a new look at classical versus modern homing missile guidance-printed
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A New Look at Classical Versus Modern Homing Missile Guidance-printedTRANSCRIPT
A NEW LOOK AT
CLASSICAL VERSUS MODERN HOMING MISSILE GUIDANCE
F. Willim ,Nesline and Paul Zarchan Raytheon Company, Missile Systems Division
Bedford, Mastachusetts 0 1730
Abstract
Modern guidance systems are generally accepted to yield better performance than classical propor- ticmal navigation systems. However, it is not always recognized that this better performance carries with it certain costs in inproved components or additional instruments. This paper compares a modern gui- asnce system, MGS, to a classical proportional navi- gationai, PIV , homing misslle guidance system in terms of perforuance, robustness, and ease of implementation. Q aan titative first order m i s s dis- tances are compared to show that MGS has the small- est miss if component tolerances can be met, but as component tolerances or measurement errors degrade, M G S degrades faster than P N until, at relatively large component or measurement erl-ors, P N has less miss distance than MGS.
Introduction
During the 1960's modern control theory was used in theoretical studies of closed form guidance laws for interceptor missiles. I t was shown that P N was an optimal soilltion to the linear guidance prob- lem in the sense of producing zero m i s s distance for the least integral square control effort with a zero lag guidance system in the absence of target maneu- ver. ( 1 ) Aithough the reasons for the popularity of PN had nothing to do with that analysis, tbis important result gave credibility to the use of modern zontrol theory as a tool that many analysts have used to derive missile guidance laws. ( 2 P 3.4) Although much has been written concerning the math- ematics of guidance, little, if any, has appeared in the open literature concerning the practical imple- mentation of a modern guidance system, MGS .
Proportional navigation has been in use for over three decades on radar, T V , and IR homin B missile systems because of its effectiveness. ( 5 , 6 Although P N was apparently known by the German scientists at Peenemiinde, no application using PN was reported. (7 ) I t was first studied by C. Yuan and others during World War I1 at the RCA Labora- tories under the auspices of the U . S . Na-j ; r ( g ) , it was extensively studied by Bennett 3nd Matthews at Hughes Aircraft and implemented in a pulse radar system(9), and it was fully developed by H . Rosen and M. Fossier for a continuous wave radar system at Raytheon Company. The latter development included a closing velocity multiplier to compensate the guidance law dynamically in flight for changing engagement geometry. After World War 11, the U . S . work on P N was declassified and first appeared in the Journal of Applied Physics. ( 10) .
The purpose of this paper is to compare both classical and modern methods of guidance in terms
Copyright 1979 by F .W. Nesline and P. Zarchan with release to AIAA to publish in all forms.
of performance and implementation. Both guidance philosophies are reviewed and typical implementations are discussed. Finally both methods of guidance are comp~red in terms of performance and sensitivity to errors in implementation.
Proportional Navigation
If two bodies are closing on each other they will eventually intercept i f the line of sight between the two does not rotate. Proportional navigation, PN, is a method of guidance in which the missile acceleration is made proportional to the line of sight rate. The geometry of ar, idealized intercept in which the missile and target are closing on each other at constant speed is shown in Fig. 1. Here movement of the missile and target causc the line of sight to rotate through a small angle, A , indicating a differential displacement, y , between target and missile perpendicular to the reference. The PN guidance law is an attempt to mechanize an accelera- tion command, nc, perpendicular to the line of sight according to
. where A ' is the effectjve navigation ratio, Vc is the closing velocity, and A i s the line of sight rate.
