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AIAA 2001-4277 AIAA Guidance, Navigation, and Control Conference and Exhibit 6-9 August 2001 Montreal, Canada

A01-37143

ZERO-MISS-DISTANCE GUIDANCE LAW BASED ON LINE-OF-SIGHT RATE MEASUREMENT ONLY Pini Gurfil* Department of Mechanical and Aerospace Engineering Princeton University, Princeton, NJ 08540

ABSTRACT This paper presents a high performance, simple and robust guidance method which utilizes line-of-sight (LOS) rate measurement only to yield zero-missdistance (ZMD) against highly maneuvering targets. The novel guidance law adopts the basic framework of proportional navigation guidance (PNG), yet instead of using an acceleration command which is proportional to the measured LOS rate, the acceleration command is applied proportionally to an equivalent LOS rate. The equivalent LOS rate is a linear combination of the measured LOS rate and higher-order LOS rate derivatives, which are estimated from the noisy LOS rate measurement using a Kalman-Bucy filter. It is shown that this methodology resembles optimal guidance, because high-order LOS rate derivatives comprise information regarding both target acceleration and the relative range. However, while optimal guidance requires a direct estimation of target maneuver and a measurement of the relative range, the new guidance method extracts this information indirectly from the LOS rate measurement. Thus, the difficulties associated with target maneuver estimation are avoided. A considerable part of this paper is devoted to a comprehensive simulation study of the new guidance law. Deterministic simulations and Monte-Carlo analyses show that excellent performance is obtained against highly maneuvering targets. 1. INTRODUCTION Synthesis of an efficient, robust and high performance missile guidance is a formidable task. Recently, new threats have appeared which render the guidance design more difficult than ever. The interception of highly maneuvering targets, such as intercontinental and

tactical ballistic missiles, constitutes a tantalizing objective of guidance engineers worldwide. The classical proportional navigation guidance (PNG) has exhibited poor performance against maneuvering targets,1"4 and thus its suitability to interception of modern evaders is questionable. Alternatives to PNG have been derived based upon optimal control theory5"8 (one-sided optimization) and differential games9 (two-sided optimization). While these optimal guidance laws (OGLs) often offer valuable operational features, such as a small miss distance, they also suffer from several inadequacies. First, there are no closed-form expressions of the acceleration command for high-order systems; realization of OGLs is complex; and the use of loworder OGLs for high-order systems creates a model mismatch that considerably reduces the robustness to system parameter uncertainties.10 However, probably the most acute drawback of OGLs is the need to estimate the target maneuver and to either measure or estimate the relative range (or the time-to-go). While the latter task can be usually reasonably fulfilled, an efficient, high-bandwidth direct estimation of target maneuver constitutes an evolving technology which has not reached its maturity yet. Although numerous efforts have been reported in the literature,11"15 the inherent time delays associated with the estimation process16 seriously degrade the guidance performance and increase miss distance. Recent studies17"19 have proposed an alternative to PNG and OGLs: The so-called "neoclassical" approach to missile guidance adopts the general framework of PNG, but uses modern concepts such as the small gain theorem and the adjoint system4' 20 to significantly enhance the performance of PNG. The new guidance method, often referred to as zero-miss-distance PNG

* Lecturer and Research Associate, Member AIAA. Email: [email protected]. Copyright © 2001 by Pini Gurfil. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.

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(ZMD-PNG) utilizes LOS rate measurement only, similarly to PNG, but renders a miss distance which is of the order of magnitude of the OGL miss distance and often smaller. ZMD-PNG is merely PNG with an additional lead controller, designed to render the total dynamics of the guidance loop positive real. ZMD-PNG have been rigorously proved to yield ZMD for any flight time against deterministic and random bounded target maneuvers and specific stochastic inputs such as activeand passive-receiver noise in radar guided missiles.18'19 However, due to the inherent phase lead required, ZMDPNG may significantly amplify the LOS rate measurement noise to infeasible level. The purpose of this paper is therefore to establish a suitable design procedure of ZMD-PNG for the general stochastic guidance system where the LOS rate measurement is corrupted by noise. The resulting ZMD-PNG law exhibits outstanding performance. The performance of ZMD-PNG is compared to PNG and OGL using both deterministic simulations and statistical Monte-Carlo tests. The results with a target performing a randomphase sinusoidal maneuver show that the ZMD-PNG law exhibits excellent performance.

