UFDC Home  Search all Groups  UF Institutional Repository  UF Institutional Repository  UF Theses & Dissertations  Vendor Digitized Files   Help 
Material Information
Subjects
Notes
Record Information

Full Text 
AN ANALYSIS OF A NEW NONLINEAR ESTIMATION TECHNIQUE: THE STATEDEPENDENT RICATTI EQUATION METHOD By CRAIG M. EWING A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA ACKNOWLEDGMENTS I would like to thank my advisor, Dr. Norman FitzCoy, for his efforts in continually pushing me to develop a better dissertation. I agree it is now something I can look upon with pride. I also thank Dr. James Cloutier, the man behind the idea. Without his help and guidance this work could never have been written. I would especially like to thank my wife, Darsi, for her neverending support through these years of work. She never let me give up. TABLE OF CONTENTS ACKN O W LED G M EN TS............................... .................................... ....................ii L IST O F T A B LE S .............................................................................. .................... v L IST O F FIG U R E S .................................................................... ..................................vi A B ST R A C T ............................................... ................................................................... x CHAPTERS 1 INTRODUCTION....................... ............................................. 1 2 LITERA TU R E SU R V EY ............................................................ ............................4 3 SCOPE OF W O RK .................................... ....................................................... 8 4 EXOATMOSPHERICGUIDANCEPROBLEM ENGAGEMENT SCENARIO ..... 10 5 EXOATMOSPHERICGUIDANCEPROBLEM FILTER DEVELOPMENT.......... 13 5.1 EK F D erivation ................................................................. ........................ 14 5.2 Bootstrap Estimator Derivation............................................... 19 5.3 SD REF D erivation ............................................................ ........................ 22 6 EXOATMOSPHERICGUIDANCEPROBLEM SIMULATION RESULTS........... 49 6.1 B aseline Perform ance.................. ...................... ................................................. 49 6.2 PDF Distribution Comparison................................................... 54 6.3 Sensitivity A analysis .............................................................. ..................... 64 6.4 SDREF Closed Loop Performance............................................... 79 6.5 Summary of Exoatmospheric Guidance Problem.............................................. 87 7 PENDULUM PROBLEM FILTER DEVELOPMENT.............................. .............. 88 7.1 SD REF D erivation .......................................... .......................................... 90 7.2 EKF D evelopm ent.............................................................. ....................... 91 8 PENDULUM PROBLEM SIMULATION RESULTS................................ ....92 iii 9 MSDREF DEVELOPMENT ................................................................. ............ 103 9.1 M SD REF D erivation........................................................................................103 9.2 Stability Proof................ .................................. .............................. 104 10 MSDREF PROBLEM SIMULATION RESULTS............................................. 112 11 CRITICAL ANALYSIS OF RESULTS ........................................ .................... 119 REFERENCES .................................................................................................... 122 BIOGRAPHICAL SKETCH................................................................................... 125 LIST OF TABLES Table page 1. Filter simulation parameters baseline conditions............................. ............ 50 2. EKF and SDREF parameters onepercent initialization error.................................. 65 3. EKF and SDREF parameters fivepercent initialization error ................................ 69 4. EKF and SDREF measurement noise parameters level 1 ....................................... 72 5. EKF and SDREF measurement noise parameters level 2..................................... 72 6. M aneuvering target parameters................................... .............................. 76 7. Headon simulation parameters.......................... ....................... 81 8. 90degree beam shot simulation parameters ......................... .......... ........... .. 81 9. Tailchase simulation parameters.......................................... 82 10. Miss distance performance for varying scenarios................... ........................ 83 11. Miss distance performance for stressing target maneuvers.................................... 84 12. Miss distance performance for varying measurement noise................................. 85 13. Miss distance performance for varying initialization error...................................... 86 LIST OF FIGURES Figure page 1. G generic intercept vehicle.......................................................... ...................... 11 2. Guidance system block diagram.......................................... 12 3. EKF position error with 3a standard deviations ................................................... 52 4. SDREF position error with 3a standard deviations ................................................52 5. Bootstrap position error with 3o standard deviations .......................... ............. 52 6. Comparison of EKF and SDREF covariance 3o standard deviations for Y position .................................................................................................... 53 7. Expanded view of Figure 6 .................................................................................. 53 8. Posterior probability density functions at time = 1, 5, and 9 seconds for X position .................................................................................................... 55 9. Posterior probability density function at time = 1, 5, and 9 seconds for Y position ...... .............. ............................................................ 56 10. Monte Carlo posterior probability density functions at time = 1, 5, and 9 seconds for X position................................................................ ....................... 58 11. Monte Carlo posterior probability density functions at time = 1, 5, and 9 seconds for Y position............................................................... .......................... 59 12. EKF estimated and true acceleration............................................ ....................... 61 13. SDREF estimated and true acceleration......................................... ..................... 61 14. Bootstrap estimated and true acceleration..................... ......................61 15. Monte Carlo posterior probability density functions at time = 2, 2.5, and 3 seconds for thrust cutoff X position...................... ........................ 62 16. Monte Carlo posterior probability density functions at time = 2, 2.5, and 3 seconds for thrust cutoff Y position............................................ .................... 63 17. EKF position errors with 3o standard deviations for 1percent initialization errors .................................................... .. .... ....................... 67 18. SDREF position errors with 3o standard deviations for 1percent initialization errors ................................................................ .... ....................... 67 19. EKF position errors with 3a standard deviations for 5percent initialization errors ................. .... .. ....................... 68 20. SDREF position errors with 3o standard deviations for 5percent initialization errors ..................................................... .. ... ....................... 68 21. SDREF position errors with 3a standard deviations after new initialization method, percent error .................................................................................... 70 22. SDREF velocity errors with 3O standard deviations after new initialization method, 1percent error .... .......... ..... ....................... 70 23. SDREF acceleration errors with 3o standard deviations after new initialization method, 1percent error ........................ ....................... 70 24. SDREF position errors with 3o standard deviations after new initialization method, 5percent error................ ............................................................. 71 25. SDREF velocity errors with 3o standard deviations after new initialization method, 5percent error ...................... ...................... 71 26. SDREF acceleration errors with 3a standard deviations after new initialization method, 5percent error .............................................................. ...................... 71 27. EKF position errors with 3o7 standard deviations 1percent error in X, 100 microradian error in Y and Z............................................. ....................... 74 28. SDREF position errors with 3a standard deviations 1percent error in X, 100 microradian error in Y and Z.............................................. ....................... 74 29. EKF position errors with 3o standard deviations 10percent error in X, 500 microradian error in Y and Z............................................... ...................... 75 30. SDREF position errors with 3cr standard deviations 10percent error in X, 500 microradian error in Y and Z.............................................. ....................... 75 31. EKF position errors with 3a standard deviations for 10 deg/sec rotating target m maneuver ................................................... ............... ...................... 77 32. SDREF position errors with 3a standard deviations for 10 deg/sec rotating target m maneuver .......................................................... ................................ 77 33. EKF position errors with 3a standard deviations for dogleg target maneuver...... 78 34. SDREF position errors with 3u standard deviations for dogleg target maneuver. 78 35. A aspect angle description ........................................................ ............................ 80 36. Pendulum setup ............................................. ............................................... 88 37. EKF and SDREF estimates vs. true theta with no initialization error..................... 93 38. EKF and SDREF estimates vs. true thetadot with no initialization error .............. 93 39. EKF and SDREF estimation error for theta with no initialization error ............... 94 40. EKF and SDREF estimation error for thetadot with no initialization error............. 94 41. Monte Carlo probability density function for theta, with no initialization error, tim e = 0.5 sec ........................................ .............. ................................... 95 42. Monte Carlo probability density function for thetadot, with no initialization error, tim e = 0.5 sec......................... ... ............................ ....................... 95 43. Monte Carlo probability density function for theta, with no initialization error, tim e = 2.0 sec ........................................ .................................................... 96 44. Monte Carlo probability density function for thetadot, with no initialization error, tim e = 2.0 sec..................................... ............................................... 96 45. Monte Carlo probability density function for theta, with no initialization error, time = 3.0 sec ................................................97 46. Monte Carlo probability density function for thetadot, with no initialization error, tim e = 3.0 sec............................................................. ...................... 97 47. EKF and SDREF theta estimation error with initialization error............................ 99 48. EKF and SDREF thetadot estimation error with initialization error ..................... 99 49. Monte Carlo probability density function for theta, with initialization error, time = 0.5 sec ................................................ 100 50. Monte Carlo probability density function for thetadot, with initialization error, tim e = 0.5 sec............................................. ............... 100 51. Monte Carlo probability density function for theta, with initialization error, tim e = 2 .0 sec ....................................................... ............. ......................... 10 1 52. Monte Carlo probability density function for thetadot, with initialization error, tim e = 2 .0 sec ............................................................. ................................... 10 1 53. Monte Carlo probability density function for theta, with initialization error, tim e = 3 .0 sec ...................................... ......................................................... 102 54. Monte Carlo probability density function for thetadot, with initialization error, tim e = 3.0 sec ...................................................... .............. ......................... 102 55. True vs. estim ated X 1 state ....................................................................... 117 56. True vs. estimated X2 state ............................................................. 