Figure 1 - Intercept Geometry
The effective navigation ratio determines both the trajectory and acceleration history of the missile. For a zero lag guidance system, PN will result in zero miss distance [ y ( t ~ ) =0] due to heading error or target maneuver for any N' (assuming infinite missile accelex ation capability) . This phenomenon i s clearly demonstrhted for the head-on case in the normalized trajectories shown in Fig. 2. Although relative tar get-missile iisplacement during the flight increases with decreasing ? I t , all flights result in zero miss distance. The effective navigation ratio
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TARGET-WSGILE SWAhnT lW W t ii TARGET MANEUVtir
Figure 2 - Typical Proportional Navigation Trajectories
m i s s distance . Normaliz 3d missile acceleration histories due to both disturbances are shown in Fig. 3. Here missile a~ celeration i s monctorlirally decreasing (except for Nt=2) for a heading errcr disturbance and monotonically increasing for a target maneuver disturbance. Figure 3 also shows that increasing N' minimizes the maximum acceleration due to target maneuver but maximizes the maximum acceleration due to heading error. In practice the navigation ratio is held fixed with acceptable values ( 3 < N'C 5) determined by noise, radome and target maneuver considerations.
A typical implementation of a P N guidance system is shown in Fig. 4 where an inertialiy . stabilized seeker is used to measure the boresight error, E . This signal, which is proportional to the line of sight rate, is low pass filtered to obtzin an estimate of the lirw of ight rate. The time constant, TN , of the noise filter r .In be fixed, as in this irnplemen tation, or time-varying to account for the range dependence of t\e measurement noise. The closing velocity c- either be estimated, as in 1R
MISSILE ACCELERATION DUE TO HEADING ERROR
applications, o r measured by a doppler radar, a s in radar homing applications. The resulting accefera- tion command which is proportional to the line of sight rate estimate is applied to an acceleration . uto- pilot that moves wing or fail control stlrfaces so as to develop the commanded acceleration.
Other guidance concepts such as those used in modern guidance systems can best be w.derstood by studyink proportional navigation. I t can be observed from Fig. 1 that PN is mathematically equivalent ;0
The espression in the brackets of Eq. ( 2 ) represents the miss distance that would result (in the absence of target maneuve-) i f the missile made no further corrective accelerations and is referred to a; the zero effort miss, ZEM. Therefore P N can be thought of
MISSILE ACCELERl
TIME =F
Figure 3 - Tvpical Proportional Navigration A c c e l ~ r a + i + - U G - f --: m e
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Figure 4 - Propor t iona l Navigation Guidance Sys tem - P N
as a guidance law in which acceleration commands Modern Guidance a re issued inversely proportions; t o the square of time-to-go and directly proportional to ' h e ZEM. If In a mc inrn guidance system, the ZEM is target maneuver , nT, is considered. the Z E M modified to take into account target maneuver ~ n d change3 and a new guidance law known a s augmen- missile guidance system dynamics. If the guidance ted proport ional navigation, APN, r e su l t s system dynamics can be r6presented by a first
order transfer function, with bandwidth w , either N ' 1 modern control theory (11) or the Schwartz Inequality
= 7- [ y + + 2 n T t g o 2 ~ t
(3 ' ( see Appendix D ) cap be used in deriving a guidance
go law wEich drives the miss distance to zerq vhile
This guidance law is compared to P N , in ,rms of trajectory and acceleration histories, for the case of a maneuvering t a rge t with t h e r e s u l t s dbp l ayed in Fig. 5. Although l ~ + f : guidance laws achieve zero miss distance, the tl L 'L :torr and acceleration histories are vastly dr s e n t . The informatirn con- cerning target maneuver enabies A P N guidallce to use up l e s s acce le ra t ion capabili ty than 9 N while keeping it c l o se r t o an intercept course . In addition the A P N acce le ra t ion hls tory is msnotonically d e - creasing unlike the monotanically increas ing h i s to ry of P N . It can be shown that f a r N' = 3 A P N i s opt i - m a l in the s ense that it achizves z e r o m i s s d i s tance utlhaing t h e l eas t integral <,quare cont ro l effort ( s t e Appendix C ).
TARGET-MISSILE SEFAhATION
I DUE TO TARGET MANFUJER
minimizihg the integral of the square of tiat acceler- a tion
[y( tF)=O subject to minimizing i,' nc2d t ] . This
law can be written a s
MISS11 E ACCELERATION DUE TO TARGET MANEUVER
TIME t f
F i g u r e 5 - Guidance Method Comparison
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where
The expression within the brackets of Eq. (4) is the ZEM and Eq. (6) shows that the effective navigation ratio is tirne-varying. The~aefoze this guidance law is a form of APN with an extra term to account for guidance system dynamics a1 i a time- varying navigation ratio.