2. BACKGROUND To establish the necessary background for the novel guidance concept conceived, we begin with a brief review of the guidance kinematics. As mentioned, ZMDPNG is actually a variant of the well-known PNG. Thus, it is useful to recall the linearized PNG model. The general formulation of a nonlinear three-dimensional PNG interception problem is complicated. However, by assuming that the lateral and longitudinal maneuver planes are decoupled by means of roll-control, one can deal with the equivalent two-dimensional problem in quite a realistic manner.1"4' 22 Furthermore, a linearized model of the two-dimensional PNG about the collision course can be developed. This model has been widely used1"4' 17~19' 22 and it has been shown to faithfully approximate the full nonlinear guidance dynamics.22 A block diagram describing the linear model is given in Fig. 1. In this system, missile acceleration au is subtracted form target acceleration aT to form a relative acceleration y . A double integration yields the relative vertical position y, which at the end of the engagement, t = tf, is the miss distance y(tf). Assuming a constant closing velocity Vc , the relative range is given by A

R = Vc-tgo, where tgo=tf-t. Dividing the relative vertical position y by R yields the geometric LOS angle X . It is assumed that X is a small angle. The missile target tracking loop is modeled in Fig. 1 as an ideal

differentiator with an additional transfer function Gl ( s ) , representing the seeker, the LOS rate measurement and the noise filtering dynamics. The target tracking loop generates a LOS rate command A,m , which is multiplied by the PN gain N' - Vc to form a commanded missile maneuver acceleration ac, with N' being the effective PN constant. The flight control system, whose dynamics are represented by the transfer function G2 ( s ) , attempts to adequately maneuver the missile to follow the desired acceleration command. Target Maneuver

Missile Acceleration

miss=X'/)

y^

1 s

y

Kinematic Integrator

1 5

,1.

Kinematic Integrator

1

/I

A ———»

s

vcr

^(5)

# , ar -P-* NVC —— £———.

Seeker Dynamics [

Kinematics

Traget Tracking Loop

G2 s)

«

Flight Control System

Figure 1: Linearized PNG block diagram.

Based on the method of adjoints4'20 and the small gain theorem, it was rigorously proven in Refs. 17-19 that zero miss distance (ZMD) for the system depicted in Fig. 1 is obtained for any flight time and any bounded target maneuver provided that the conditions formulated in the following theorems are satisfied. Theorem 1: Let [PR] be the class of positive real A

transfer functions, and denote G(s) = Gl(s)G2(s). If 3K(s) ("the guidance controller") such that K(s)G(s) I s e [PR], then 3aM < oo such that

Theorem 2: Let the missile maneuver acceleration aM

be limited, \aM \ < aMmax . Denote (i0 = aMm I aTm . If 3K(s) such that K(s)G(s)/se {PR}, and in addition r>2|Li 0 /(|Li 0 -l),then y(tf) = 0 V^, V|ar|
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The positive realness requirement necessitates the guidance controller to be a proportional-derivative (PD) phase-lead controller of the form18 (1)

where m denotes the relative degree of G(s). Obviously, the design principles formulated in Theorems 1 and 2 posses a significant drawback: In addition to the realization difficulty, the noise amplification of the PD guidance controller (1) may be prohibitively high. Thus, when the LOS rate measurement is corrupted by noise, the guidance controller must be modified into a lead-lag network so that a certain extent of filtering is used. This results in a suboptimal operation, since the ZMD property does not hold when the positive realness condition is violated. However, when the filtering, or "roll-off, is performed at high frequencies, a near-ZMD performance can be achieved. The major question is, therefore, how to design an appropriate guidance controller for the stochastic case, which will provide the necessary noise filtering from one hand, but will not deteriorate the miss distance on the other hand. In order to quantify this trade-off, we first discuss the physical interpretation of the deterministic ZMD-PNG. This will provide us with the necessary insight to extend the ZMDPNG law to the stochastic case. The deterministic ZMD-PNG is implemented as described in Fig. 2a. The kinematic LOS rate A, is measured by the seeker, whose output is the measured LOS rate, Km . A,m constitutes an input to the guidance controller K(s). At the guidance controller output, we have A .

_

..

_

(m+l)

where feC,m

(3)

•__

(fzm r = £ Tz< • TZ, • *z, • - • Tzm = IITz and Cnp is the usual combinations space of choosing p

terms out of n. Note that the variable (hm)eq has LOS rate units. It represents the sum of the measured LOS rate

Am and higher-order derivatives of A,m multiplied by the coefficients of the polynomial K(s). Thus, (km)eq can be viewed as an equivalent LOS rate. The equivalent LOS rate is multiplied by the PN gain NVC to form a commanded missile acceleration. In this sense, the ZMD-PNG law is a generalization of PNG; instead of applying an acceleration command which is merely proportional to the measured LOS rate, the acceleration command is proportional to an equivalent LOS rate. The superb performance of ZMD-PNG, illustrated in Refs. 17-19, is therefore a result of extracting additional information from the LOS rate measurement. In other words, a prediction procedure is employed to obtain the future state of the target. This can be immediately seen by recalling that4 Mt) = {[2-N'G(s)].Ut) + aT/Vc}/tgo (4) Which implies

