and (b) DFT conscan, for T = T/4. 19

31 Azimuth pointing error (deg) Elevation pointing error (deg) (a) Azimuth pointing error (deg) Elevation pointing error (deg) (b) Figure 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 square wave. 2

32 Azimuth pointing error (deg) Elevation pointing error (deg) (a) Azimuth pointing error (deg) Elevation pointing error (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

33 Azimuth pointing error (deg) Elevation pointing error (deg) Azimuth pointing error (deg) (a) Elevation pointing error (deg) (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

34 Azimuth pointing error (deg) Elevation pointing error (deg) Azimuth pointing error (deg) Elevation pointing error (deg) (a) (b) Figure 3.1: 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 T = T/4. Note that sliding window least squares conscan loses sight of the target. 23

35 Azimuth pointing error (deg) Elevation pointing error (deg) Azimuth pointing error (deg) Elevation pointing error (deg) (a) (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 that in the second case the tracker loses sight of the target. 24

36 Azimuth pointing error (deg) Elevation pointing error (deg) (a) Azimuth pointing error (deg) Elevation pointing error (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

37 Azimuth pointing error (deg) Elevation pointing error (deg) Azimuth pointing error (deg) (a) Elevation pointing error (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

38 Azimuth pointing error (deg) Elevation pointing error (deg) Azimuth pointing error (deg) (a) Elevation pointing error (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 T = T/4. Note that sliding window least squares conscan loses sight of the target. 27

39 Azimuth pointing error (deg) Azimuth pointing error (deg) Elevation pointing error (deg) (a) Elevation pointing error (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

40 Azimuth pointing error (deg) Elevation pointing error (deg) (a) Azimuth pointing error (deg) Elevation pointing error (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 target when using Rosette scan. 29

41 3.3 Fly-by Trajectory To simulate a fly-by, assume the missile flies from north to south at a constant altitude of 11 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 2 seconds. Figure 3.17 illustrates the simulation. Figures plot the pointing error of each tracking algorithm when the missile spins at, 2, and 5 Hz. Since the elevation angle is nearly constant for this scenario, only the azimuth pointing error will be considered. N 2 km 15 km v = 112 m/s 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 3

42 LS pointing error (deg) DFT pointing error (deg) Figure 3.18: Pointing error for least squares conscan and DFT conscan for a non-spinning fly-by missile. LS pointing error (deg) DFT pointing error (deg) Figure 3.19: Pointing error for sliding least squares conscan and sliding DFT conscan for a non-spinning fly-by missile. 31

43 Sine mixed pointing error (deg) Square mixed pointing error (deg) Figure 3.2: Pointing error for analog conscan, mixing with a sine wave and a bandpassed square wave, for a non-spinning fly-by missile. Lissajous pointing error (deg) Rosette pointing error (deg) Figure 3.21: Pointing error for Lissajous scan and Rosette scan for a non-spinning fly-by missile. 32

44 Azimuth Pointing Error (deg) Azimuth Pointing Error (deg) Figure 3.22: Pointing error for least squares conscan and DFT conscan for a fly-by missile spinning at 2 Hz. 1 LS pointing error (deg) DFT pointing error (deg) Figure 3.23: Pointing error for sliding least squares conscan and sliding DFT conscan for a fly-by missile spinning at 2 Hz. Note that sliding window least squares conscan loses sight of the target. 33

45 Sine mixed pointing error (deg) Square mixed pointing error (deg) Figure 3.24: Pointing error for analog conscan, mixing with a sine wave and a bandpassed square wave, for a fly-by missile spinning at 2 Hz. Note that in the second case the tracker loses sight of the target. Lissajous pointing error (deg) Rosette pointing error (deg) Figure 3.25: Pointing error for Lissajous scan and Rosette scan for a fly-by missile spinning at 2 Hz. 34

46 Azimuth Pointing Error (deg) Azimuth Pointing Error (deg) Figure 3.26: Pointing error for least squares conscan and DFT conscan for a fly-by missile spinning at 5 Hz. Note that DFT conscan loses track of the target. 5 LS pointing error (deg) DFT pointing error (deg) Figure 3.27: Pointing error for sliding least squares conscan and sliding DFT conscan for a fly-by missile spinning at 5 Hz. Note that sliding window least squares conscan loses sight of the target. 35

47 Sine mixed pointing error (deg) Square mixed pointing error (deg) Figure 3.28: Pointing error for analog conscan, mixing with a sine wave and a bandpassed square wave, for a fly-by missile spinning at 5 Hz. Note that in both cases, the tracker loses sight of the target. Lissajous pointing error (deg) Rosette pointing error (deg) Figure 3.29: Pointing error for Lissajous scan and Rosette scan for a fly-by missile spinning at 5 Hz. 36

48 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 3.3 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 Amplitude Variance Hz 5 Hz Scan frequency (Hz) Figure 3.3: Average amplitude variance per scan as a function of scan frequency, for roll rates of 2 Hz and 5 Hz. Figures illustrate the difference in tracking in the fly-by scenario when the scan frequency is 5 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 5 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

49 .4 LS pointing error (deg) DFT pointing error (deg) Figure 3.31: Pointing error for least squares conscan and DFT conscan for a fly-by missile spinning at 5 Hz, for a scan frequency of 5 Hz. The number of samples per cycle remains constant at 4. Compare with Figure LS pointing error (deg) DFT pointing error (deg) Figure 3.32: Pointing error for sliding least squares conscan and sliding DFT conscan for a fly-by missile spinning at 5 Hz, for a scan frequency of 5 Hz. Compare with Figure