117 57. True vs. estimated X1 state, nonzero initial conditions....................................... 118 58. True vs. estimated X2 state, nonzero initial conditions.......................................118 Abstract of Dissertation Presented to the Graduate School of The University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AN ANALYSIS OF A NEW NONLINEAR ESTIMATION TECHNIQUE: THE STATEDEPENDENT RICATTI EQUATION METHOD By Craig M. Ewing August 1999 Chairman: Norman FitzCoy Major Department: Aerospace Engineering, Mechanics, and Engineering Science Research into nonlinear estimation techniques for terminal homing missiles has been conducted for many decades. The terminal state estimator, also called the guidance filter, is responsible for providing accurate estimates of target motion for use in guiding the missile to a collision course with the target. Some form of the extendedKalman filter (EKF) has become the standard estimation technique employed in most modem weapon guidance systems. EKF linearization of nonlinear dynamics and/or measurements can cause problems of divergence when confronted by highly nonlinear conditions. The objective of this dissertation is to analyze a new nonlinear estimation technique that is based on the parameterization of the nonlinearities. This parameterization converts the nonlinear estimation problem into the form of a steadystate continuous Kalman filtering problem with statedependent coefficients. This new technique, called the statedependent Ricatti equation filter (SDREF), allows the nonlinearities of the system to be fully incorporated into the filter design, before stochastic uncertainties are imposed, without the need for linearization. The SDREF was investigated in three problems: an exoatmospheric, terminal homing, ballisticmissile intercept problem; a highly nonlinear pendulum example; and an algorithmic loss of observability problem. The exoatmospheric guidance problem examined nonlinear measurements with linear dynamics. To investigate the SDREF when used with a combination of nonlinear dynamics and nonlinear measurements, a highly nonlinear, twostate pendulum problem was also examined. While these problems were useful in gaining insight into the performance characteristics of the SDREF, no formal proof of stability could be determined for the original formulation of the estimator. The original SDREF solved an algebraic SDRE that arose from an infinitetime horizon formulation of the nonlinear filtering problem. A modification to the SDREF formulation was developed that led to a differential SDREF, and a proof for local asymptotic stability was achieved. The modification removed the infinitetime horizon assumption and integrated the coupled statedependent state and covariance equations. This new form of the estimator is called the modified SDREF (MSDREF). A problem involving algorithmic loss of observability was then examined. This problem shows a performance advantage when using parameterization versus linearization as in the EKF technique. CHAPTER 1 INTRODUCTION The extended Kalman filter (EKF) has become an accepted standard for both tactical and strategic interceptor guidance problems. While some problems lend themselves to this type of estimation strategy, systems with highly nonlinear dynamic environments or misunderstood uncertainties can cause divergence of the filters. Many alternative nonlinear estimation techniques have been researched in an attempt to overcome this problem. The linearization of the nonlinear dynamics and/or measurements is one area of concern. The EKF linearizes about the current estimate of the state. If the state estimate is fairly accurate, this method has been shown to be effective. If, however, the estimates are poor, linearization can cause errors in the filter, which can lead to degraded performance or possible divergence of the state estimate. It is this linearization that motivates the following research. This dissertation investigates a new nonlinear estimation algorithm, which originates from a recent nonlinear regulator design technique. It uses parameterization to bring the nonlinear system to a linearlike structure with statedependent coefficients (SDC). The filtering equivalent to the regulator technique can be realized in a recursive nonlinear estimation algorithm, which has the structure of the steadystate continuous Kalman filter. The Kalman gain is found by solving the statedependent Ricatti equation at each filter update time. The filtering technique, called the statedependent Ricatti equation filter (SDREF), 2 allows the incorporation of nonlinear dynamics and/or measurements in the filter design without the need to linearize the equations. The SDREF was analyzed in a several ways to determine the benefits and issues associated with this type of structure. By using a sampleddata Bayesian filter, known as the bootstrap filter, the "true" probability density function (pdf) of the SDREF states was monitored. This allowed an investigation into the differences in pdf evolution for linearizing versus parameterizing filters. Finding the true pdf can be difficult; however recent work, by the author on the bootstrap estimator, proved valuable for this analysis. The bootstrap technique uses thousands of samples of the pdf along with Bayes' rule, to estimate the true pdf of any given state. With this filter and a large number of samples, the "true" state pdf can be closely approximated. An EKF was evaluated in a similar manner for comparison purposes. The SDREF was exercised in three problems. A realistic and detailed evaluation of the filter was accomplished by examining an exoatmospheric guidance problem. The SDREF, bootstrap filter, and EKF were coded into a sixdegreeoffreedom (6DOF) FORTRAN simulation for analysis in an exoatmospheric terminal homing engagement using passive and active sensor information. The sensor measurements included azimuth, elevation and range. The characteristics of the SDREF and EKF were evaluated in several different engagement scenarios. The SDREF was developed in a Cartesian coordinate frame, which causes the measurements to be nonlinear, while the dynamics remain linear. The second problem was that of a highly nonlinear, twodimensional pendulum problem. Evaluation of the SDREF when used with nonlinear dynamics and nonlinear 3 measurements was examined. The SDREF and EKF were coded into the visual simulation environment VisSim. The performance of the filters and their pdf's were compared. This academic problem was well suited to exploring any differences that nonlinear dynamics may have on the performance or pdf characteristics of the SDREF. No formal proof of stability could be determined using the above formulations of the SDREF, so an alternate filter formulation was sought. The original SDREF was based on an infinitetime horizon formulation of the nonlinear filtering problem that resulted in an algebraic SDRE. A MSDREF was developed that utilizes a differential SDRE. Because the Ricatti equation is state dependent, it must be integrated in conjunction with the state estimates. A proof of local asymptotic stability was able to be determined. The new MSDREF was examined in a problem consisting of loss of algorithmic observability; and the results indicated a significant performance advantage over the EKF. This dissertation fully develops the requirements for the SDREF and MSDREF. It includes an investigation of controllability and observability, which are pointwise and in the linear sense and pertain to a valid SDRE solution. An exhaustive investigation of the pdf characteristics, noise sensitivity, initialerror robustness, and closedloop performance of the SDREF using the EKF as a comparative source was performed. A theoretical basis for the MSDREF, when used as an observer, was shown. The following research provides theoretical and practical information demonstrating that the SDREF and MSDREF are viable candidates for nonlinear estimation, and for some problems can out perform the EKF. CHAPTER 2 LITERATURE SURVEY The literature abounds with research of various nonlinear estimation techniques, most of it mathematical and not application oriented. An elegant presentation of the history of airtoair target state estimation techniques by Cloutier et al.' outlines a wide variety of techniques, which include the Kalman filter, EKF, secondorder Gaussian filter (SOGF), recursive nonlinear filter, recursive maximumlikelihood filter, and the assumed density filter. These filters have also been examined in the context of single and multiplemodel adaptive filters. Adaptive filters may make use of certain properties of the residual or covariance to continuously change filter parameters to gain better filter performance. On the other hand, multiplemodel filters use banks of filters to model many possible target dynamic situations. Various methods are then used to either select the results of one filter or combine that of several. With the advent of the Strategic Defense Initiative (SDI), a new set of problems arose when estimating the motion of a boosted intercontinental ballistic missile (ICBM) with the use of passiveonly measurements. While the EKF is still the most widely accepted estimator for this type of engagement, it has been shown to have degraded performance when stressed by highly maneuverable targets.24 This is the same problem that tactical filters experience in maintaining track of highly maneuverable aircraft or missiles. When tracking a highly nonlinear maneuver, state estimates can become poor and the linearization assumption begins to break down. The specific problems associated 4 with guidance filters for the exoatmospheric intercept problem have been addressed in several research programs noted in the following paragraph. Schmidt2 investigated an extension of work by Daum in which a closedform solution to the FokkerPlanck equations for propagating the pdf was developed. A target maneuverdetection algorithm was added to this formulation to enhance performance. The emphasis in this research was to more accurately model the target acceleration based on the rocket equation, instead of a Markovprocess. The target model caused the state equations to become nonlinear as well. The results of the work did show an improvement over the EKF technique; however this was accomplished, in part, through the use of the maneuverdetection algorithm. Gido34 developed a nonlinear estimation algorithm, which numerically implements the theoretical work of Kolmogorov. The algorithm places a grid over the pdf and uses the FokkerPlanck equations coupled with Bayes' rule to provide the state estimates. While this method showed improved performance over the EKF, it is computationally intensive. In the author's masters' thesis at the University of Florida,5 another novel nonlinear estimation technique was investigated. It involved using thousands of samples of the pdf, along with the state dynamics and Bayes' rule to propagate and update the pdf .7 The technique places no restrictions on the type of statistics used to model noise. It also makes no linearization assumptions. While computationally intensive, it is directly applicable to a massively parallel processing architecture since the same operation is done to each sample. While the bootstrap method may not be seen in any missile guidance system design in the near future, it does provide a bruteforce method for obtaining a reasonable representation of the true state PDF for a given problem. It is in this context that the filter 6 is used for the current research. While many of the above filters were shown to have performance superior to that of the EKF, realistic implementation may be difficult. Other techniques that have been proposed to solve the bearingsonly guidance problem include the modified gain pseudomeasurement filter (MGPMF)8 and modified gain EKF (MGEKF).9 Both use pseudomeasurements to avoid the need for linearization around a "nominal" trajectory, such as is found in the EKF. These methods are restricted to a special class of nonlinear functions, which can be manipulated into a pseudo measurement form. The construction of implementable nonlinear filters for nonlinear stochastic systems remains a challenge. Numerous filtering algorithms, based upon series expansions to approximately realize the conditional mean, have been suggested (see the bibliography of Bucy et al.10) Once again, these techniques represented a heavy computational burden, even for loworder dynamical systems. The SDREF may alleviate some of the problems discussed so far. The technique is derived from a new nonlinear regulator design methodology known as the state dependent Ricatti equation (SDRE) control method.1112 The objective is to take the nonlinear system described by x= f(x)+g(x)u (21) and transform it to the statedependent coefficient form x= A(x)x + B(x)u (22) where f (x)= A(x)x B(x)= g(x) (23) 7 using direct parameterization. Once in this form the SDREF is obtained by considering the statedependent coefficients to be fixed and applying the linear continuous Kalman filter to the "frozen" problem. The result is an algebraic SDRE whose structure is the same as the steadystate Ricatti equation. The emphasis of this research is to develop and analyze a guidance filter based on this technique. Initial work in the development of the SDREF'3 demonstrates the filter in a simple