(5) Hence, the use of A (and possibly higher-order derivatives, if needed) as an additional signal for generation of the commanded maneuver acceleration is equivalent, in a sense, to using information regarding both target maneuver and relative range (or time-to-go), as required by optimal guidance methods. However, ZMD-PNG extracts this information indirectly from the LOS rate measurement, whereas optimal guidance laws need direct estimation of target maneuver and either measurement or estimation of the relative range. Obviously, merely an estimation of LOS rate derivatives is far simpler, and as subsequently shown, yields good performance. 3. ESTIMATION OF HIGH-ORDER LOS RATE DERIVATIVES The measurement of LOS rate is corrupted by noise, usually as a result of sensor imperfections (such as rategyro noise). A simple differentiation of the measured LOS rate to generate high-order derivatives is therefore infeasible. Rather, high-order derivatives should be estimated from the LOS rate measurement. To this end, consider the measurement equation

*«=*,+v

(6)

where v is a zero-mean Gaussian white noise and hs is the noise-free LOS rate measurement. It is required to recover high-order derivatives of Xs out of the noisy measurement (6). The highest order derivative required is determined by m, the relative order of G(s) (see Eq. 1). A diagram of the estimation process, which constitutes the stochastic ZMD-PNG, is depicted in Fig. 2b. The input to the estimator is the noisy measurement

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A, m . The estimator provides the estimated high-order „

(m+l)

derivatives, hs,ks,...,Xs , which are then combined

using some weighting vector h to yield the estimated equivalent LOS rate:

mainly from the firm relationship between the target acceleration and the high-order LOS rate derivatives, as pointed out in Eq. (4), and the probabilistic interpretation of EGA, described as follows. Let the highest order LOS rate derivative, denoted (m+l)

A,5 , be a zero-mean random variable uniformly

"I

(m+l)

distributed

within

the

limits

± ( X , ) max .

The

(m+l)

(7)

probability of attaining an extremum value is 7Tn (m+l)

is therefore given by

The variance of

(m+l)

2

(m+l)

Qualitatively speaking, since Km is the only measurement, the higher the order of the required LOS rate derivative is, the larger the estimation error will be. Fortunately, the relative order of the overall loop dynamics is often small; thus, practically speaking, there is usually no need to estimate LOS rate derivatives

beyond the LOS jerk (X 5 ). Nevertheless, we consider hereafter the general estimation procedure. The issue of determining the weighting vector h will be addressed in the sequel. Deterministic ZMD-PNG

[1 + 471

]/3

(8)

Using a standard Wiener-Kolmogorov whitening procedure,24 a shaping filter of the exponentially (m+l)

correlated A,c can be derived:

_

(9)

^ (t)-

where TE is the random process correlation time and w(Y) is a zero-mean Gaussian white noise with power spectral density (PSD): q=2vl+l)/i3E (10)

The dynamics of the ECH model can be thus written in the state space representation: Seeker Dynamics LOS Rate Measurement No se

Flight Control System

o.

tm

A(0 + -1/T,

Stochastic ZMD-PNG

where A r =[ks, X 5 ,..., Ks ],

^

A/V

p N.Gai

Seeker Dynamics

°b^ „, x * Gi(,v)

identity matrix, 0^x/ is a kxl

(12)

Guidance Controller K(s)

In order to estimate high-order LOS rate derivatives, we use a variant of the well-known exponentially correlated acceleration (EGA) approach, which is a kinematicmodel estimator23 first introduced by Singer11 for estimation of the target maneuver. Several appropriate modifications will be performed to adapt this approach to the estimation of high-order LOS rate derivatives. The new technique will be referred to as exponentially correlated high-order LOS rate derivative (ECH) estimation. The application of the EGA approach to estimation of high-order LOS rate derivatives stems

zeros matrix and w(t)

is a zero mean Gaussian white noise with PSD q ,

Flight Contro System

Figure 2: Implementation of ZMD-PNG. (a) The deterministic case, (b) The stochastic case.

is an mXm

The initial conditions satisfy E[A(0)] = A(0) (13) cov[A(0)] = P0 The measurement equation (5) is re-written as follows: X m = [1,0,...0]A + v(t) = crA + v(t) (14) where /Tl ijf/M — (1

(15) and r is the PSD of the measurement noise. It is important to note that r as used in the estimator design need not necessarily be equal to the actual LOS rate measurement noise. Also, we assume that the initial

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conditions, measurement noise and process noise are uncorrelated, £[A(0)w(0] = £[A(0)v(f)] = E[w(t)v(t)] = 0 (16) The ECH model formulated in Eqs. (!!)-( 16) can be

used to obtain an estimated state vector A using the well-known time-varying Kalman-Bucy filter (TVKF),21

= P(t)c/r

(17) T

T

T

dynamics, is adding a first-order lag. This, of course, results in miss distance increase, due to the violation of the positive realness condition stated in Theorems 1 and 2. The purpose of the weight vector h is to partially recover the miss distance performance obtained prior to the filtering process. In practice, h quantifies the tradeoff between miss distance and noise filtering. A method for selecting this vector is suggested next. First, note that (20) can be partitioned as

n2(s)

F(J)=

= AP(t) + P(t) A + qgg - P(t)cc P(t) I r

(23)

to

where A) 7 )] (18) Note that the ECH constitutes a kinematic-model estimator. Thus, 3P>0 such that \imP(t)-^P (Ref. t—>°o