This guidance law, unlike P N , requires inbor- mat ion concerning time- lo- go, t ,, , guidance system bandwidth, o , and achieved rni&ile acceleration, n~ . Range measurements and additional filtering are required to estimate tgo and acc?krometer measurements must now be fed into the guidance sys tern. The states required for the implementation of this guidance law (y, +, n ~ ) can not be obtained from a simple low pass filter as was done s-ith P N , but must be e stimatsd,
A simple form of a Kalman estimator can be derived by considering the two most importart stochastic disturbances in a guidance syste~, , random target maneuver and glint noise, T resultir e; Kalman filter is stationary and represented
transfer function
with characteristic frequency, w given by 0 '
where r P , and b, are estimates of the spectral density levels opthe target rnaneu-. cr process noise and glint meec~urernent noise respectively (See
Appendices A and B for derivations vra Wiener and Kalrnan filter formulations). Thus the characteristic frequency of the filter increases with increasing process noise and decreases with increasing measure- ment noise.
A typical implementation of a modern guidance system appears in Fig,. 6. Here the line of sight angle is reconstructed from s seeker measurement of the boresight error and by integrating the rate gyro measurement of the seeker dish rate. This angle is then converted to relative target-missile position, y*, by the rnultiplicatiun of the r a g e measurement. Tine signal is then sent through the Kalman filter in order to obtain estimates of the necessary states for the implementation of the modern guidance law. These states are multiplied by control gabs, which a r e funetiana, of the estimated ti&e to go and autopilot bmdwidth, in order to genes - ate an acceleration command, This command is applied to an a-eeleration autopilot in order t o develop the commanded acceleratim.
In summary the implementation of MGS requires some additional information which is not required by a dasaical PN guidance system. Estimates of range, measurement and process noise statistics are needed for the %npl.emen%.ation of the Kalman filter whde estivates of time-to-go , guidance system bandwidth <.ad missiie acceleration are needed for the implementation of the guidance law.
Performance Comparison
Both classical and modern guidance zethods are now compared in terms of performance as measured by the rms (mot mean square) miss distance. The comparison is made utilizing the linearized, but realistic model of the kinematic homing loop shown in Fig. 7. Here autopilot dynamics are represented by a first order t r ans f~ . function and o d y 2-2 two most important s t ochas~c error sources are considered, namely glint noise and random target maneuver. The seeker, noise filter and guidance dynamics have been previously presented in Figs. 4 and 6. In this case the Kalman filter of MGS is optimal since it is ;wrfectly matched to the wreal worldM in that is has an exact dpnmical m o d d of the system along with pr f=Lect knowledge of the naeasurlerment and process
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Figure 7 - Kinematic Homing Loop
noise statistics. With this methodology any deteriora- tion in MGS performance will be caused solely by the guidance law.
Imbedded in the hGS guidance law is a dynamical m o d i l of the actual system. If this model is inaccu- rate or if the tgo estimate is lacking or inaccurate, M G S performance degrades. If the estimated time to go, f 0, is considered to be a simple funct;on of tgo as skown in Eq . 9,
then the influence of bias errors, B , and scale factor errors, A , on MGS system performance can be investigated . Typical miss distance results, shown in Fig. 8, indicate that errors in t' result in rapid performance dekradation of MGS!*I~ fact, ~iegative bias e r r ~ r s lead to guidance system instabilities of MGS. P N performance, superimposec on Fig, 8 is not sensitive to these e r r o r s . For this example P N achieves a miss of 4. 3 ft. Therefore if the required miss distance was less than 4. 3 ft, only MGS could meet that specification and then only if bias e r r o r s could be kept below 0. 2 s and scale factor e r r o r s were between 0.68 and 1.35. If t.he required miss were greater than 4. 3 ft, P N could meet the specification, but MGS could only meet the specification i f bias and scale factor e r r o r s could be kept below the values of the curves in Fig. 8. Of course, P N does not use these quan- tities at all and therefore i s not sensitive to such e r ro r s . In summary, the implementation of MGS places requirements not only on the algorithm for calculating t but also on the specialb-filtering needed to esk%ate range and range rate.