twodimensional pendulum problem. Other applications include induction machine estimation,14 simultaneous state and parameter estimation, 5 and gyroless satellite angular rate estimation.16 The current research extends their work to include an exhaustive investigation into the internal properties of this filter in a strategic interceptor guidance problem, as well as in the frictionpendulum problem. Insight into the internal representation of the pdf when using parameterization versus linearization is examined, as well as the necessary conditions for stability. The first example presented here uses only nonlinear measurements, while the pendulum problem discussed herein adds the complexity of nonlinear dynamics as well. A third example demonstrates the unique parameterization property of the SDREF when confronted with a loss of algorithmic observability. CHAPTER 3 SCOPE OF WORK This chapter outlines the scope of the work in this dissertation. Throughout the design and analysis of the SDREF, the EKF was used to compare performance issues between linearized and parameterized methods. Similarly a sampleddata bootstrap filter was used to compare probability distribution functions between the different nonlinear filters in the exoatmospheric guidance problem. The three filters are theoretically derived and applicationspecific equations are formulated for each problem. Controllability and observability requirements for the SDREF algorithm are analyzed and determined. Several performance measures are investigated that include sensitivity to initial errors, noise parameters, and target maneuvers. In addition, a somewhat nonstandard methodology, the bootstrap filter, is used to investigate the generally unobservable evolution of the pdf within the filter. The pdf was examined at distinct time steps to determine whether the parameterized filter pdf is different from the linearized filter. All three filters were coded in FORTRAN and installed in a 6DOF simulation for evaluation in an exoatmospheric terminal homing engagement. To examine the effects of both dynamic and measurement nonlinearities, the SDREF and EKF were also coded in VisSim (a visual simulation tool for a simple, but highly nonlinear, pendulum problem). Performance was measured in a manner similar to the exoatmospheric guidance problem. Because no formal proof for stability could be found for the SDREF, modifications were made that allowed a detailed proof of local 8 9 asymptotic stability to be found. The MSDREF is fully derived and an examination of its performance against the EKF for a problem consisting of loss of algorithmic observability is also explored. CHAPTER 4 EXOATMOSPHERICGUIDANCEPROBLEM ENGAGEMENT SCENARIO The filters were initially analyzed in the context of an exoatmospheric terminal homing engagement of an interceptor tracking a thrusting ICBM for approximately the last 10 seconds before impact. The interceptor was closing on the target at approximately 10 kilometers per second. The target may perform thrust cutoff or other evasive maneuvers, which provide for a stressing nonlinear engagement scenario in which to test the filters. At the start of the engagement, the guidance filter was given erroneous initial state and covariance estimates. The sensor is assumed to have acquired the target and the guidance filter is ready to estimate the relative motion between the two vehicles using passiveonly measurements. The estimates were used by a standard augmented proportionalnavigation (PRONAV) guidance law to command accelerations that keep the interceptor on a collision course with the target. No errors associated with the attitude control system were used as this would complicate the analysis and cloud the performance comparison. Figure 1 shows a model of the interceptor, a generic 4kilogram vehicle configured with divert and attitude control thrusters. A generic, SS18like ICBM model was used to generate the target motion. The interceptor may have many error sources, some of which include handover errors, pointing errors, thrust errors, centerofgravity location errors, and seeker noise. Initially many of these noise sources were not used so that the filter characteristics could be more easily discerned. Several engagement closing 10 11 angles were examined to allow for more stressing conditions (i.e., tail chase, noseon, beam shot). Once proper filter operation was assured, noise sources were added to more realistically emulate the true intercept scenario, as well as to identify weaknesses in filter performance. Back View SDivert Thrusters Top View Z Divert Thrusters Back View Attitude Control Thrusters X Z Z Figure 1. Generic intercept vehicle. The simulation used was the Guided Simulation Program Exoatmospheric (GSP EXO).' It is a FORTRAN simulation developed by the General Electric Company for the Air Force. Figure 2 shows a block diagram representation of the subsystems modeled. Figure 2. Guidance system block diagram To allow for rapid and efficient evaluation of filtering concepts, the author has modified the guidance system. It has a MonteCarlo capability with an outer loop around the MonteCarlo loop, to allow for parameter trades. Analysis was performed using a personal computer; however, analysis can be accomplished on other platforms, including the SUN, VAX, and CRAY. CHAPTER 5 EXOATMOSPHERICGUIDANCEPROBLEM FILTER DEVELOPMENT The following paragraphs outline the formulations of the EKF, bootstrap filter, SDREF, and MSDREF that were used for this research. A brief theoretical description of each filter is given, followed by the applicationspecific equations. There are three filter types under consideration in this research paper. The mathematical derivations for all of the estimators are fully developed and all assumptions regarding noise characteristics and linearization are explained. The filters are presented in coordination with the problems that were used in examining their performance. The SDREF theory and derivation used in the exoatmospheric problem is shown in this Chapter, the SDREF pendulum problem and derivation is presented in Chapter 7, while the MSDREF stability proof and analysis is found in Chapter 8. Many variations of SDREF are possible due to the infinite number of parameterizations available. Initially, simple parameterizations were used. Another important factor in evaluating performance differences between the three filters was the selection of the noise characteristics. The EKF performance may be enhanced in some cases by artistically "tuning" the noise parameters within the filter. To assure an unbiased comparison between the filters, noise parameters were set to the same nominal values for all filters. Due to several limitations on the bootstrap filter, developed in a thesis by this author, it was used only for pdf comparisons in the analysis of the strategic guidance problem. The bootstrap filter needs further development to be useful in stressing 13 14 engagements as the number of samples required for adequate performance makes it intractable to run for large numbers of comparison. The SDREF and EKF were analyzed in great detail to assess key performance characteristics including convergence, stability, and state estimation error. A key performance critique was the comparison of the evolution of the SDREF and EKF pdf's to that obtained by using the bootstrap filter. Interesting insight into the differences between how the SDREF and EKF model nonlinearities was observed. There are many issues involved with the SDREF development, including correct parameterization of the nonlinear measurement equations, investigation into the requirements for solution of the SDREF, and the stability of the SDREF. These areas were explored to determine an accurate set of state and measurement equations and a feasible method to solve them. An initial non stressing scenario was selected to establish a benign comparative baseline performance assessment. The pdfs were compared using this baseline scenario as well as a nonlinear maneuvering target scenario. Once the baseline was established, the filters were tested against more stressing engagements and intercept angles. Performance was measured by examining the state estimate error and 3sigma standard deviations, as well as comparing the state estimate to the true state. The author has found that this is an accurate method of evaluating filter performance. 5.1 EKF Derivation The derivation for the EKF can be found in many sources,1819 and is presented here for completeness. The dynamics for the intercept problem at hand are linear, while the measurement equations are nonlinear. For this reason a combination of linear and EKF equations were used and implemented in discrete form. Given a linear, timevarying differential equation for a dynamic system x(t) = F(t)x(t) + G(t)u(t) + L(t)w(t) (5.1) where x(0) is given, u(t) is a deterministic input, and w(t) is a random disturbance, the corresponding discrete time solution is t, x(t) = D(t, t_,)x(t_) + fQ(tr .,)[G()u(r) + L(r)w(r)dr. (5.2) 1. If u and w are assumed piecewise constant over the sampling interval, this can be written as Xk = ~ kXkI + Tkluk +Ak ,_,wk (5.3) where rk_ k = J (tk,r)G(T)udr (5.4) and AkIW, =j (tk,r)L(r)wdr. (5.5) 1ti However if F, G, and L are also considered constant over the sampling interval, AT, then 4(At) = e" (5.6) F(A) = D(At)[l, D'(At)]F'G (5.7) A(Ar)= 'D(At)[,, '(At)]F'L. (5.8) Assuming then that E{xo) =m, E{(xxo)(xxo)T) = P E{w)= 0 E{wwwkT)=Q; and E{ww,_,} =0 the state propagation equation becomes xk = (), xk + rk,Uk_ + A,_(0) (5.9) The covariance matrix can be propagated in a similar manner by P. = l,P k, l'r+ k (5.10) with Qki = AkQAk. (5.11) To relate the covariance matrix of random sequences to the spectral density of the random process for the continuous disturbance input w(t) we note that E{w(t)wT ()}= Q'c (t)S(t z) (5.12) The equation for Qk,_then becomes Qk1= i (tk')L(r)Q'c ( r)r(t t)dr (5.13) k; This now represents the integrated effect of the continuous disturbance input w(t). Thus the two equations necessary for the propagation of the state and covariance are xk = kl,_l + k_Uk_, (5.14) P'k = kIPk_ ITi' + Qki (5.15) where Ok1 is defined by eFM and 1_, is define in equation (5.7). The measurement process for the filter is now determined. Given the continuous, nonlinear measurement equation z(t) = h(x(t))+ v(t). (5.16) The discrete version can be written as z, = h(xk(t))+v (5.17) 17 A complete derivation of the EKF update equations can be found in,18 however the final solution for the calculation of the Kalman gain, and the update of the covariance and state are k (+)= ()+ K [zk hk(k())] (5.18) K, = 'k()HkT (i())[Hk(ik())Pk ()H'(k (())+Rk (5.19) Pk(+) = [I KkH,(())]Pk()[ KkHk (k ())I + KRKkT (5.20) where ik () and Pk ()come propagation equations (5.14) and (5.15) and Hk (xk = (5.21) with E{v } =0 and E {vkvk } = Rk. Now (5.14) and (5.15) and (5.18) through (5.20) define the full set of EKF equations. For the exoatmospheric intercept problem the state consists of sxR sYR SZR xR xk = YR (5.22) VZR axr ay, az, where the states are relative position, relative velocity, and target acceleration. The system equations are sR =VR (5.23) vR = AT A,, (5.24) ar, = Aa + w (5.25) Putting this in state space notation, Ss 1 0 S 0 0 0 000 S1=0 I0 v, + 00 I 0 a co 0 + 0 0 (5.26) a, 0 0 AJla, 0 0 o0 0 0 L o wr From this equation the state transition matrix is computed as I IAt I(e "+ At 1)/A' 0(At)= 0 1 I(le)/IA (5.27) 0 0 le' where I is the 3x3 identity matrix. Assuming the commanded acceleration remains constant over the filtering interval IAt2A ,I 2 SD(tk,,r)G(r)udT= AtAcoA (5.28) 0 The state propagation equation becomes At AcaI + tAO1 (5.29) 0 The covariance propagation proceeds in a similar fashion. The covariance can then be propagated by P = [ <]_, []T + Q. (5.30) The measurement equation is zk = hk (,(tk) + v, (5.31) Range where h (i(tk))= zE / Range Range = x + + R (5.32) y, I Range and vk = N(O, R) (5.33) is the Gaussian (normal), random vector with zero mean, and covariance Rk. This completes the derivation and equation formulation for the ninestate EKF needed for the interceptor guidance system. Given an initial estimate of the state x(0) and the covariance matrix P(0) equations (5.14) and (5.15) and (5.18) through (5.20) can be solved in a recursive fashion to achieve an estimate of the relative vehicle motion. 5.2 Bootstrap Estimator Derivation The bootstrap filter is derived using the recursive Bayesian estimation relationships. To begin the discrete dynamic state equation is given by Xk =fk (Xk, iWk) (5.34) where fk is the system transition function and w,_, is a random, zero mean, whitenoise sequence representing a known calculable probability distribution function. At discrete times, an observation becomes available z, = h(x,,vk ) (5.35) where vk is a zero mean, whitenoise sequence representing a known distribution function. It is further assumed that an initial pdf p(xolYo) of the state vector is available. The objective of this process is to construct the pdf of the current state xk given all the available information; i.e., p(xkIY). 