23), and the steady-state Kalman-Bucy filter (SSKF) is given by: k = Pc/r

where dSSKF (s) is the filtering dynamics of the SSKF, and nt(s) is the numerator polynomial of the i-l order LOS rate derivative estimator. In the deterministic case, we had dSSKF (s) = I , since no filtering was used. To satisfy the positive realness condition in the deterministic case, formulated by Theorems 1 and 2, h = [1, Aj , . . . hm ]T would have been required to satisfy

(19)

T

T

AP + PA - Pcc P/ r = It is of prime importance to note that in the specific application discussed here, there is no significant

advantage of using the TVKF rather than the SSKF. With this notion in mind, we adopt the SSKF, and obtain the vector of filter transfer functions between the m+l components of the vector A and the input X m , (m+l)

to

A* to

where f\ F — f\

K. * t.

(21)

bF=k

4. CHOOSING A WEIGHTING VECTOR Eq. (20) present a closed-form expression for the estimates XS9XS9...9XS

. However, in order to obtain

the guidance controller whose output is

, these

estimates should be combined using some weighting vector h, (see Eq. (7)). Thus, in the stochastic case we have

(22) By observation, the denominator of K(s) is of order m +1 and the numerator is of order m. Thus, the net effect of the ECH filter, in terms of the overall

The same procedure can be adopted in the stochastic case. That is, first we ignore the presence of the denominator polynomial and assume dSSKF (s) = I . Next, we find h such that (24) is satisfied. In a sense, this concept represents a variant of the separation principle, where the estimation and control designs are done separately and then interconnected. The overall system robustness reduction that stems from using the separation principle as a design methodology reflects itself in our case by the slight increase in miss distance compared to the deterministic case. 5. ILLUSTRATIVE EXAMPLE In the previous sections, a new guidance law, ZMDPNG, was synthesized. The purpose of this section is to investigate the performance of ZMD-PNG when implemented in a real-life electro-optical missile. 5.1 Guidance Law Synthesis

In order to design a ZMD-PNG law, transfer functions of the flight control system, G2 (s) = aM (s) I ac (s) , and

the tracking \oop,Gl(s) = A,W /A , are required. These transfer function can be found in two steps. First, the nonlinear terms are left out. Second, the resulting highorder linear models are reduced using state truncation method such as balanced realization. It is important to stress that this procedure is used for the guidance design only, not for the overall performance evaluation of the missile, where the complete, detailed non-linear stochastic models are used. We start with model reduction of the complex flight control system, described in Appendix A, which has 9 zeros and 13 poles. Using balanced realization state

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truncation, and the parameter values given in Appendix C, the following reduced-order transfer function is obtained:

(s 1 23.3 + 1)(5/ 1.93 + 1) An additional state truncation yields

The resulting transfer functions vector is

Us) 0.0037s2 + 0.087s+ 1

(25) F(S) =

1

Us)

0.0001s3 + 0.038s2 + 0.087s+ 1 Us)

(26) 0.56^ + 1 The simplified model (26) constitutes an adequate approximation to the overall flight control system dynamics, both in the frequency and time domains. It is subsequently used for ZMD-PNG design. The tracking loop overall transfer function, Gl (s) , is obtained in a similar manner. Using the numerical values of Appendix C, neglecting the FOV saturation and the pure tracking delay, we have

0.0814s 2 +s 0.876s2

(35)

The next step is to choose the weighting vector h. To this end, we follow the design guideline discussed in Section 4 (Eqs. 23, 24) and choose h = [1, 0.3, 0.2]r (36) which yields the guidance controller 0.20352+0.3875 + 1 K(s)=hTF(s)=•(37) 0.000 b3 +0.03852 +0.0875 + 1 The final step is choosing N'. To this end, we utilize c w <27) the design principle given in Theorem 2. Assuming that ' the missile-target maneuver ratio is JI0 = 2, requires The overall transfer function of the guidance loop is N' > 4. We chose N' = 5 , to account for possible therefore: uncertainties in //0 (that is, the case where the actual 1 G(s) = Gl(s)G2(s)=(28) missile-target maneuver ratio is smaller than 2). In (0.15 +1)(0.565summary, the ZMD-PNG command to the flight control The relative order of G(s) is m=2, so the highest order system is LOS rate derivative required is the LOS jerk., A, 5 . 0.20352 +0.3875 + 1 ac=5-Vc (38) Assume that the correlation time of Ks equals the total 0.000153 +0.03852 +0.0875 + 1 time constant of the reduced-order system (28), i. e. The performance of the guidance law (38) will be 1E - 0.66 . Applying (11), we write the ECH model for compared to OGL and the classical PNG. The this case: commanded OGL maneuver acceleration is:5 "01 0 "0 ac = N'&[y+yfgo (39) 0 0 1 0 w(t) (29) 0