Mcdern Guidance Sys tem, unlike Proportional Navigation, attempts to compensate for autopilot dynamics by the use of a dynamic lead t e r m in the missile acceleratioi~ command. In order to implement this concept, MGS requires an accurate neasurement of the achieved missile acceleration and an estimate of the autopilot bandwidth. W A ~ . If this measure- ment is perfect and if the dynamic model within MGS is perfectly matched to the "real world", optimum performance can be obtained. However, if for example, we assume that missile acceleration is measured perfectly, but the estimate of autopilot
' ' '" ! - :- rrvr-r qrrnrCJim +n Frl 7 0 - . A- - -- - -
then the influence of scale factor errors, C, or .MCS system performance can be investigated. Fig. shows that system instabilities result if the scale factor falls below 0.6. As before, PN performa is not sensitive to this error source. In this i
for miss distances below 4.3 f t , only MGS can . I et the requirements if the scale factor is greater than 0.6. If the required miss distance is g r-ea ter than 4.3 f t , P N can always meet thc req&ement, but MGS can only meet the specification if the r ale factor is greater than 0.6. In summary, the impbementation of MGS places reqtiirements on allowable errors in dynamic modeling .
The nonhernispherical shape of the missile radome causes distortion of the incoming radar . beam. A s the radar beam passes through the radome a refraction effect takes place and the net result is an error in the a ~ g l e of the apparent target. The radorne error slope, R , is a measure of the distortion taking place and is a function of the gimbal angle, among other things. (6) The guidance system designer attempts ;o specify the manufacturing tolerances and the limjts on tlre permissible variations of R, This error rrource i r particularly important at high altitudes where the missi le turning rate t ime constant, T , i s large. This t ime constant in con- junction witg large radome refraction slopes can cause guidance system inatabiiity. It is, therefore, of considerable practical importance to s e e how P N and MGS performance degrade in the presence of radome slope errora . Typical high altitude per - formance results for both guidance systems a r e shown in Fig. 10. The results indicate that the P N guidance system has m o r e of a tolerance to radome e r r o r s than does MGS. In this regard P N is more robust.
Figure 10 indicates that the MGS implementation is more sensitive to negative radome slopes than positive slopes. In a practical deqjgn, the seeker stabilization loop gain could bp adjusted to bias the radome, thus i ~ s u r i n g only positive slopes. It is for this reason that the allowable radome slope range is more of an important measure than the average radome slope. In fact, the allowable radome slope range is one important measure used in guidance svstem desipn to crnc!qy ~ - n q ~ f l - c + - - - - ; r , g tn1eran~-= nv
the radome. The average rms miss distance due to a radome slope range can be calculated from the
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C
t u
V) u, Y
af u, I:
Figure
MGS
Figure 8 - E r r o r s in Estimating Time To Go Seriously Degrade MGS Performance
9 - E r r o r s in Estimating System Dynamics Can Lead t o MGS Instability
I I I
.1 -.06 0 .OS .1 Normrlkod h d o m Slogl,
Figure 10 - Radorne E r r o r s hfluence System Performance
information provided in Fig. 10. Typical resu l t s acceleration ratio for both guidance system showing the sensitivity of both guidance e ystcmrs . implementations, i t is evident that PN requires to radome slope range i s displayed in Fig. 1 1. This f igure shows that when the radome is taken into con- sideration; MGS can only offer superior mi s s d i s - tance per formance if the allowable radome slope range is l e s s than 0.075. Otherwise P N yields smal le r average r m s m i s s distances. Neverthe - less, if a radome slope range less than 0.075 can be met, MGS yields less m i s s distance.