20 The prior distribution before the update is defined as p(xk I Y) = p(xk I x) p(xk 1 k)dxk (5.36) where the probabilistic model of the state evolution can be obtained through the use of the system state equations and the known statistics of wk as P(Xk xk1) )=p(xk Ixl, wk) p(wk_xkI) dw,_ (5.37) Because the process noise is assumed to be independent of any previous state values, p(wk_ xk1) = p(wk_). If x,_1 and wk_ are known, then x, can be arrived at deterministically. The update stage occurs when a measurement becomes available. The a priori density may be updated according to Bayes' rule as p(xIY )= P(Yk Ix) P(xklYkI) (5.38) P(YlY ,) where the normalizing denominator is P(YlYk) )= p(yYk x,) p(xk IYkY) dx (5.39) and the conditional pdf, p(ylxk), is defined by the measurement model and the known statistics of v,. The method by which the following theory is implemented in the bootstrap filter will now be defined. The bootstrap filtering algorithm provides the mechanism for defining a set of random samples {x, (i); i = 1,..., N} from the pdf and obtaining the distribution p(xk5, ,,_) .57 The propagation stage assumes a number of N random samples have been taken from some initial pdf p(xk _, lY). Each random sample for each state is then passed through the system model to obtain a set of a priori samples at time step k xk (i)=k 1 f (i),wkl(i)),i= 1,... N (5.40) where the asterisk represents the a priori sample. The update stage of the filter takes each of the a priori samples and computes a set of a posteriori samples from which the estimate can be obtained. To do this, the likelihood of each a priori sample l(xk'y ) is calculated using the measurement equations and a normalized weight is given to each sample by P(yk Xk(j)) This defines a discrete distribution over the a priori density. The normalized values of q, can be resampled using a uniform distribution. A larger value of q, allows more samples from that bin to fall into the a posteriori pdf. The state associated with each new sample selected is assigned to the new a posteriori state {xk (i); i = ...,N. This a posteriori set of samples is approximately distributed as p(xk lY). From this the mean and variance for each state estimate can be computed. Because this filter makes no assumption of linearity or Gaussian statistics, it was useful in providing a close approximation of the true pdf. This semitrue pdf can then be used to investigate the effect of linearization and parameterization on the evolution of the pdf. 22 5.3 SDREF Derivation The SDREF is the nonlinear "dual" of a nonlinear regulator control design technique, which uses a parameterization technique to bring the nonlinear dynamic and measurement equations x=f(x) (5.42) z = h(x) (5.43) and transform them into a linearlike statedependentcoefficient (SDC) form x= F(x)x (5.44) z = H(x)x (5.45) The distinctive assumption in the SDREF technique is that the SDCs can be treated as "frozen" over a small interval to give adequate piecewise performance. The algebraic Ricatti equation is solved at every filter measurement update and the linear Kalman filtering algorithms applied. The technique was first applied to the following nonlinear regulator problem.13 The task is to minimize I= 1 xQ(x)x + uTR(x)u dt (5.46) with respect to the state x and control u, subject to the nonlinear differential constraint S= f(x) + g(x)u (5.47) The SDREF approach for obtaining a sub optimal solution is to use direct parameterization to bring the nonlinear dynamic equation to the SDC form i= A(x)x+B(x)u (5.48) 23 where f(x) = A(x)x and B(x) = g(x) From this point the SDREF can be solved for P as Ar(x)P + PA(x) PB(x)R~(x)BT(x)P+Q(x)=0 (5.49) and the nonlinear feedback controller can be constructed as u= R(x)BT(x)P(x)x (5.50) The SDREF technique is shown'3 to have desirable properties of local asymptotic stability and suboptimality given mild conditions of stabilizability and detectability. It also shows that if A,(x) and A,(x) are distinct SDC parameterizations, they can be combined to yield a third parameterization by A(x, a) = a4, (x)+(l a)A2(x) (5.51) This technique of combining parameterizations allows an infinite number of parameterization possibilities to enhance controller design. An example'3 of the SDC form and SDREF is repeated to introduce the method. Consider the stochastic nonlinear system = f(x) + r(w) (5.52) z= h(x) + v (5.53) where w is Gaussian zeromean white process noise with E[w(t)wT(t + r)]= Q(t)8(r) and v is Gaussian zeromean white measurement noise with E[v(t)v'(t + )] = R(t)8(r). After bringing the system to the SDC form x= F(x)x+Fw (5.54) z= H(x)x +v (5.55) 24 the SDREF is given by the steady state, linear, continuous Kalman filter equations as x = F()i+ + K (i)[z(x) H(i)5] (5.56) where K (j) = PHT ()R' (5.57) and P is the positive definite solution to the SDREF F(i)P+ PFT(i) PHT(i)R'H(i)P+F'Q = 0 (5.58) The preceding method was applied to a simple twodimensional pendulum problem.13 Results were comparable to those from an EKF. The main thrust of the research was to develop a SDREF for the linear dynamic and nonlinear measurement equations associated with the strategic, terminal homing guidance problem and investigate filter characteristics using the EKF and bootstrap filter for comparison and analysis. 5.3.1 AngleOnly Measurement SDREF The following derivations are performed for the bearingmeasurements only SDREF. It shows that this form of the SDREF is not detectable and therefore violates the necessary conditions to assure the positive definite solution of the Ricatti equation. Both the continuous and discrete forms of the filter are given, as well as stabilizability and detectability analyses for each type. The filter derivation begins with the definition of the state equations x= Fx + Gu+ Lw (Continuous form) (5.59) Xk =lkIX_1 +rFk_l1+Ak_Iwk_1 (Discreteform) (5.60) and the measurement equations z=h(x)+v (Nonlinear form) (5.61) z = Hx+v (Linear form) (5.62) The states to be estimated are SXR SYR SZR VXR xk = VYR (5.63) VZR vz, ayr TaZ az, where the states are relative position, relative velocity, and target acceleration. The system equations are sR = vR (5.64) R = AT (5.65) a, = A + WT (5.66) Putting this in state space notation R = 0 L v + I aco, + O WT (5.67) Ir 0o JAlaJ / i = [F] Ix)+ [G] u + [L]w The measurement equation is Zk =h(x)+v, (5.68) 26 where, based on the smallangle approximation, h(x) = Rane Range = J2 +y2 R2 (5.69) Range and vk = N(0,R,) The SDREF approach for obtaining a sub optimal solution is to use direct parameterization to bring the nonlinear dynamic equation to the SDC form x= F(x)x+G(x)u (5.70) where f(x) = F(x)x and z= H(x)x + v (5.71) From this point the SDREF can be solved F(x)P+PFT(x)PHT(x)Rl(x)H(Hx)P+LQLT(x)=0 (continuous) (5.72) QPTgrPOPHr(R+HPHT)'HPOT+Q_, =0 (discrete). (5.73) Since the system dynamic equations are linear, the only parameterization needed is for the measurement equation. From the definition z = Hx + v (5.74) the linear form for the SDREF can be cast as H [ R 0 0 0 0 0 0 J (5.75) 01/R 0 0 0 0 0 00 5.3.1.1 Continuous SDREF Using the abovedefined state and measurement equations with A =0.1, the following matrices are used for the filter equations 27 0 0 0 1 00 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 00 1 0 0 F= 0 0 0 0 0 0 0 1 0 (5.76) 0 0 0 0 00 0 0 1 0 0 0 0 0 0 .1 0 0 0 0 0 0 0 0 0 .1 0 0 0 0 0 0 0 0 0 .1 The time constant for the Markov process A, was based on an expected target acceleration of approximately 981 (m/s2). 0 0 0 0 0 0 0 0 0 1 0 0 G= 0 1 0 (5.77) 0 0 1 0 0 0 0 0 0 0 0 0 The process noise used for this problem is 1 g on the xtarget acceleration and 3 g's on both y and z target accelerations. The disturbance input is E[w(t)] = 0 (5.78) [(000)= (( )] (579) 000 000 G= 0 1 0 (5.77) 0 0 1 000 000 000 The process noise used for this problem is 1 g on the xtarget acceleration and 3 g's on both y and z target accelerations. The disturbance input is E[w(t)]=0 (5.78) E[w(t),w'(V)= Qc(t)3(tr)] (5.79) 28 which for small values of At can be expressed as 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 Qk1 =LQcL At= 0 0 0 0 0 0 0 0 0 At (5.80) 0 0 0 0 00 0 0 0 0 0 0 0 0 0 96 0 0 0 0 0 0 0 0 0 866 0 0 0 0 0 0 0 0 0 866 where 866 (m/s2) is the variance of 3 g's of acceleration noise in the y and z directions. The parameterized measurement matrix can be computed as 0 0 0 0 0 0 0 0 (x)= Range (5.81) 0 a 0 000000 Range The standard linear, continuous Kalman filtering equations follow as = F(i) + K, (G)[z() H(i)i] (5.82) where K, (i) = PHTR,' (5.83) and P is the positivedefinite solution to the statedependent Ricatti equation F(i)P + PFT() PHT(i)RK,_1H(i)P+Q, = 0 (5.84) where Qki = LQc'LrAt. Ricatti equation theory provides for a legitimate positive semi definite solution if the following conditions are met (TD,Q,PER",HT ERm,ReR RrnQ=QT>0,R=RT>o,m and (F,H) is pointwise detectable in the linear sense and(C, F) is pointwise 29 stabilizable in the linear sense for all x e 9, where C is a fullrank factorization (FRF) of Qk given by (C'C)= Q and rank(C)= rank(Qk) (5.86) For the given linear dynamics we can construct the controllability matrix as Con=[C FC ...F"'C] (5.87) If Con is of full rank, controllability and therefore stabilizability is assured. If it is not fully controllable, a minimum condition of stabilizability must be met. Putting the system equations into the controllability canonical form, the uncontrollable poles can be checked for stability.20 Using a similarity transformation, the controllability canonical form can be found as F=TFT', C =TC (5.88) where T is unitary and the transformed system has a staircase form = 0] C_ 0o where (F,,G.) are controllable and (F,,) is uncontrollable. The poles of the uncontrollable subspaces are then checked to insure stability. If they are found to be stable, then stabilizability is verified. Using the noise matrix in equation (5.13) and performing the FRF for C, it can be shown that the controllability matrix has full rank and therefore stabilizability is verified. 30 Similarly, the observability Grammian matrix is constructed as H HF Obs= (5.90) HFn1 If Obs is of full rank for all x, then full pointwise observability in then linear sense is assumed. Once again pointwise detectability needs to be assumed for the Ricatti equation solution to be positive semi definite. If it is not of full rank the similar observability canonical form can be used to again check the pointwise stability of the unobservable poles. This canonical form is given by =[F FI, H=[0 Ho] (5.91) Using the above referenced matrices for Fand H, it can be shown that the observability matrix is not of full rank. Therefore, pointwise stability of the unobservable poles must be checked. It was found that the poles were not pointwise stable and therefore point wise detectability was not verified. A similar analysis was then performed on the discrete form of the filter. From this point on, only the discrete form of the SDREF is analyzed. This form was chosen due to the use of discrete measurements that are only available at a subset of the integration step size of the simulation. The discrete filter is better suited for this type of simulation and would most likely be used in any realistic guidance software. 5.3.1.2 Discrete SDREF The discrete form of the SDREF is derived using the following formulas. Since F is a constant matrix, the state transition matrix can be found using Laplace transforms as I IAt J(e +At1)/1 D(At) = 0 I I(1e ')/A (5.92) 0 0 le When using 2 = 0.1 and At = 0.025, the following transition matrix is computed 1 0 0 .025 0 0 .000312 0 0 0 1 0 0 .025 0 0 .000312 0 0 0 1 0 0 .025 0 0 .000312 0 0 0 1 0 0 .0249 0 0 S= 0 0 0 0 1 0 0 .0249 0 (5.93) 0 0 0 0 0 1 0 0 .0249 0 0 0 0 0 0 .9975 0 0 0 0 0 0 0 0 0 .9975 0 0 0 0 0 0 0 0 0 .9975 Assuming the commanded acceleration remains constant over the filtering interval At2A co 2 Fuk ,,= (tk,T )G()ud= AtAcoml (5.94) 0 The state propagation equation becomes [ At21 2 x [E]x, + AtI Ao (5.95) 0 Also, E[wwT] = QS(t ) (5.96) and Qk, = I(tk ,T)L()Q'c (r)L (r)c'(tk ,r)dv (5.97) 1I Using 1 g of acceleration noise on the x target acceleration and 3 g's in y and z and performing the above integration, the process noise matrix is obtained. 9.46e9 0 0 9.36e7 0 0 4.99e5 0 0 0 8.54e8 0 0 8.44e6 0 0 4.45e4 0 0 0 854e8 0 0 8.44e6 0 0 4.45e4 9.36e7 0 0 9.98e5 0 0 .006 0 0 Qk = 0 8.44e6 0 0 9.0e4 0 0 .054 0 0 0 8.44e6 0 0 9.0e4 0 0 .054 4.98e5 0 0 .006 0 0 .478 0 0 0 4.5e4 0 0 .054 0 0 4.32 0 0 0 4.5e4 0 0 .054 0 0 4.32 The discrete Ricatti equation solver requires the following conditions to be met to insure that a legitimate positive semi definite solution to the Ricatti equation is obtained: (O, Q,Pe Rn',HT R Re Rm", Q=Q > R = R >0,m and (D, H) is pointwise detectable in the linear sense, and (C, () is pointwise stabilizable in the linear sense for all x e Q, where C is a FRF ofQ,_ and is defined as (CTC)= Qk and rank(C)=rank(Qk_) (5.99) It is also must assumed that Q is invertible without need for further modifications to the solver. If all assumptions are met, then the Ricatti equation will have a unique non negative definite solution throughout 2. 33 Using the above referenced matrices, the same analysis used for pointwise stabilizability of the continuous system was performed. The controllability matrix of the pair (C,D) was found to be of full rank and therefore the system is stabilizable. Again an analysis similar to the continuous version was performed using the pair ((D, H) and the pointwise observability matrix was again found to be not of full rank. The canonical form was calculated and again the poles of the unobservable subspace were determined to be pointwise unstable and therefore the system could not be declared pointwise detectable in the linear sense. Because the system has been shown to be pointwise undetectable, the Ricatti equation solver cannot provide the positive semi definite solution to the Ricatti equation. Consequently, another approach to the problem must be examined. One way to make the system detectable is to allow a measurement in the range or xdirection. This is shown in the next section. 