0

-1.51

1.51

Where tgo is the estimated time-to-go, aT is the

r

where A = [A,5, A, 5 ,A, 5 ]. The PSD of the process estimated target maneuver, TD is the so-called "design" A ^ noise is evaluated using (10). To this end, let time constant and ^=fgo /1D. Due to the fact that the 3 $s)max = 750deg/sec ,7imax =0.5 (30) most common OGL was originally conceived for firstwhich yields: order dynamics, there is a considerable mismatch q= 45000 deg2/sec9 (31) between actual missile dynamics and the OGL. To deal The measurement equation is given by with this mismatch, it was suggested10 to use a "design" time constant, TD , which is about 1.5 times larger than X m =[l,0,0]A + v(0 (32) the equivalent time constant of the system. In our case, with 2 2 the equivalent time constant is 0.66 sec, so TD = 1 sec . = 0.04deg /sec (33) It is also assumed that the time-to-go is estimated Note that, generally speaking, the parameters in (30)exactly, i. e. tgo = tgo. For the estimation of target (33) constitute tuning parameters, that reflect the usual trade-offs associated with Kalman filter design. The maneuver, the following model was adopted:16 SSKF estimator is initialized as follows: aT=e-"°-aT (40) (34) A(0) - 0

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In (40) it is assumed that the target maneuver estimator constitutes a pure time delay of Ta seconds. We chose

Evidently, both ZMD-PNG and OGL yield superior performance comparing to PNG. Note that with PNG, the actual maneuver acceleration saturates 1.5 seconds ia - 0.2 sec , which is a rather optimistic value, since before impact, which seriously increases the miss often the estimation delay is even larger. The PNG distance. However, with ZMD-PNG this saturation is design is far simpler. The sole degree-of- freedom is N' . avoided, as expected,18 and the miss distance is much We chose N' = 5 , so that with PNG we have smaller. Note also that ZMD-PNG requires a smaller maneuver effort than OGL, because the OGL used here ac=5-Vc-Xm (41) is actually suboptimal due to the high-order system The target maneuver simulated to test the performance dynamics. The three guidance laws examined yielded of the three guidance laws was a random phase the following miss distance: sinusoidal maneuver. In addition, the starting time of the = -0.108m, y(tfJ)\ I OGL =0.114m, ,,~^ maneuver is unknown; It can be initiated at any given ZMD-PNG time point within the flight time of the interceptor. Thus, -60.2m the target maneuver can be described by PNG Thus, while ZMD-PNG and OGL render a similar miss aT (0 = aTo sin(cor/ + (|)) (42) distance, PNG induces a considerably larger miss. Where aT is the maneuver magnitude, COT is the We proceed with a thorough statistical examination of the guidance laws performance using a Monte-Carlo frequency and § is the phase, assumed uniformly technique. In each simulation run, parameter values, as distributed between 0 and 271, which is completely well as the seed used to generate the noise signals, are equivalent to a random maneuver initiation time. In this randomly selected according to pre-specified example, the numerical values chosen where aT = lOg probability distribution functions. After a large database and COT =1.7 rad/szc. of simulation runs has been created, the results, especially the miss distance, are statistically analyzed. 5.2 Guidance Law Performance In this example, each simulation run used random parameter values (based upon the data given in The performance of ZMD-PNG, PNG and OGL against Appendix C) and a random target maneuver phase. For the random phase sinusoidal target maneuver includes each flight time, 300 simulation runs were performed. both deterministic simulation runs and a Monte-Carlo The procedure was repeated for flight times ranging analysis. For the deterministic case only, a constant from 2 sec to 10 sec. In order to evaluate miss distance flight time, tf - 5 sec , and a constant target maneuver statistics, the mean value of the absolute miss distance, phase, <|) = 0 , were chosen. The time histories of the defined by actual and commanded accelerations for (44) Vc = 1000 m/s are depicted in Fig. 3. 20 10

Jo -10 -20 (0

0.5

1

1.5

2

2.5

3

3.5

3

3.5

4

4.5

time [sec] 40

5

were examined. In addition, standard deviations of miss distance for each flight time was considered. The results are depicted in Figs. 4, 5. Fig. 4 compares the mean value of absolute miss distance, and Fig. 5 compares the standard deviation of miss distance. PNG was not considered here, because the deterministic examination showed that it yielded miss distance which was larger than the miss obtained with ZMD-PNG and OGL by an order-of-magnitude.

20

-20

-40

0.5

1.5

2

2.5

4.5

time [sec]

Figure 3: Comparison of actual and commanded missile acceleration.

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guidance laws (OGLs). ZMD-PNG yields small, near zero, miss distances against highly maneuvering targets for a wide range of flight times. This conclusion was established based upon Monte-Carlo analysis of miss distance. The acceleration commands generated by ZMD-PNG, although noisier than acceleration commands of OGL, are reasonable in magnitude, and are actually smaller than acceleration commands of an OGL designed for a first order system.

0.9

-300 Monte-Garlo Runs

0.8

"V"-' b_-cr

o J0.3

50.2 0.1

8

9

10

Flight Time [sec]

Figure 4: Mean values of the absolute miss distance with ZMD-PNG are smaller than with OGL.