Finally, missile acceleration saturation, one of the most important guidance system nonlinearities in the performance comparison, is considered. Since MGS predicts intercept using estimates of target acceleration and measurements of missile acceleration, it requires less acceleration than P N tn hi t 3 m q n ~ v i ~ r ~ r i n n t q r - m t T n P i n 1 3 . rr thC-n
u C u
r m s m i s s is plotted versus the missi le- to-target
a larger acceleration advantage over the target than MCS to achieve a specific m i s s distance. For example, to achieve an RMS m i s s of less than 7 f t , PN requires a 5 to 1 acceleration advantage over the target whereas MGS requires only a 2. 3 to 1 advan- tage. This reduced accelerstion requirements extends the missile's zone of effectiveness against maneuvering targets and is the major advantage of MCS over P N .
Summary
A modern guidance system and a propnrtional navigation guidance system designed to meet the same miss distance specification yield different i-~!t;.:-clf~*i~:: :f ryb-.r;--+--=, -rid each subsvstem mUSt U l e e t a ClUielttilL be^ UA i r y w ~ ~ i ~ ~ ~ i i i a .
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MGS 12 -
C
I- Lr. V
cn h - V) H r
I I
t 1 I 1 0 - 0 5 ,075 .1 .15
NORMALIZED RADOME SLOPE RANGE
Figure 11 - P N Allows Grea ter Flexibility in Radome Design
~ISSlLL T O TARGET ACCELERATION R A T I O
Figure 12 - MGS Requires Less Acceleration Due to Target Maneuver
modern guidance system imposes more severe requirements on radome refraction slope and on knowledge of the system dynamics, but i t does not require as much maximum missile normal acceleration to intercept an accelerating target. If the miss distance specification is extremely small, only MGS can do the job. Thus the additional instrumentation and subsystem requirements is the price that must be paid to meet severe miss distance requirements . If the miss distance specification i s such that both M G S and P N can do the job, then instrumentation specifications can Ie relaxed in favor of more missile normal acceleration capability. When sufficient missile acceleration is available, PN offers the least stringent instrumentation requirements. Thus, if component tolerances can be met, MGS has the smallest miss distance, but as component tolerances or measurement errors degrade, the performance of MGS degrades faster than that of PN until at relatively large component or measure- ment errors , P N has less miss distance than MGS.
( 3) Willems, G. , "Optimal Controllers for Homing Missiles with Two Time Cw. i tants, 'I U . S . Army Missile Commani, 3;;~ort No. RE-TR-69-20. Redstone Arsenal, Alabama, October 1969.
( 4) Price, C . F. , "Optimal Stochastic Guidance Laws for Tactical Missiles", The Analytical Sciences Corporation, Report No. TR- 170-2, Reading, Mass. , September 1971.
( 5) Phillips, T. L . , "Anti-Aircraft Missile Guidance" Electronic Progress published by Raytheon Co. , March-April 1958, pp 1-5.
( 6 ) Nesline , F . W. , "Missile Guidance for Low Altitude Air Defenset', A U A Guidance and Control Conference, Palo Alto, CA, August 1978.
(7) Benecke, T. and Quick, A.W. (Eds) , "History of German Guided Missiles Development, " ACARD First Guided Missiles Seminar, Munich, Germany, A p r i l 1956.
( 8) Yuan, C .L., "Homing and Navigational Courses of Automatic Target- Seeking Devices, 'I RCA Laboratories, Repcrt No. PTR- 12C, Prhceton , N e w Jersey, December 1943.
( 9 ) Bennett, R.R. and Mathews, W . E . , "Analytical Determination of Miss Distances for Linear Homin g Navigation Systems, Technical Memorandum No. 260, Hughes Aircraft Company, Culver City, California, March 1952.
( 10) Yuan, C .L . , "Homing and Navigational Courses of Automatic Target-Seeking Devices, " Journal of Applied Physics, December 1948. .
( 11) Cottrell, R .G . , "Optimal Intercept Guidance for Short-Range Tactical Missiles", AIAA Journal, Vol. 9, July 1971, pp 1414-1415.
( 12) Fitzgerald; R . J. , and Zarchan , P . , "Shaping Filters for Randomly Initiated Target Maneuvers " , AIAA Guidance and Control Conference, Palo Alto, CA, August 1978.