5.3.2 SDREF With Range Measurement This section analyzes the same filter described above except that a range measurement has been added to the measurement equation. By adding the range measurement the system becomes both stabilizable and pointwise detectable under the conditions set forth above. Two types of range measurements were evaluated to provide adequate observability for the system. 5.3.2.1 Intensity range measurement The range measurement used is the source radiant intensity of the target arriving at the focal plane and is I= Range k (5.100) Range where Ik =intensity (watts) ( watts i J, = source radiant intensity watts (sr/pm) A = bandwidth (,um) r = radius of seeker aperature (m) R = range from source to seeker (m) v, = white Gaussian noise This gives the measurement equation: z/Range h(x)= ylRange Range=x x/+yR2 +zR (5.101) JA2tmr2 /Range2 Putting equation (5.101) into the SDC form 0 0 000000 Range H= 0 1 0 0 0 0 0 0 0 (5.102) Range Rang 2r 0 0 0 0 0 0 00 Range2 *XR The measurement noise for the range measurement is somewhat more complicated as it is range dependent. The actual intensity measurement has Gaussian noise from the focal plane electronics, which can cause error in the measurement. However, it is assumed this error is small in comparison to the fluctuations of the source radiant intensity and is 35 assumed to be zero. Jz is, however, comprised of both the deterministic source radiant intensity and a random part given by JA = J, + I_ (5.103) The resulting intensity contribution from the random fluctuations in the plume core can be calculated from equation (5.101) as I = J (5.104) Range2 The variance of this random component of intensity is obtained by taking the expected value of the square of I, (i.e., E{I, }). The equation for the expectation operation gives E i(J,,AAIr2' 2 a E{J' 1 (5.105) Range1 J Range2 If p= E(J2,, ), then the range dependent measurement variance term for the filter becomes (pAArr2 2 SRange 2 (5.106) Using appropriate values for these constants, a measurement matrix can be given by 2.5e9 0 0 Rk= 0 2.5e9 0 (5.107) 0 0 8.9e 7 Using the abovedetermined matrices, the SDREF was again calculated. The filter responded well during the first three to four seconds of flight and then diverged. The 36 problem lies in the instability of the Ricatti equation solution. In examining the point wise closedloop eigenvalues of the system (i.e., the spectrum of DT HT(R + HPHT) HPQ (5.108) with x, in this case range, frozen), the system would be asymptotically pointwise stable in the linear sense if the closedloop eigenvalues all lie inside the unit circle. The eigenvalues of the SDREF were computed for each filter update stage. They began near the unit circle (i.e., 0.9992) and grew to become unstable as the simulation ran. The final step in the Ricatti equation solution is the solution of an nt order linear matrix equation. A number was computed to determine the condition number for the inverse matrix. It could be seen that the condition number increased (i.e., illconditioning) as the closed loop eigenvalues approached the value of one. The relatively small size of the range variance causes R' to become very large, and therefore ill conditioned. Once the closed loop eigenvalues became larger than one, the problem diverged quickly. Because of this numerical instability, a different range measurement was tried. 5.3.2.2 Distance range measurement The new range measurement is the distance measurement in meters. This value is assumed to be available from the signal processing algorithms. The new measurement is Range =x, + yR + (5.109) 37 Therefore, the new measurement equation becomes z Range h(x) (5.110) Range Range and the measurement matrix becomes 0 0 1 0 0 0 0 0 Range H= 0 0 0 0 0 0 0 (5.111) Range Range 0 0 0 0 0 0 0 0 X, Using these values in the SDREF, satisfactory results were obtained. (These results are given in Chapter 7.) It was decided that this would be the formulation that would be used to further test this new filtering technique. Before these results are presented, however, an investigation into the theoretical stability of the SDREF is presented. 5.3.3 Theoretical Investigations One of the main theoretical criteria for filtering techniques is the stability of the algorithm. The following section reviews several theoretical stability investigations. While these investigations were unsuccessful in proving stability, it is hoped that they provide valuable insight into the SDREF. It is these attempts that led to the development of the MSDREF shown Chapter 9. While the methods outlined below approach the problem from several different theoretical aspects, all were unsuccessful in showing local stability for the SDREF. Because the true and estimated states x and i both appear in the error equation 38 e = (F(x)x F(.)! K(x)[H(x)x H(i)]) (5.112) each of the approaches fail when presented with the cross terms that are created. It is hoped that this section will prove useful to future stability investigations. The approaches are shown starting with simple linearization techniques and progress through more complicated and sometimes restrictive theorems. The first of these is a simple Taylor series expansion of the parameterized equations. 5.3.3.1 Taylor series expansion The following stability analysis will exploit a Taylor series expansion about the local zero point. Consider the stochastic, nonlinear system = f(x)+g(w) (5.113) z= h(x)+v (5.114) where w is Gaussian zeromean white process noise with E[w(t)wr(t + r)] = Q(t)8(T) and v is Gaussian zeromean white measurement noise with E[v(t)v'(t + )] = R(t)8(T). After bringing the system to the SDC form = F(x)x+Fw (5.115) z = H(x)x+v (5.116) the SDREF is given by the linear, continuous Kalman filter equations having state dependent coefficients S= F(i) + K(i)[z(x) H(i)i] (5.117) where K(i)= P(i)HT(i)R' (5.118) and P is the positive semi definite solution to the algebraic SDRE F(i)P(i)+P(i)FT(i)P(i)HT(i)R'H(i) P(i)+ FQF =0 (5.119) 39 Conjecture: Assume that R > 0, Q >0, f(x), and h(x) RK, and that the parameterizations F(x) and H(x)can be shown to satisfy that (F(x), H(x))is a pointwise detectable pair in the linear sense and (C(x), F(x)) is a pointwise stabilizable pair in the linear sense, where CC = Qk,. Also assume F(x) e C" and H(x) e C~ are analytic, where C~ is the class of functions whose derivatives of all orders are continuous, and that f(0) = h(0) = 0. Then the SDREF is locally asymptotically stable. Outline of Attempted Proof: For ease of notation the proof will be examined in scalar form. Given that the scalar system dynamic equations are x = F(x)x+Gu+ Lw z=H(x)x+v and (5.120) x = F(i)i + Gu +K[z H(x) ] and defining the estimation error rate as e = x the above equations can be written as = (F(x)x F(i) K(x)[H(x)x H(i)i]) (5.121) Now expanding h(x) and H(x)x in a Taylor series expansion about zero h(x) = H(x)x (5.122) A(0) d h ) h (0) i (x0) (0) (0) + H.O.T H(0)+ )+ +.O.T x x= 10 A X=o 2! L Ix=0 ax=0 21 Also for f(x)= F(x)x f(x) F(x)x (5.123) 3() d2f ( 2O)2 F dF(O) d2F (x0)2 Po)+ (xo)+ if + 0) o L H O.T (0o )+ +H.o.T.= F(O)+ ((x)+" 2 o.r. x SI o= 2 L A X=O a 0 2! Similar equations can be written for h(I) = H().i and f (x) = F(i)i. Equating like terms A(0) A(0) h() = H(0)  and O(o) F(O)= (O) Using this and recalling that K = PHTR', E(t) = (F(0) PHT(O)R'H(O)))E(t)+ H.O.T. (5.124) Herein lies the problem, the higher order terms contain both x and i. These terms cannot be guaranteed to approach zero, much less go to zero faster than the first term. If the error dot equation could be expressed in this linear form a local stability proof could be achieved in the following manner. For ease of notation, let F(0) = F, H(0) = H, and e(t) = e. Following a similar derivation,'9 a Lyapunov function is chosen V(e) = eIE (5.125) where I is the positive definite information matrix, the inverse of P. From the definition of the derivative of the inverse, I = P = IPI (5.126) By substituting this into the matrix Riccati equation, a differential equation j = IF + FI +ILWLTI HTN'H (5.127) is yielded that has the steadystate solution 0= IF +FrI +ILWLI HTN'H (5.128) 41 The time derivative of the Lyapunov function is V= 2erI = 2eT[F KH]e = 2eT[F PHTN'H]e (5.129) = er[(IFHTN'H)+(IF HTNH)]e Using (5.128) this can be written as V= 4ILWLrI+H TN H]E (5.130) As the term in square brackets is positive definite at all times, V < 0 at all times. This would verify the local asymptotic stability of the SDREF for some small neighborhood about the origin. The estimation error would approach zero as time approaches infinity. 5.3.3.2 Using a Taylor series expansion about the true states This approach varies slightly from the above, in that a Taylor series expansion of the estimated dynamics and measurement equations are linearized about the true states. Using the same state and measurement equations (5.1135.120) and the error equation (5.121) in the preceding proof, F(i) and H(i) are expanded as F"(x) F() = F(x)+F'(x)(ix) + (.x)2+...H.O.T. (5.131) 2 and H()= H(x)+H'(x)(x)+H(x x)2+...H.O.T. (5.132) 2 Using equations (5.131) and (5.132) equation (5.121) is written as S= {F(x) KH(x)}e+ {F'(x) KH'(x)} d t(e2) (5.133) Once again the problem with this approach can be seen. The only way it can be shown that the second term disappears or is negative is for ex to be zero. This can only be shown to be true for the linear case. 42 5.3.3.3 Taylor series expansion about the error Again, using the same system equations (5.1135.121) as in the first Taylor series approach, this second approach linearizes about the error. Linearizing the dynamic and measurement equations F"(E) F(x)= F(e)+F'(e)(x )+ (xE)2+...H.O.T. (5.134) 2 F"'(e) F'(x)= F'(e)+F"(E)(x e) + (x )2+...H.O.T. (5.135) 2 H"(e) H(x)=H(e) +H'(E)(x ) + (x e) +...H.O.T. (5.136) 2 H"(e) H'(x)= H'(e)+ H"(e)(xe)+ (xe)2+... HO.T. (5.137) 2 After substituting these into error equation (5.121), = (F KH) +(F' KH')(x e)e +(F' KH')ei +(F" KH")(x E), +(F"KH")(x E)2 (5.138) 2 +(F"'KH"')(x e)2 + (e2) 2 Again, the error equation is only stable if it can be shown that ex > 0. This can only be shown to be true for the linear case. 5.3.3.4 Residual error stability approach The following approach illustrates, under conditions of pointwise detectability in the linear sense that ensure a legitimate solution of the SDRE, that the SDREF can be shown to be asymptotically stable.21 The system dynamics and filter equations are given in discrete form as xk+i = 4kXk +Wk (5.139) Zkt+ = h(xkl)+ Vk+ (5.140) Xk+i/k b (5.141) "k+i =k+k + Kk+I[Zk+ Hk+ (k+Ik)k+l/ ] (5.142) Kk+I Pk+1 (k+I)Hk+T (k+,)Rk+,' (5.143) with E[vk+l vk+I ]= Rk+l (5.144) E[wwk+WT] = k+ (5.145) where P, (ik+) is the solution of the SDRE. These are the equations necessary for the discrete SDREF. It is also assumed that vk+5 and w.k+ are random, zero mean inputs and therefore, the stability of the SDREF is independent of these error sources. The state estimation error both after and before the update is defined respectively as k k+ k+ (5.146) xk+/k = Xk+l Xk+/k (5.147) Also, for ease of notation Hk+ (k+= Hk (5.148) Pk+ (k= P+. (5.149) e,+ = Zk+l Hk+ (k+ )k+l (5.150) And finally a candidate Lyapunov function Vk, is defined as vi = pk+ pk Xk+ (5.151) The conditions for which Vk+~ is a decreasing sequence must be determined, therefore showing the exponential stability of the SDREF. 44 In the preceding stability approaches, linearization of the filter equations has not shown stability of the filter. This attempt constrains the state transition matrix and measurement noise matrix so as to characterize the error due to linearization. The measurement residual error can be characterized as ek+1 =zk+ h(ik+1/k) (5.152) e+ = Hk+(l k+ )Xk+l Hk+ (k+ l)k+l (5.153) Rewriting H,, (i+,) using a Taylor series expansion, for each output error prediction component of e+,, namely e, ,k+, there always exists a residual r,,k+jdue to the first order linearization in order to obtain an exact equality. Therefore, e,.k+1 H,,k (x+ k+I)x+ Hi.k+ (xk+) + H'i,+ (xk+1 )(X+ X )+. (5.154) and the residual error is rk+= {Hi, (Xk+1)(ikl Xk)+. } +l (5.155) Then the exact measurement error can be defined as e,.k+ = Hik+1 +1k + r.,k+ (5.156) or written in another form Hik+3lk+1k = ai.k+lei,,k+ (5.157) where a,,,k is an unknown timevarying scalar related to ri,k+ by ,k++ (1 a,k+l )e,k+l (5.158) Introducing the signal vector ak+lek+ = Hk+15k+/k (5.159) where ak+, e RP' is an unknown time varying diagonal matrix defined as ak+1 = diag(a,k+1, .. a.,~k+1) (5.160) If both sides of equation (5.142) are subtracted from xkl,, + = k+I/k Pk+IHk+ITRk+I'ek+1, (5.161) Substituting this into equation (5.151), and using the fact that Pk+1 =Pk Vk+1= .k+k iT Pk+11 k+l/k Ktk T Hk+RIT+ k+1 (51 ek +lt R+1 Hk+1 k+l/tk ++l tR+ Hk+,Pk+H l R,+ ek+l Because the covariance matrix is not propagated during the time between updates, it is assumed that the a priori covariance before update is equal to the previous updated covariance, Pk+,1k = P (5.163) Therefore, Pk+' =Pk' (5.164) Before proceeding further, the inverse relationship is defined as Pk+ = Pk+,,11 + Hk+,TRk+,'Hk+5 (5.165) Once again a similar problem arises. Equation (5.165) is not derivable from the basic premise of the SDREF update equations.21 If this equation was available, it could be shown that Vk,, is a decreasing sequence. Then Vk+I = Vk+I1, Xk+U III T H Rk+1 ek+1 (5.166) ek+, Rk+1, Hk+,,k+11k +e,, +RLHk+ Pk+ Hk+IT Rk+1 ek+ and reviewing that Hk+lk+1= ak+, (5.167) and XTk+IHHTk+i = eTk+laT k+ (5.168) Equation (5.166) is written as Vt+l = Vk+l/k ek+,1 t+a k+l atk+lek ek+ ak+lRk+lHk+lek+1/k k e+TRk1 'ak+ek+ (5.169) +ek+l Rk+l lHk+I Pk+lHk+I Rk+,lek+l And using the fact that Vk+,1/ = TFkTPk+,,kI'Fk (5.170) with Equations (5.167168), Equation (5.169) is rewritten Vk+1 V = eV+lT k a+l Rk+,1ak+l akt+iRk+., Rk+,,ak+, +Rk+1lHH+lP+1Htk+TRk+'I }ek+, (5.170) +;5FT(FkTPIF Pk1 Therefore under the condition of pointwise detectability in the linear sense, and certain restrictions on R and Q (i.e., Rk > 0 and Qk >0) then, VI, <0 and the system is asymptotically stable as the values in parentheses are < 0. 