3

4

5

6

7

8

Flight Time [sec]

Figure 5: Standard deviations of the miss distance with ZMD-PNG are smaller than with OGL. Clearly, ZMD-PNG yields smaller mean and root-meansquare (RMS) miss than OGL. This is true for all flight times greater than 2 seconds. Moreover, with ZMDPNG the miss distance is more robust to variations in flight times and in missile parameters. These observations imply that the newly developed guidance law can deal adequately with highly maneuvering targets. This merit is achieved with neither estimating target maneuver nor estimating time-to-go.

ACKNOWLEDGEMENTS This work was carried out as part of the author's doctoral studies in the Faculty of Aerospace Engineering at the Technion, Israel Institute of Technology. Some of the ideas in this paper were inspired by discussions with Dr. H. Weiss from RAFAEL. The author would also like to gratefully acknowledge the useful remarks and support of his Ph.D. advisors, Dr. M. Jodorkovsky from RAFAEL and Prof. M. Guelman from the Technion. APPENDIX A: MODEL OF THE FLIGHT CONTROL SYSTEM The flight control system used in this example was adopted from Ref. 26 and is depicted in Fig. 6a. This pitch-plane three-loop control system comprises a rate loop, a synthetic stability loop and an accelerometer feedback loop. The input to the accelerometer feedback loop is the commanded acceleration ac, which is generated by the guidance law. The output is the required acceleration aR, which is limited due to aerodynamic or structural constraints, to yield the actual acceleration aM . The autopilot of this loop is the gain

KA. The feedback signal (^^) m is generated by an accelerometer, which is located at the point X ACC , thus sensing the following acceleration magnitude: a A ( t ) = aM(t) + (XACC-XCG)Q(t) (Al) Where XCG is the location of the center-of-gravity and 0 is the body pitch angle. The signal aA is the output of the aerodynamic transfer function aA /8 , with 5 being the fin deflection angle. 8 is also the output of the body pitch rate control loop, whose input is the commanded pitch rate qc, generated by the synthetic stability loop.26 The autopilot of the pitch rate loop is the gain Kq. This gain generates a commanded fin deflection

6. CONCLUSIONS This study presented a new guidance law, called zeromiss-distance proportional navigation guidance (ZMDPNG), which applies an acceleration command proportionally to an equivalent line-of-sight (LOS) rate instead of merely the LOS rate. The equivalent LOS rate angle 5C , which constitutes an input to the fin actuators. is a linear combination of the measured LOS rate and The aerodynamic transfer function aR/8 then yields high-order derivatives of the LOS rate estimated from the required output acceleration aR . The values of the the LOS rate measurement using a Kalman-Bucy filter. 26 The new guidance law does not require either autopilot gains and the other parameters are given in estimation of target maneuver or measurement of the Appendix C. The aerodynamic model is obtained relative range, as needed when implementing optimal assuming a short-period approximation. For an axisAmerican Institute of Aeronautics and Astronautics

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symmetric missile, this model yields the following aerodynamic transfer functions:26

8

dl2s +dns + l

(A2) The definitions of the coefficients are given in (26). The numerical values of the aerodynamic coefficients are given in Appendix C. The coefficients are assumed normally distributed, with mean and standard deviation as given in Appendix C. We proceed with a description of the fin actuator control loop. The purpose of the fin actuators control loop is to track a commanded fin deflection angle 8C to yield an actual deflection angle 8 . The most commonly used fin actuators control loop model assumes second order dynamics of the form C*

]f

_:_ -

5C

servo_____________

7/K

(A3)

f +2- C»

is the natural frequency, £>servo *s a damping

where (!)„

coefficient and Kservo is a scale factor. In practice, the fin actuator control loop is rather complex. There are acceleration, rate and position limits:

K S / N MO = S8^ sat \

(A4)

50)

(&A)B is a measurement bias, assumed a Gaussian constant, and na is the measurement noise, assumed zero-mean, white and Gaussian. Similarly, the rate gyro model is given by K RG Where co

is the natural frequency of the rate gyro,

^RG is a damping coefficient, KRG is a scale factor, qB is a measurement bias, assumed a Gaussian constant, and nq is the measurement noise, assumed zero-mean, white and Gaussian. Numerical values are given in Appendix C. APPENDIX B: MODEL OF THE ELECTROOPTICAL TARGET TRACKING LOOP The layout of the tracking loop, depicted in Fig. 6b, was adopted from Ref. 27. This tracking loop is based upon a rate-gyro stabilized platform, where the camera is mounted on gimbals, whose movement is (ideally) isolated from the motion of the missile. The location of the target within FOV limits is measured by an electrooptical tracker, which is an implementation of a correlation algorithm that utilizes the sequence of images generated by the visual. Thus, the tracker generates a measurement £m of the tracking error £. The FOV is a nonlinear saturation effect given by27 x, \x\ < 1 (Bl) fov(jc)=' 0,