(13) Wiener, N., "Extrapolation, Lnterpolation and Smoothing of Stationary Time Serj esI1, MIT P r e s s , Cambridge, Mass., 1949.
( 1 4 ) Newton, G.C., Gould, L.A. , and Kaiser, J .F . , "Analytical Design of Linear Feedback Controls", John Wiley and Sons, Inc., New York, 1957.
( 15 ) Kalrnan, R.E. , and Bucy, R . , "New Results in Linear Filtering and Prediction, " Journal of Basic Engineering (ASME) , Vol . 83 D , 1961, pp 95-108.
( 16) Gelb, A . , ed . , "Applied Optimal Estimation, " MIT Press, Cambridge, Mass. , 1974.
References
(1) Bryson, A.E. and Ho Y .C. , "Applied Optimal Control, " Blaisdell Publishing Company, Waltham , Mass. 1969.
( 17) Kliger , I. - " A Simple Derivation of Certain Optimal Control Laws I' , Raytheon Memorandum SAD- 1230, Bedford, ,Maas., Nov. , 1970.
( 2) Willerns, G . , "Optimal Con trollers for Homing Missiles, " U. S. Army Missile Command, Report N o . RE-TR- 68-15, Redstone Arsenal,
- C . n ."-!:t32,:, 2 2 . ' + -mhr . r - -rn,, . f - .
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Appendix A - Wiener Gpthnal Filter
The disturbances entering the guidance system are consilered to be white glint noise with spectral density @ly and random target maneuver. In this paper the maneuver is considered t r~ be a step func- tion whose initiation time is uniformly distributed over the flight time. I t can be shown( 12) that integrated white noise has the same autocorrelation function as this maneuver process. The optimal filter with transfer function, Ho, c z : ~ be derived by either Wirner or Kalman filter thcory.
The G'riener filter formula?lon is based upon the diagram of Fig. A-1. The prablern is to find H, which will nkinimize the integral of the mean square
error signal : minimize /*eLd t ] , 0
The optimal transfer function, Ho, can be found from the explicit solution of the Wiener-Hopf integral equation
where WS and WN are the spectral densities of the signal and noise, (WS+WN )+ represents that part which has all its poles and zeroes in the left half plane, (WS+WN)' represents that part which has all its poles and zeroes in the right half plane. The expression f 1 + is the component of ( 1 which has all its poles in the left half plane. In order to obtain [ ] + we expand ( . ] in partial fractions and throw away all the terms corresponding to poles in the right half plane. From Fig. A-1 the output spectral densities of the signal and noise, WS and
can be expressed in terms of the input spec- :% densities. and QN, and the shaping network transfer function as
1 herefore
If we define
then Eq. (A-4) can be factored
Therefore
Substitution of Eqs. (A-2 \, (A-7) and (A-8) into (A-,I 1 yields
Us = White Noise with Power Spectral Density O, Un = Whit. Noiu with Povnr Sp.ctral Wnsity q,,, YT = Actual Targot Position
VT* = Measured Target Position
VT = Estimated Tarpat Position
e = Error in Estimate
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The expression in the brackets of Eq. ( A - 9 ) can be expanded by partial fractions yielding
(A- 10)
.nating those terms wi th poles in the right half A .ane leaves
which simplifies to
Substitution of E q . (.4- 12) into ( 4 -9 ) yields the optimal transfer function
Appendix B - Kalman Optimal Filter
The same transfer function can also be obtained by the Kalman formulation.
The state and measurement equations can bf: derived from the plant, shown in Fig. B- 1, and are
The Kalman filter equation is
where the Kalman gains, 5, are determined from the following matrix Riccati equations
I I I PROCESS I
i PLANT : MEASUREMENT 0
I NOISE I I
NOISE I I
I I I ' us
P
1 Y 1 I -----+--- - . I I s s I I , 4
I I
= White Noise wi?h Power Spectral Density tS = White Noise with Power Spectral Density *,,, = Target Acceleration
Y = Target Rate
Y, = Target Position
Y,* = Measured Target Position
Fig. B- 1 Kalrnan filter formulation.