50 Sine mixed pointing error (deg) Square mixed pointing error (deg) Figure 3.33: Pointing error for analog conscan, mixing with a sine wave and a bandpassed square wave, for a fly-by missile spinning at 5 Hz, for a scan frequency of 5 Hz. Compare with Figure Lissajous pointing error (deg) Rosette pointing error (deg) Figure 3.34: Pointing error for Lissajous scan and Rosette scan for a fly-by missile spinning at 5 Hz, for a scan frequency of 5 Hz. Compare with Figure

51 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 simultaneous 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]. Azimuth 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 estimates 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 4

52 Σ Σ 1 2 Σ Σ Front + + Elevation Difference + + Azimuth Difference Sum 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 compared 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 4 data points. 41

53 Monopulse error estimate (deg) Azimuth angle error (deg) Figure 4.2: Normalized S-curves for a monopulse tracker with a carrier frequency of 5135 MHz. The single dotted line is at the half-power beamwidth..1 Conscan (Least Squares) Conscan (DFT) Sliding Window Conscan (LS) Sliding Window Conscan (DFT) Analog Conscan (Sine mixed) Analog Conscan (Square mixed) Lissajous Scan Rosette Scan Monopulse Azimuth Pointing Error (deg) Figure 4.3: A comparison of monopulse tracking and the various methods of sequential lobing for a non-spinning fly-by missile. The scan frequency is 5 Hz. 42

54 .2 Conscan (Least Squares) Conscan (DFT) Sliding Window Conscan (LS) Sliding Window Conscan (DFT) Analog Conscan (Sine mixed) Analog Conscan (Square mixed) Lissajous Scan Rosette Scan Monopulse.15 Windowed Azimuth Pointing Error (deg) Figure 4.4: Windowed mean pointing error for monopulse tracking and the various methods of sequential lobing for a fly-by missile spinning at 5 Hz. The scan frequency is 5 Hz. 43

55 .7 Conscan (Least Squares) Conscan (DFT) Sliding Window Conscan (LS) Sliding Window Conscan (DFT) Analog Conscan (Sine mixed) Analog Conscan (Square mixed) Lissajous Scan Rosette Scan Monopulse Windowed Azimuth Pointing Error Variance (deg 2 ) Figure 4.5: Windowed pointing error variance of monopulse tracking and the various methods of sequential lobing for a fly-by missile spinning at 5 Hz. The scan frequency is 5 Hz. 44

56 Upon observation of this information, it can be seen that in the scenario of a nonspinning 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

57 Chapter 5 Conclusion The simulations demonstrate how tracking performance ties into the trade-offs regarding 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]. Simultaneous 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 conscan, 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 performance. 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

58 Tracking Algorithm Hardware Complexity Computational Complexity Table 5.1: Summary of Results RMS error for non-spinning target (deg) RMS error for target spinning at harmonic rate (deg) Error variance for target spinning at 5 Hz (deg 2 ) Conscan (LS) Low Medium Conscan (DFT) Low Medium Sliding Window Low Medium Conscan (LS) Sliding Window Low Medium Conscan (DFT) Analog Conscan (sine-mixed) Low Low Analog Conscan (square-mixed) Low Very Low Lissajous Scan High Medium Rosette Scan High Medium Monopulse Medium Medium.19.2 None 47

59 Bibliography [1] M. Skolnik, Ed., Radar Handbook, 2nd ed. New York, NY: McGraw-Hill, 199. [2] R. Hall and D. Wu, Modeling and design of circularly-polarized cylindrical wraparound microstrip antennas, in Antennas and Propagation Society International Symposium, vol. 1, 1996, pp [3] C. Brockner, Angular jitter in conventional conical-scanning, automatic-tracking radar systems, in Proceedings of the IRE, 1951, pp [4] J. Damonte and D. Stoddard, An analysis of conical scan antennas for tracking, in IRE International Convention Record, vol. 4, 1956, pp [5] N. Levanon, Upgrading conical scan with off-boresight measurements, IEEE Transactions on Aerospace and Electronic Systems, vol. 33, no. 4, pp , [6] D. Eldred, An improved conscan algorithm based on a Kalman filter, TDA Progress Report , pp , February [7] A. Guesalaga and S. Tepper, Synthesis of automatic gain controllers for conical scan tracking radar, IEEE Transactions on Aerospace and Electronic Systems, vol. 36, no. 1, pp , 2. [8] L. Alvarez, Analysis of open-loop conical scan pointing error and variance estimators, TDA Progress Report , pp. 81 9, November [9] M. Rice, Digital Communications: A Discrete-Time Approach. Upper Saddle River, NJ: Pearson Prentice Hall, 29. [1] A. Mileant and T. Peng, Pointing a ground antenna at a spinning spacecraft using conical scan-simulation results, in IEEE International Conference on Systems Engineering, 1989, 1989, pp [11] W. Gawronski, Modeling and Control of Antennas and Telescopes. New York, NY: Springer, 28. [12] G. Brooker, Conical-scan antennas for W-band radar systems, in Proceedings of the International Radar Conference, 23, 23, pp [13] R. Beard and T. McLain, Small Unmanned Aircraft: Theory and Practice. Princeton, NJ: Princeton University Press,