5.3.3.5 Lyapunov approach using the expectation operator This stability approach uses a similar method.9 Given the system equations x+, = A, ,+w, (5.171) = + K,[H(x,)x, H(x,)x,] + Kv, (5.172) the error can be updated and propagated by the equations e, x, i (Updated error) (5.173) x x, (Propagated error) (5.174) Using equations (5.1715.172) e, = x, x, K,[H(x,)x, H(,),] K,v (5.175) = A,_e, + w,_, (5.176) After some manipulation of these equations, the error update equation can be developed eF1= A, , Kix,+A,K,HA, H K,v,+w, (5.177) where H, = H(x) and H, = H(() If the following Lyapunov function is defined V,+ = e,j.e+, (5.178) it can be shown that E{V, (eF.)V,(e,)} 0 (5.179) Substituting equation (5.177) into equation (5.179), the following equation can be computed E{V,,,((,) V()}= eE{ATA I}e, +E{2"ATAKH,x2xTATAKHx} +E{.2xTArAKH2x + 2xTATAKH2Y} T T KT AT Y1 (5.180) +E(2xTHTKTATAKH (5.180) + ExTHTKTATAKHlx} + E{xTH2,TKTAAKH2x} In this equation several terms exist that cannot be shown to be negative definite. These terms are brought about by the same problem found in the previous approaches evaluated to show stability. They are cross products created by the presence of both xand i in the error equation. This same approach was also tried with the Lyupanov function V=eTe (5.181) and Equation (5.182) was developed. E{V,, (e,)V(e,)} = e,TE{ATA e, + E{2A'rAKH,x 2x ATAKHx} + E(2 TArAKH2i + 2xTA'AKH2i} +E{2xTHTKTATAKH2} (5.182) +E{XTH,TKTKKH,x} + E(IrH2T'KKH2z) Once again cross terms are generated and cannot be shown to be negative definite. All of the previous attempts at stability proofs fail for the same reason: The cross term xi, produced in the error equation, raises significant difficulties in finding a closed form differential equation for e. Attempts to limit the problem to a scalar were not able to resolve the cross coupling problem. While several standard and nonstandard approaches were tried, a satisfactory stability proof could not be verified. It is hoped that these investigations can prove useful in future attempts to determine filter stability. CHAPTER 6 EXOATMOSPHERICGUIDANCEPROBLEM SIMULATION RESULTS Chapter 6 investigates the comparative performance of each filter against a variety of engagement conditions and noise parameters. Before any study on filter performance was done, a baseline performance analysis was completed, which allowed a detailed investigation of the SDREF against one engagement scenario. Once the baseline performance was established, further investigation into filter sensitivity, robustness, and closedloop performance was performed. A study of pdf evolution for each filter is also shown. 6.1 Baseline Performance To establish a baseline performance measure, an engagement representing the final 10 seconds of a closing interceptor/boosting ICBM engagement was chosen. Table 1 details the major initial filter parameters. The aspect angle is defined as the initial Euler rotations between the axes of the interceptor and the target. For example, an aspect angle of (30,30,30) would signify that the interceptor's initial velocity vector has been rotated 30 degrees about each axis. An aspect angle of (0,0,0) would be a headon engagement. The (30,30,30) aspect angle was chosen so that all axes of the filter would be exercised in the engagement. SIGP and SIGVP are the standard deviations for the random initialization errors on the "true" position and velocity states respectively. When SIGP 50 and SIGVP are zero, the filter has been initialized with the "true" initialintercept conditions. SIGXHAT are the values used to initialize the covariance matrix for the EKF and the bootstrap filter. XHAT is the initial state given to the filters. The measurement noise for the xdirection is a rangedependent noise that is given as a percentage of the range (i.e., as the interceptor gets closer to the target the noise on the range measurement decreases). All filters were made to operate with the same initial conditions and statistics so that comparisons could be made. A perfect filter was used to guide the vehicle to impact, while the EKF, SDREF, and bootstrap filter were run open loop. The bootstrap filter was run with 10,000 samples. Table 1. Filter simulation parameters baseline conditions Parameters (units) X Y Z Aspect Angle (degrees) 30 30 30 SIGP (m) 0 0 0 SIGVP (m/s) 0 0 0 a,2 (m2) 1.0E5 1.0E5 1.0E5 SIGXHAT O 2 (m2/s2) 1.0E3 1.0E3 1.0E3 aA_2 (m2/s4) 1.0E2 1.0E2 1.0E2 R (m) 1.0E5 10.0 10.0 XHAT V (m/s) 1.0E4 335.9 223.9 A (m/s2) 55.2 57.40 38.2 SIGQ (EKF) (m2 /s4) 196.0 196.0 196.0 SIGQ (SDREF) (m2 /s4) 1.0E6 1.0E3 1.0E3 Meas Noise (%,pm,/pm) 0.1 50.0 50.0 51 The following graphs reflect filter performance. Figures 3 through 5 show the positionstate estimate errors with 3v+. standard deviations for each filter. Velocity and acceleration performance was similar between the EKF and SDREF. The bootstrap acceleration estimates are poor due to an inability to forward measurement information to the second derivative states. The problem can be remedied by the use of millions of samples or other adhoc improvement schemes; however, computer limitations and the scope of this dissertation do not allow investigation at this time. Figures 3 through 5 give initial insight into the relative performance of each of the filters for the given scenario. All of the filters adequately estimate the state to the precision necessary for an intercept to occur. With state estimate errors remaining within 3a standard deviations, the filters perform well. The bootstrap filter does show a tendency to keep a wider variance than the other two filters. However, as discussed in paragraph 6.2, when the EKF and SDREF are run in Monte Carlo fashion, they too have wider variances. The results shown here are for one run, while noticing that the bootstrap filter is inherently Monte Carlo, even for one run. The performance curves in Figures 3 through 5 verify proper filter operation and establish a baseline performance from which more indepth analysis proceeds. It is interesting to note in Figures 6 and 7 that the standard deviations for the EKF and SDREF are approximately identical after the initial transients in the EKF die out. This indicates that once the EKF has settled and the estimates are being computed accurately, the EKF covariance propagation and update are equivalent to the algebraic Ricatti solution being computed in the SDREF. This relationship is only shown for one state; however, similar results were found for all states. Figure 3. EKF position errors with +3" standard deviations. S. 1n ,~~0~~~ I I I =I Figure 4. SDREF position errors with 3ustandard deviations. 1 & I I I I '* E n ~m Pm i 0. Figure 5. Bootstrap position errors with 3astandard deviations. ' " I < ! ' r k 10   Figure 6. Comparison of EKF and SDREF covariance 3cr standard deviations for Y position. 5 EKF EKF 1 SDREF SSDREF 15 _____ ___ 15 0. 0 1 5 0.0 0.5 1.0 Time (sec) Figure 7. Expanded view of Figure 6. 1.5 20 54 6.2 PDF Distribution Comparison Using the baseline scenario, a comparison of the posterior (after the measurement update) probability density functions was investigated, to determine whether parameterizing versus linearizing the nonlinearities has an effect on the shape of the pdf. The scenario given in Table 1 was run once and the state estimate and standard deviation were computed three times during the engagement. Samples were taken at 1, 5, and 9 seconds. The pdfs for the EKF and SDREF were computed using a Gaussian distribution function based on the mean and standard deviation at each time increment. The resulting curves, Figures 8 through 9, do not represent the "true" pdf as nonlinear dynamics are involved. However, they do show the basis of the theoretical filter formulation. ("True" pdf functions, calculated using Monte Carlo simulations, are discussed in following paragraphs.) The pdf for the bootstrap filter was generated by computing a histogram of 1,000 samples, curve fitting a function using a cubic spline, and normalizing the area under the curve. Figures 8 through 9 show the onerun posterior pdf functions for the filters. The first time increment is taken 1 second into the flight. The filters are still trying to converge and therefore the distributions are large. This is demonstrated in Figure 8 by comparing the narrow spike of the highly converged bootstrap filter to the wide, non converged pdfs for the other filters. If consideration is given to the scaling on the xaxis, no concern should be given to the fact that the distributions are not near the true state. It can be seen that the estimates are close to the "truth". The xposition state shows a more accurate representation of the pdf than the y and zposition states. This is due, in part, to measurement noise decreasing as the engagement progresses, and a more accurate True Sta p 0.025  oostrap Time= 1 Second EKF SDRE 0.020 0.015 S0.010 0005 0.000 88500 89000 89500 90000 90500 91000 X Position (m) 0.05 Time =5 Seconds True sate 0,04 0.03 S0.02 0.01 0.00 ... . ... 50000 50200 50400 50600 50800 5 X Position (m) 0.16 0'14 Time 9 Seconds True St   EKF 0.12 0,10 t 0.08 008 0 0.06 0.04 1000 12180 12200 12220 12240 12260 12280 X Position (m) Figure 8. Posterior probability density functions at time = 1, 5, and 9 seconds for X position. I Y Position (m) True State 5 T t Boolstrap S EKF Time = 5 Seconds SDREF 1 421 420 419 418 417 416 415 Y Position (m)  Bootstrap EKF Time = 9 Seconds SDREF True State 174.0 1735 173.0 Y Position (m) Figure 9. Posterior probability density functions at time = 1, 5, and 9 seconds for Y position. 0. 0. 0.0 5 4 3 2 knowledge of the true state is obtained. Another interesting factor shown in Figure 9, is the compression of the EKF and SDREF distributions. This is typical of the "falling asleep" problem associated with many filters. As the state estimation error becomes small, the covariance matrix also becomes small, and the filter reacts slowly to new data, which takes the estimate away from the converged answer. The bootstrap filter, however, keeps the distribution open wider. Even using the onerun pdfs, the EKF and SDREF pdf shapes appear to be similar. To determine the "true" pdf functions for the EKF and SDREF, the filters were run in a 100run Monte Carlo experiment. Filter state estimates were saved at the same time increments. The samples from each of the runs were used to generate the pdf curves in Figures 10 through 11. The same histogram method described in the previous paragraph for the bootstrap filter was used to plot the curves. (The bootstrap filter did not need to be examined in this fashion, as its formulation is inherently Monte Carlo.) The largest differences can be seen in Figure 11. In the later time samples (5 and 9 seconds), the previously tight distributions have flattened out, while the wide distributions have tightened up. This is to be expected, as the onerun case is only one realization of the Monte Carlo runs. Summarizing the above analysis shows that the EKF and SDREF pdf's do not resemble the "true" pdf. However, they are similar to each other. This may be explained by the fact that the EKF linearization method provides an accurate filtering methodology as long as the filter state estimate is close to that of the "true" state. If a target maneuver is introduced that causes the state estimate to be far from the "truth", then it is assumed 0 2798 89740 897 0 89780 89800 X Position (m) 89820 89840 89860 X Position (m) 12220 12240 12260 X Position (m) Figure 10. Monte Carlo posterior probability density functions at time = 1, 5, and 9 seconds for X position. True Sale BoosIrap EKF Time = 1 Second SDRE ,4 ' 0.05 0.04 0.03 0.02 0.01 0.00 0.35 0.30 Trne State oottp Time= 1 Second SORE 0.25 020 0.15 0.10 0.05  0.00 270 260 Y Position (m) 0.5 True Slate 0.4 Time = 5 Seconds Sa 03 S0.2 0.1 0 0 .. . 425 420 415 410 40 Y Position (m) Bootsrap EKF Time = 9 Seconds SDREF True State 0 175.0 174.0 173.0 1720 Y Position (m) 171,0 170.0 Figure 11. Monte Carlo posterior probability density functions at time = 1, 5, and 9 seconds for Y position. 