whereS max ,S max ,8 max are the maximal values of fin angle, rate and acceleration, respectively. Numerical values are given in Appendix C. In addition, there are disturbances such as fin bias, 80, and fin position measurement noise n5. It is assumed that 8B is a Gaussian random constant and that U8 is a zero-mean Gaussian white noise. Numerical values are given in Appendix C. Our next goal is modeling of the flight control sensors. The missile flight control system uses an accelerometer as a feedback sensor located in the outer acceleration control loop, and a rate gyro in the inner rate control loop. The accelerometer is modeled as

where CACC is

co a

(A5) is the accelerometer natural frequency,

damping coefficient, KACC is a scale factor,

The resulting tracking error satisfies

(B2) The tracking error £ L is transformed into a commanded LOS rate Xc by the tracking loop controller, which is, in our case, the gain KT (see Fig. 6b). The commanded LOS rate constitutes an input to the seeker, that generates a measured LOS rate, A,m . This signal is integrated to yield a measured LOS angle X m , which is subtracted from the true (kinematic) LOS angle A, to yield the tracking error £ . Since the electro-optical tracker is merely a computational algorithm, it can be best modeled by a pure delay:

e m (0 = «"itTe(0 + we

(B3)

where TT is the tracking delay and ne is the tracking noise, assumed zero-mean, white and Gaussian. Numerical values are given in Appendix C. The seekef

.g

as follows. 27

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(c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

i

i\>m — o2

->

""

Parameter

i^r 'i xi^ w 'i n~i \u*r /o/i\)

/

""

, /«J + 2.C^ r -*/»,,_ + l

^

is the natural frequency of the seeker, ^seeker

^-w,

is the damping coefficient, XB is the LOS rate measurement bias,

r^ servo

assumed a Gaussian constant, and n^ is the LOS rate measurement

]£

Where as usual (On

noise,

assumed

Ji

^>--

zero-mean,

white

J

K

. —*®—* v - *€>—* Synthetic | Stability Loop

d

c

Gaussian. aK

--+ y

\Q m „ _ „ _ _ _ ?

Accelerometer

Q . o

^aA

aR \

' /— aM :

—-+\"/f -— ^

Fin actuators

SB n

s

^max max

aA ^ S

max

e )—^> ±+% >O

Tracker

i,

^^-'A^ K *

-

Field-of-View Saturadon

4

It ^

^>

Seeker

Control Loop

0>n n RG

^ ^

Controller

Rate gyro

Figure 6: Block diagrams of (a) the flight control system and (b) the electro-optical target tracking loop.

Aerodynamics

I/ sec2

Mean value (nominal) -250

Standard deviation 16.7

I/ sec2

280

18.7

za Mzs

I/ sec

-1.6

0.107

I/ sec

0.23

0.0153

<

I/ sec

-1.5

AX

m m/sec

0.7 500

m/sec

1000

VM

vc *, ^« ^i

deg

0

0.0167

deg

20

deg/se c deg/se c2

230

rad/ sec

Accelerometer

13.55

Table Cl: Parameter values for the illustrative

3

0.65

0.0108

1

0.0267

deg/ se

0

0.0167

rad/ sec

300

3

0.65

0.0108

1

0.0167

milli-g

0

1

milli-g

0

1

msec

25

mrad

0

deg

1.5

I/ sec

10

rad/ sec

150

1.5

0.7

0.0117

K

AC

K), n

a

TT n

e

^max

CO seeker

-

300

n

KT

0.1534

17000

0.0667

b ACC

Seeker

rad/sec

0.167

0

n

deg- sec/ w 0.0007804

0

deg/ se

n

Target tracking loop

^0

Autopilot

deg

qB CO ACC

2

sec

0.0167

RG

o

ILLUSTRATIVE EXAMPLE The following table contains parameter values that were used in the illustrative example.

Ma M8

1

K

APPENDIX T! PARAMETER VATITES FOR THE

Units

0.01

b/?G

1 s

Parameter/ Disturbance

0.6

servo

Acceleration Limit

Rate Loop

(aA)m

Acceleration Loop

K

and

d

Mean value Standar deviation (nominal) 2 200 rad/ sec Units

r^ seeker

0.05

4

deg/ sei

0

0.0667

n

deg/ se

0

0.0167

i

Table Cl (cont.): Parameter values for the illustrative example

10 American Institute of Aeronautics and Astronautics

(c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

REFERENCES ^archan, P., "Representation of Realistic Evasive Maneuvers by the Use of Shaping Filters", /. Guidance and Control, Vol.2, No.4, July-August 1978, pp. 290-295. ^archan, P., "Complete Statistical Analysis of Nonlinear Missile Guidance Systems-SLAM", /. Guidance and Control, Vol.2, Jan. 1979, pp. 71-78. 3 Zarchan, P., "Proportional Navigation and Weaving Targets", J. Guidance, Control and Dynamics, Vol. 18, No. 5, Sep.-Oct. 1995, pp. 969-974. 4 Zarchan, P. Tactical and Strategic Missile Guidance, Progress in Aeronautics and Astronautics, Vol.124, AIAA, Washington, DC, 1990, pp. 37-58; Chapter 3. 5 Cottrel, R. G., "Optimal Intercept Guidance for Short Range Tactical Missiles", AIAA Journal, Vol. 9, July 1971, pp. 1414-1415. 6 Holder, E. J., and Sylvester, V. B., "An Analysis of Modern Versus Classical Homing Guidance", IEEE Transactions on Aerospace and Electronic Systems, Vol. 26, No. 4. July 1990, pp. 599-605. 7 Rusnak, I., and Meir., L, "Optimal Guidance for High Order and Acceleration Constrained Missile", / Guidance, Control and Dynamics, Vol. 14, No. 3, May 1991, pp. 589-596. 8 Stockum, L. A., and Weimer, F. C, "Optimal and Suboptimal Guidance for a Short Range Missile", IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-12, No. 3, May 1976. 9 Gutman, S., "On Optimal Guidance for Homing missiles", /. Guidance, Control and Dynamics, Vol. 2, No. 4, July 1979, pp. 296-300. 10 Weiss, H., and Hexner, G., "Modern Guidance Laws with Model Mismatch", in Proc. of the IF AC Symposium on Missile Guidance, January 1998, TelAviv, Israel. 11 Singer, R. A., "Estimating Optimal Tracking Filter Performance for Manned Maneuvering Targets", IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-6, No. 4, July 1970. 12 Tang, Y. M., and Borrie, J. A., "Missile Guidance Based on Kalman Filter Estimation of Target Maneuver", IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-20, No. 6, November 1984. 13 Hepner, S. A. R., and Geering, H. P., "Observability Analysis for Target Maneuver Estimation via Bearing-Only and Bearing-Rate-Only Measurements",/. Guidance, Control, and Dynamics, Vol.13, No. 6, Nov.-Dec. 1990, pp. 977-983. 14 Hepner, S. A. R., and Geering, H. P., "Adaptive TwoTime-Scale Tracking Filter for Target Acceleration Estimation", J. Guidance, Control, and Dynamics, Vol.14, No. 3, May-June 1991, pp. 581-588.

15

Chang, W. T., and Lin, S. A., "Incremental Maneuver Estimation Model for Target Tracking", IEEE Transactions on Aerospace and Electronic Systems, Vol. 28, No.2, April 1992, pp. 439-451. 16 Shinar, J., Shima, T. and Kebke, A., "On the Validity of Linearized Analysis in the Interception of Re-entry Vehicles" in Proceeding of the 1998 AIAA Guidance, Navigation and Control Conf., Boston, MA., paper CP-4303. 17 Gurfil, P., Jodorkovsky, M., and Guelman, M., "Simple Guidance Law against Highly Maneuvering Targets", in Proc. of the 36th AIAA Conf. on Guidance, Navigation and Control, Boston, MA, Aug. 1998, pp. 570-580. 18 Gurfil, P., Jodorkovsky, M., and Guelman, M., "Design of Non-Saturating Guidance Systems", /. Guidance, Control and Dynamics, Vol. 23, No.4, July

2000, pp. 693-700. Gurfil, P., Jodorkovsky, M., and Guelman, M., "Neoclassical Guidance for Homing Missiles", to be published in /. Guidance, Control and Dynamics, Vol. 24, No. 3, May 2001. 20 Zadeh, L. A., Desoer, C.H., Linear System Theory-The State Space Approach, McGraw-Hill, New York, 1963, pp. 337-369; Chapter 6. 21 Kalman, R. E., and Bucy, R. S., "New Results in Linear Filtering and Prediction Theory", Transactions of the ASME, J. Basic Engineering, Vol. 83D, pp.95108, March 1961. 22 Shinar, J., and Steinberg, D., "Analysis of Optimal Evasive Maneuvers Based on a Linearized TwoDimensional Kinematic Model", J. Aircraft, Vol. 14, No. 8, August 1977, pp. 795-802. 23 Bar-Shalom, Y., and Li., X. R., Estimation and Tracking: Principles, Techniques and Software, Artech House, Boston, 1993, pp. 625-655; Chapter 9. 24 Fitzgerald, R. J., "Shaping Filters for Disturbances with Random Starting Times", /. Guidance and Control, Vol. 2, No.2, March-April 1979, pp.152-154. 25 Ohlmeyer, E., J. "Root-Mean-Square Miss Distance of Proportional Navigation Missile Against Sinusoidal Target", /. Guidance, Control, and Dynamics, Vol.19, No. 3, May-June 1996, pp. 563-568. 26 Nesline, F. W., and Nesline, M. L., "Homing Missile Autopilot Response Sensitivity to Stability Derivative Variations", in Proc. of the 23rd IEEE Conf. on Decision and Control, Las Vegas, December 1984. 27 Shneydor, N. A., Missile Guidance and Pursuit : Kinematics, Dynamics and Control, Horwood Publishing, 1998, pp. 125-148; Chapter 7. 19

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