Y OF THE OR1 GlNAL PAGE I S 1306
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The transfer function between the position estimate output and the position measurement input can easily be obtained from Eq. ( B - i l l as where
A 1 + 2s/wO+ 2 s 2 1 9 2 (E- 12) YT - 7- 2 3 3 y- T
1 + 2 S / 0 0 + 2s iw02 + s I'J,
T i s is identical to the transfer function obtained by the Wiener filter approach.
Recognizing that the covariance matrix, P-, is symmetric, the scalar %quations representing the steady state solution (E = 2) can be written from {B-4) as
Appendix C - Matrix Approach to Modern Guidance
Optimal guidance laws are generally derived on the basis of modern control theory. The linear model of Figure C-1, in which missile dynamics are represented by a single lag, is used for the appli- cation of modern control theory to the guidance problem. In this case we are interested in deriving a guidance law whish will achieve zero miss distance while minimizing the integral of the square of the
t f , guidance commands [minimize J rlLdt subject to v (t,)=01. 0 =
The solution to minimizing the quadratic per- formance index
After some algebra, the solutions to E q . (B-7) can L.e substituted into Eq. (B-5) yielding the steady state Kalrnan gains
subj ect to the linear differential equation
4 = c ? r + C n c (C-2) \
with zero miss distance f y ( t )=0 1 is well known ( 1) F Defining
and is given b y
t h e gain matrix becomes
where R is obtained from the differential equation
The filter equations are obtained by substituting Eq. (B-10) into Eq, (B-3) yielding
4 . . 10. -- w- -- . - - - - -
-I'- -- -- - - - - CE "15 REPROOU
h &%-u--b
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I MISSILE 1
I 4 LINEARIZED
I DYNAMICS 1 KINEMATICS I I . !
nc = Commanded Missile Acceleration
nL = Achieved Missile Acceleration
Y = Target Acceleration
Y = Relative Target - Missile Rate
Y = Relative Target - Missile Separation Fig. C- 1 Guidance Problem Formulation
The solutior, to Eq . (C-4) is
where T is the normalized time to go until intercept given by
The integral appearing in Eq . (C- 3) therefore becomes
Sdbstituting Eq. (C-8) into Eq. ( C - 3 ) yields the optimal closed loop guidance law
where
If missile guidance dynamics are neglected (a+-), application of ~ ' ~ o ~ i t a l d Rule reduces the guidance law to
This guidance law is augmented proportion a1 navigation with an effective navigation ratio of three.
Appendix D - Scalar Approach to Modern Guidance
The Schwartz Inequality can also be used in the derivation of optimal guidance laws. [ For the zame problem considered in Appendix C , we can express the system state vector of Eq. (C-2) at the terminal time according to the matrix super- position integral
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where Q, is the fundamental matrix. Equation (D-1) actually represents a set of scalar equations for each of the different states. For the guidance problem, the first state is the object of our control and can be expressed as
In order to demonstrate the power of Eq. (D- 8) in solving for optimal guidance laws the example of Appendix C is reconsidered. The fundamental matrix can be found from Eq.. (C-5) to be
(D- 2 )
For zero miss distance Eq. (D-'2) can be rewritten as
and
Application of the Schwartz Inequality yields
where
t = t F - t go
The integral of Eq. (D-5) becomes
(D- 12)
The integral of the square of the acceleration can be minimized by insuring that the equality sign of E q . (D-5) holds. According to the Schwartz Inequality, this occurs when
Substitution of Eq. (D--6) into Eq. (D-3) yields
(D- 13)
Substitution o f Eq. (D-13) into Eq. (D-8) yields
Therefore the acceleration is given by
where
and
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Thus modern control theory and the Schwartz Inequality can be used in the derivation of certain types of guidm-ce laws. The mathematical solutions of both approaches are in fact equivalent as can be seen by rewriting E q . (C-3) and (D-8).
(D- 16)
A careful examination of the example used in both approaches shows the following equivalences
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