60 that the distributions between the EKF and SDREF differ due to the different theories of linearization versus parameterization. The following paragraphs explore this scenario. The filters were once again run in a MonteCarlo simulation. However, this time the target performs a thrust cutoff maneuver, 2 seconds after launch, which causes the target acceleration, in all axes, to drop to zero for the rest of the flight. This is representative of timing the engagement too close to booster burnout. The filters are still operating under the assumption of a GaussMarkov target acceleration model, even though no target acceleration is present after 2 seconds. The same initial conditions listed in Table 1 were repeated. Figures 12 through 14 show the divergence from the "true" accelerationstate for the filters. For previously discussed reasons, Figure 14 shows the bootstrap filter performs poorly when tracking the acceleration states. The deviation from the "true" state is significant after the maneuver occurs. This is a good test to determine the differences between linearization and parameterization. Once again the Monte Carlo pdf functions were plotted in Figures 15 through 16 Samples were taken at 2, 2.5, and 3 seconds into the engagement. This allows the pdf to be examined before and during the most stressful time of the maneuver. Suprisingly, the EKF and SDREF pdf shapes do not differ significantly. It can be seen in Figure15 that although the "true" probability has become bimodal, the shapes are still similar. While the singlerun EKF and SDREF distributions can not be bimodal, Figures 15 through 16 represent Monte Carlo results. The results from the pdf comparison experiments indicate that there is no appreciable difference in pdf shapes between the EKF and SDREF. The conclusion may 61  I I Figure 12. EKF estimated and true acceleration. Figure 13. SDREF estimated and true acceleration. Figure 14. Bootstrap estimated and true acceleration. 0 0997 79920 799 0 74900 75000 75100 X Position (m) 75200 75300 70100 70200 X Position (m) Figure 15. Monte Carlo posterior probability density functions at time = 2, 2.5, and 3 seconds for thrust cutoff X position. True Sta Time = 2 Seconds / :' .i  0.030 0.025 0.020 N 0.015 0.010 0.005 0U000 I8ZU 79 /840 79860 79880 X Position (m) Time= 2.5 Seconds  ooltrap True Bate. E \ 74800  0.35 0.30 0.25 0.20 , 0.15 oto 010  005 0.00 390 Tue Time = 2 Seconds  ootstrap EKF REF Y Position (m) 405 400 Y Position (m) 0.40 Bootstrap 0.35 Time =3 Seconds EKF SDREF 0.30 0.25 True State 0.20 S0.15 0.10 / 0.05 0.00  404 402 400 398 396 394 Y Position (m) Figure 16. Monte Carlo posterior probability density functions at time = 2, 2.5, and 3 seconds for thrust cutoff Y position. be drawn that the two filters operate with identical statistical models although the filters' method of handling and operating on the stochastic processes are significantly different 6.3 Sensitivity Analysis The following section examines the SDREF sensitivity to measurement noise, initialization error, and the robustness properties in the presence of stressing target maneuvers. Only the EKF and SDREF are used for this area of the study. Further bootstrap filter development is needed to make it useful in less than ideal conditions. The EKF and SDREF were tuned separately to give them the best chance of performing well under these varying conditions. Therefore each filter may have been started with different initial conditions and statistical information. The following studies are accomplished using perfect state information to guide the vehicle, while the EKF and SDREF run open loop. Allowing the guidance algorithms access to perfect state information removes the closedloop guidance algorithm effects from the analysis of filter performance. 6.3.1 Initialization Error To test the response to initialization errors, the filters were given erroneous initial state estimate errors of 1 and 5 percent. The EKF initial covariance was set to approximate this error. The SDREF, however, has no direct means of initializing the covariance matrix as it is solving for the covariance matrix at each filter update time. This will be shown to be a detriment for this particular implementation of the SDREF. Table 2 gives the initial filter conditions for the EKF and 1 percent initial error. Measurement noise was held constant for both cases, as was the measurement noise matrix given to both filters. Range measurement noise is again range dependent and is 65 given as a percentage of "true" range. The initial setup for the SDREF used the same parameters as those of the EKF except that higher values of process noise were tried to increase the covariance response. The only tuning parameters in the SDREF are Qk, R, and the nonlinear parameterization methodology. The measurement noise matrix,R, is not directly related to the initialization error; therefore, only the other two tuning parameters can be used. Because this implementation of the SDREF has linear dynamic equations the only method available for tuning is to adjust Q,,. This is an unacceptable way of tuning the filter for initialization errors as it causes the covariance estimate to remain artificially high for the entire flight. The SDREF as previously derived, does not permit any initialization of the covariance matrix, as it is simply calculated from the solution of the SDRE. This deficiency is examined later in this section. Table 2. EKF and SDREF parameters onepercent initialization error Parameters (units) X Y Z Aspect Angle (degrees) 30 30 30 SIGP (m) 1000 35 30 SIGVP (m/s) 100 3.4 2.2 o72 (m2) 1.0E6 1.0E3 1.0E3 SIGXHAT 0,2 (m2 /s2) 1.0E4 50.0 50.0 oA2 (m2/s4) 1.0E1 1.0E1 1.0E1 R (m) 1.0E5 10.0 10.0 XHAT V (m/s) 1.0E4 335.9 223.9 A (m/s2) 55.2 57.4 38.3 SIGQ (EKF) (m2 / s) 196.0 196.0 196.0 SIGQ (SDREF) (m2 /s4) 1.0E6 1.0E3 1.0E3 Meas Noise (%, im, tm) 0.1 50.0 50.0 Figures 17 and 18 show the positionstate response of the EKF and SDREF to a 1 percent initial error. (Note the oscillations in the SDREF estimate during the initial settling portion of the flight.) Velocity and acceleration trends were similar. The filters were then run with a 5percent initial error. (The initialization parameters for the EKF and SDREF are given in Table 3.) Because of the type of engagement and large distances and closing velocities involved, a fixedpercentage type of initial error induces large disturbances in the xdirection. Therefore, little attention is given to the performance in this direction, as these magnitudes of error are not likely to occur. Figures 19 through 20 show the performance for a 5percent initial error. The inability to properly adjust the initial covariance matrix in the SDREF causes a large oscillatory error in the early portion of the estimation problem. This error was unacceptable and a new initialization procedure was investigated. The basic problem with the initialization of the SDREF is that the initial covariance matrix is computed using the solution of the algebraic Ricatti equation. This indicates that no large errors are expected. Therefore, there is no method of supplying the SDREF with an initial covariance error of the magnitude of the expected initialization errors. Because of this the SDREF computes a smaller magnitude covariance matrix and therefore a smaller gain to be used in the state update. This is why the filter oscillates during the initial portion of the flight. If the SDREF could be "jumpstarted" with a high initial covariance, the majority of the initialization error could be taken out in the first time step. 67 PPL 0 2 4 6 8 10 12 F1 0 2 4 8 10 1 Time (sec) Figure 17. EKF position errors with 3ustandard deviations for 1percent initialization error. 2. 1000 0 2 4 8 10 0 2 4 6 0 10 4o0 20 0 2 4 8 Time (sec) Figure 18. SDREF position errors with 3astandard deviations for 1percent initialization error.  ~YU INU  111~1 II 10 E400 01s 1)0 so S o g 0 150 .60 40 S so 100 F o 0 4 5 a 10 04 6 0 Time (sec) Figure 20. SDREF position errors with 13standard deviations for 5percent initialization error. I I I I ~ uD E 'm m 0 0 I 0 12 Io o 4 e S 10 0 0 0 10 12 Time (sec) Figure 19. EKF position errors with 3cstandard deviations for 5percent initialization error. ^ Table 3. EKF and SDREF parameters fivepercent initialization error Parameters (units) X Y Z Aspect Angle (degrees) 30 30 30 SIGP (m) 5.0E3 168.0 112.0 SIGVP (m/s) 5.0E2 16.8 11.2 C2 (m2) 25.0E6 3.0E4 3.0E4 SIGXHAT o2 (m2 / 2) 25.0E4 3.0E2 3.0E2 CA2 (m2 / 4) 1.OEl 1.LEl 1.OEI R (m) 1.0E5 10.0 10.0 XHAT V (m/s) 1.0E4 335.9 223.9 A (m/s2) 55.2 57.4 38.2 SIGQ (EKF) (m2/s4) 196.0 196.0 196.0 SIGQ (SDREF) (m2 / s4) 1.0E6 1.0E3 1.0E3 Meas Noise (%,am,Aum) 0.1 50.0 50.0 If the gain calculation used in both the EKF and SDREF is examined, K= = p (_)Hk[H p, (_)H + R,] it can be seen that the covariance used at time step zero, Pk (), is needed in the calculation. If it is assumed that the value of Pk that the SDREF computes is Pk(+) (i.e., after the updated measurement has occurred), then the SDREF is initialized with P, () to be used at time step zero. Using this methodology, the SDREF was initialized with the same covariance as the EKF and the SDREF process noise was also reduced to the same level as that of the EKF. This change in methodology had positive impacts on the performance of the SDREF when confronted with initialization errors. Figures 21 through 26 show the new performance of the SDREF using the same initialization errors as in the baseline runs. It can be seen that the initial oscillatory motion in the state estimate errors has been corrected and the filter compares favorably with that of the EKF. "I I i Time (sec) Figure 21. SDREF position errors with 3astandard deviations after new initialization method, 1percent error. I I Time (sec) Figure 22. SDREF velocity errors with 3oastandard deviations after new initialization method, 1percent error. Time (sec) Figure 23. SDREF acceleration errors with 3astandard deviations after new initialization method, 1percent error. Time (sec) Figure 24. SDREF position errors with 3cstandard deviations after new initialization method, 5percent error. I: I It o I I Time (sec) Figure 25. SDREF velocity errors with 3astandard deviations after new initialization method, 5percent error. j ^I I ITime (s )  I ... m I Jc Time (sec) Figure 26. SDREF acceleration errors with 3ostandard deviations after new initialization method, 5percent error. i This new initialization methodology not only improves the performance in the presence of initial errors, but also allows a convenient way of initializing the SDREF. 6.3.2 Measurement Errors This section investigates the performance of the EKF and SDREF when subjected to various degrees of measurement noise. The relative performance of the two filters using the baseline measurement noise was shown in Figures 3 through 5. Using the same baseline initial conditions given in Table 1, two other levels of measurement noise were added (Tables 4 and 5). Table 4. EKF and SDREF measurement noise parameters level 1 Parameters X Y Z Measurement Noise (%,rad,rad) 1.0 100.0E6 100.0E6 Filter Noise Variance (.01* Range)2 1.0E8 1.0E8 Table 5. EKF and SDREF measurement noise parameters level 2 Parameters X Y Z Measurement Noise (%,rad, rad) 10.0 500.0E6 500.0E6 Filter Noise Variance (0.1 Range)2 2.5E7 2.5E7 The second level of noise is large for a typical engagement; however, it exercises the robustness of each filter. Both filters were given measurement noise covariance matrices that matched the noise characteristics given to the sensor. The position state estimate plots are given in Figures 27 through 30. There is not a significant difference in filter performance using the large noises. As expected, the estimates for both filters are noisier and more erratic. The acceleration state estimate has become even worse under these conditions. This estimate was initially poor, and with noisy measurements the filters do not have enough information to estimate the target acceleration to the desired degree of accuracy. It can also be noted that the SDREF estimation error under the second level of noise seems to be worse than the EKF estimate. While these figures only represent one run, the SDREF seems capable of adequate performance even under stressing noise conditions. 6.3.3 Target Maneuvers This section investigates the openloop performance of the EKF and SDREF against two stressing target maneuvers. The capabilities of the filters against a thrust cut off maneuver have been previously discussed. In this section two other nonlinear maneuvers were tested. The first is a 10degrees per second rotating target maneuver. This maneuver allows the target acceleration vector to rotate approximately 110 degrees during the engagement. This is a moderately high target maneuver, but it is possible for an ICBM to perform this type of action in an active countermeasure scenario. The second stressing engagement consists of a dogleg maneuver in which the target acceleration vector rotates 90 degrees out of plane and then reverses itself to rotate back inward 90 2OO 300 soo 10 x 100 BOO 30 20 3O 10 i o N 0 30 Time (sec) Figure 27. EKF position errors with 3oastandard deviations 1percent error in X, 100 microradian error in Y and Z. Time (sec) Figure 28. SDREF position errors with 3ostandard deviations 1percent error in X, 100 microradian error in Y and Z. I ?A..S I I t0 0 iiiir i  sQ    . mo=^^ E m ;   a 2  *l . S. .ao ,'  x ) : == =  o   .80   .100 *_ ' . ............ .. 0 2 10 1 so 2w 015 ,0 2 ^     4 S 1 Time (sec) Figure 29. EKF position errors with +3" standard deviations 10percent error in X, 500 microradian error in Y and Z. 0 2 4 6 6 10 1o Time (sec) Figure 30. SDREF position errors with 3astandard deviations 10percent error in X, 500 microradian error in Y and Z. E 1 IcT I 1 a ra 12 76 degrees. The initial conditions used for this analysis were the same as the baseline scenario except for the process noise characteristics given to the filters (see Table 6). Table 6. Maneuvering target parameters Parameters (units) X Y Z Aspect Angle (degrees) 30 30 30 SIGP (m) 0 0 0 SIGVP (m/s) 0 0 0 o2 (m2) 1.0E5 1.0E5 1.0E5 SIGXHAT o2 (m2 / 2) 1.0E3 1.0E3 1.0E3 Cr2 (m2 / s4) 1.0E2 1.0E2 1.0E2 R (m) 1.0E5 10.0 10.0 XHAT V (m / s) 1.0E4 335.9 223.9 A (m/s2) 55.2 57.4 38.2 SIGQ (EKF) (m2 /s4) 866.0 866.0 866.0 SIGQ (SDREF) ( / s4) 1.0E3 1.0E3 1E3.0 Meas Noise (%,nm, rn) 0.1 50.0 50.0 The process noise levels were tuned to allow the filters to open up and "accept" the unexpected target maneuvers. While this allows for better tracking, it does lead to a noisier state estimate in both filters. Figures 31 through 34 show the relative performance of each filter against these two scenarios. As can be seen from these figures, there are no appreciable differences in the state estimates between the two filters. They each track this maneuverable target, however the state estimate error is noisier and larger, as expected. It should be noted that once the initial transient terms in the EKF covariance have settled, they are similar in shape and value to the "steadystate" covariance terms of the SDREF. 45 ..8 0 12 0 2 4 6 S 10 02 Time (sec) Figure 31. EKF position errors with 3astandard deviations for 10 deg/sec rotating target maneuver.  20 0 2 4 0 6 10 ... .. _ 0 4 6  Time (sec) Figure 32. SDREF position errors with 3ostandard deviations for 10 deg/sec rotating target maneuver. .10 4 10 12____________ B8 10 lo lo 0 0 4 4 8 10 12 o 04 6 l 0 Time (sec) Figure 33. EKF position errors with 30 standard deviations for dogleg target maneuver. 8 o . Ii K ;. / , Avv '. A .1 Time (sec) Figure 34. SDREF position errors with 3+o standard deviations for dogleg target maneuver. IL1 w~ ~nu iraaoa I to 79 6.4 SDREF Closed Loop Performance 6.4.1 Other Engagement Aspect Angles Up to this point the EKF and SDREF performance in an openloop environment have been examined. The following analysis examines the closedloop performance of each filter against various engagement scenarios. This includes a variety of engagement aspect angles, target maneuvers, initial condition errors, and measurement noise levels. In this section, however, performance in terms of probability of hit and percentage of fuel used to achieve the hit are measured. Measuring the percentage of fuel used, allows a comparison of how noisy the state estimates are, as increased error in the beginning of the engagement results in fuel being wasted performing unnecessary maneuvers. This analysis was performed using 31 Monte Carlo run statistics. The first analysis focused on three engagement scenarios. Up to this point a 30,30,30degree closing aspect angle was used. A headon, 90degree beam shot, and a tailchase engagement are now added. Figure 35 shows the three engagements and the corresponding simulation parameters are given in Tables 7 through 9. Y (0,0,0) Aspect Angle Head On (0,0,0) Aspect Angle Head On 13* (0,0,90) Aspect Angle Beam Shot Y (0,150,0) Aspect Angle Tail Chase Figure 35. Aspect angle description. x r t Table 7. Headon simulation parameters Parameters (units) X Y Z Aspect Angle (degrees) 30 30 30 SIGP (m) 0 0 0 SIGVP (m/s) 0 0 0 r2 (m2) 1.0E5 1.0E5 1.0E5 SIGXHAT O2 (m2 / s2) 1.0E3 1.0E3 1.0E3 oaA (m2/s4) 1.0E2 1.0E2 1.0E2 R (m) 1.0E5 10.0 10.0 XHAT V (m/s) 10000.0 335.9 223.9 A (m/s2) 55.2 57.4 38.2 SIGQ (EKF) (m2 / s4) 866.0 866.0 866.0 SIGQ (SDREF) (m2 /s4) 1.0E3 1.0E3 1.0E3 Meas Noise (%,/pm, mn) 0.1 50.0 50.0 Table 8. 90degree beam shot simulation parameters Parameters (units) X Y Z Aspect Angle (degrees) 0 0 90 SIGP (m) 0 0 0 SIGVP (m/s) 0 0 0 62 (m2) 1.0E5 1.0E5 1.0E5 SIGXHAT O,2 (m /s2) 1.0E3 1.0E3 1.0E3 ___ (m /s4) 1.0E2 1.0E2 1.E2 R (m) 1.0E5 10.0 10.0 XHAT V (m/s) 1.0E4 498.2 0.0 A (m/s2) 0.0 88.3 0.0 SIGQ (EKF) (m2/s4) 196.0 196.0 196.0 SIGQ (SDREF) (m2 /4) 196.0 196.0 196.0 Meas Noise (%,mn, Um) 0.1 50.0 50.0 82 Table 9. Tailchase simulation parameters Parameters (units) X Y Z Aspect Angle (degrees) 0 150 0 SIGP (m) 0 0 0 SIGVP (m/s) 0 0 0 a,2 (m2) 1.0E5 1.0E5 1.0E5 SIGXHAT O,2 (m / s2) 1.0E3 1.0E3 1.0E3 o2 (m2 / s) 1.0E2 1.0E2 1.0E2 R (m) 1.0E5 10.0 10.0 XHAT V (m/s) 1.0E4 0.0 237.9 A (m/s2) 76.5 0.0 44.1 SIGQ (EKF) (m2 /s4) 196.0 196.0 196.0 SIGQ (SDREF) (mi / s4) 196.0 196.0 196.0 Meas Noise (%,/Um, pm) 0.1 50.0 50.0 The 90degree beam shot is the most stressing of these engagements as the entire target acceleration profile is out of plane. The headon case is one of the easier engagements when using a range measurement, and the tail chase lies inbetween these two in terms of estimation difficulty. Table 10 displays the results from the Monte Carlo analysis against these engagements. Probability of hit (PHIT) is the probability that the interceptor hit the target. Circular error probable (CEP) is the probability that 50percent of the misses lie within this bound, 95percent miss includes ninetyfive percent of the misses, and worst miss displays the worst of the Monte Carlo runs. The mass usage numbers reflect the same quantities except in terms of percent of divert fuel used. The vehicle has a mass of 4 kilograms, 1.5 of which is devoted to divert fuel. When this limit is exceeded, the interceptor will continue to fly towards the target with no further guidance corrections, typically resulting in large miss distances. Since these engagements had good initialstate estimates and the target is not maneuvering, all of the runs were hits; however as shown in the following sections that this is not always the case. Table 10. Miss distance performance for varying scenarios Parameter (units) 00 90 1500 EKF SDREF EKF SDREF EKF SDREF PHIT 1.0 1.0 1.0 1.0 1.0 1.0 CEP (m) 0.185 0.262 0.432 0.625 0.315 0.265 Mean Miss (m) 0.213 0.272 0.546 0.667 0.347 0.354 95% Miss (m) 0.364 0.426 1.212 1.209 0.654 0.671 Worst Miss (m) 0.421 0.599 1.828 1.679 0.803 0.895 Mass CEP (%) 16.70 27.61 67.38 72.67 43.42 48.06 Mass Mean (%) 16.90 28.27 66.98 72.76 43.11 48.40 Mass 95% (%) 21.03 32.78 69.25 75.22 46.10 51.62 Mass Worst (%) 22.70 33.16 71.20 75.51 48.57 52.59 From Table 10, no significant difference in the closedloop performance of these two filters under the above scenarios is shown. All performed adequately and ensured a hit against the target in all cases. Fuel usage was also not a decisive factor as both filters differed by only a few percentage points. 6.4.2 Stressing Target Maneuvers To further test filter performance, two other target maneuvers were considered. These maneuvers are on the outer edge of filter capability of properly tracking the "true" state. They were included to investigate whether the parameterization of the SDREF allows an advantage over the linearization of the EKF in highly nonlinear environments. The first maneuver is a rotating target, however in this engagement the engagement aspect angle was a 90degree beam shot and the rotation rate was increased to 20 degrees per second. Therefore, the target will have completed an approximately 200degree turn during the length of the engagement. The second maneuver consisted of a dogleg evasive maneuver. This maneuver is similar to the previously described dogleg maneuver except that it was performed in the 90degreebeamshot scenario, causing it to become more difficult for the filter to estimate. Table 11 displays the results of Monte Carlo runs. The process noise levels for each filter were tuned to give the best performance against these scenarios. Table 11. Miss distance performance for stressing target maneuvers Parameters (units) 200/Sec Dogleg EKF SDREF EKF SDREF PHIT 0.935 0.806 0.871 0.742 CEP (m) 1.61 2.20 1.14 2.85 Mean Miss (m) 1.83 200.20 1.50 3.05 95% Miss (m) 4.30 7.44 3.94 6.95 Worst Miss (m) 6.00 6121.89 5.59 7.36 Mass CEP (%) 100 100 100 100 Mass Mean (%) 99.94 99.99 99.87 99.98 Mass 95% (%) 100 100 100 100 Mass Worst (%) 100 100 100 100 As shown in Table 11, the SDREF performed slightly worse against the targets than the EKF. The one large miss distance for the SDREF was caused by poor tracking, which caused the interceptor to run out of fuel early and coast to a large miss distance. The purpose of this section was to stress the filters to their extreme operational limits and investigate the behavior. These types of target maneuvers would be considered 85 unlikely. However, they do provide a convenient way to determine the robustness of each filter design. Even at these extremes, the SDREF was almost able to meet the EKF performance measures. 6.4.3 Measurement Errors This paragraph discusses filter sensitivity to various levels of measurement error. For this analysis the range measurement error was held constant at 1 percent due to the size of the range error when dealing with large engagement distances. Azimuth and elevation noises were included for 100, 500, and 1000 microradian errors. The results displayed in Table 12 were done using a 31run Monte Carlo experiment. Table 12. Miss distance performance for varying measurement noise Parameters (units) 00rad 500rad 1000rad EKF SDREF EKF SDREF EKF SDREF PHIT 1.0 1.0 0.968 1.0 0.935 0.806 CEP (m) 0.455 0.460 1.39 1.41 3.09 2.22 Mean Miss (m) 0.468 0.568 1.47 1.45 3.17 3.14 95% Miss (m) 0.775 1.218 2.47 2.60 5.66 6.29 Worst Miss (m) 0.935 1.569 2.55 3.08 6.03 8.18 Mass CEP (%) 77.13 76.86 84.19 81.83 89.79 86.99 Mass Mean (%) 77.57 76.96 84.66 82.36 90.23 87.05 Mass 95% (%) 80.67 78.95 88.27 85.12 94.46 89.40 Mass Worst (%) 81.23 79.55 90.92 86.31 96.66 92.67 Once again, the results between the two filters are similar. Even with highmeasurement noise levels, both filters performed adequately, missing only one or two times in the Monte Carlo run. The fuel usage numbers are also comparable. These results verify the results of the pdf comparison between the filters, as the probability distributions are similar between the EKF and SDREF. Therefore, it can be assumed that the filters are Monte Carlo run. The fuel usage numbers are also comparable. These results verify the results of the pdf comparison between the filters, as the probability distributions are similar between the EKF and SDREF. Therefore, it can be assumed that the filters are using approximately the same set of distribution functions, even though they are achieved in different ways. 6.4.4 Initialization Errors Using the new initialization method described in section 6.3.1, a closedloop analysis of the EKF and SDREF was performed. The initial conditions used for the open loop initialization study were used for this investigation. The filters were run closedloop in a 31run Monte Carlo simulation. Table 13 displays the results of these runs. Using this new initialization method not only shows better performance in the singlerun, open loop case, but performs equally as well in the closedloop analysis. Probability of hit, miss distance, and fuelusage numbers show that the SDREF performance is similar to that of the EKF. Table 13 Miss distance performance for varying initialization error 1% Error 5% Error Parameters (units) E SDREF EKF5% Errr PHIT 1.0 1.0 1.0 1.0 CEP (m) 0.436 0.377 0.336 0.389 Mean Miss (m) 0.464 0.380 0.423 0.432 95% Miss (m) 0.849 0.750 0.858 0.770 Worst Miss (m) 1.01 0.793 1.13 1.12 Mass CEP (%) 75.13 79.18 72.53 92.03 Mass Mean (%) 75.00 79.45 72.53 92.03 Mass 95% (%) 77.34 81.92 74.56 93.20 Mass Worst (%) 78.12 84.78 75.55 93.42 87 6.5 Summary of Exoatmospheric Guidance Problem The exoatmospheric guidance problem analysis demonstrated, with some minor caveats, the SDREF was capable of satisfying the interceptor guidance problem. The assumptions of detectability in solving the SDRE required the addition of a range measurement. An ad hoc initialization algorithm needed to be developed to properly initialize the SDREF. With these two restrictions in place, however, the SDREF performed similarly to the EKF for the exoatmospheric guidance problem for several engagements, noise sources, and initialerror combinations. CHAPTER 7 PENDULUM PROBLEM FILTER DEVELOPMENT To evaluate the effect of nonlinear dynamics on the SDREF performance, a simple twodimensional pendulum problem was constructed. The scenario consists of a simple rigid pendulum with friction about the connection point, see (Figure 36). The measurements are assumed to come from an accelerometer attached to the bob, measuring g. \0 /. Figure 36. Pendulum setup. 89 The pendulum is given some initial position and velocity and the friction term damps the oscillations. This system, while simple, demonstrates nonlinear dynamics and nonlinear measurements together in the same problem. The system dynamics for the pendulum problem are given by the 2"d order nonlinear differential equation B=gsinO BO (7.1) L where BO is the friction term. B was set to be 0.5. Both the SDREF and EKF were derived using the stochastic nonlinear system .i = f(x) +w z= h(x)+v where w and v are Gaussian, zeromean whiteprocess and measurement noise, respectively, given by E[w(t)w'(t + r)= W(t)3(r)] (7.3) E[v(t)v(t + r)= R(t)S()] If x, = x (7.4) then the state equations for the pendulum system are xi = x g (7.5) 2 = in(x,)Bx2 (7 L The measurement equation is z = sin(x1) Bx2 L 
Full Text 
xml version 1.0 encoding UTF8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID E3A4DIPYE_PKLNE8 INGEST_TIME 20130123T15:27:35Z PACKAGE AA00012992_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES 