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1 Adaptive Target State Estimation Using Neural Networks By P. K. Menon and V. Sharma Optimal Synthesis Inc. 47 San Antonio Road, Suite 2 Palo Alto, CA Abstract Development of an adaptive target state estimation algorithm for use with advanced missile guidance laws is presented. The target state estimator employs a linear neural network as the decisionmaking element in a nine-state dynamic model of the target. A Kalman filtering algorithm is used to estimate the neural network weights and the target states. The estimator performance is evaluated in a point-mass nonlinear simulation of missile-target engagement for several different engagement scenarios. This simulation incorporates error models of the seeker and the onboard inertial navigation system. Comparison of the neural network target state estimator performance with a conventional target state estimator reveals that the adaptive estimator provides more accurate estimates of the target states with minimal lag. Introduction Target state estimator is an important subsystem in advanced missile guidance systems. Target state estimators are required for two reasons. Firstly, the measurements provided by on-board seeker are often corrupted by noise, and are not in a form usable by the guidance law. Secondly, advanced guidance laws require additional information about the target such as its acceleration components, which cannot be provided by the on-board sensors. The first issue can be addressed using signal-processing approaches such as low-pass, band-pass and high-pass filtering. However, if the measurements are given in terms of quantities not directly usable by the guidance law, nonlinear transformations will have to be made on the measurements. In this case, the noise in the sensors will also get nonlinearly transformed, and simple linear filtering strategies can result in non-robust estimators. Robustness of target state estimators can be improved by formulating the state estimation problem using modern robust estimation theory [1]. The second issue is more difficult to resolve because it involves the validity of the hypotheses about the target behavior. For instance, in the target state estimators discussed in the literature, it is customary to assume that the target acceleration behaves as a random walk, so that the target is assumed execute short circular arcs of random duration and radius of curvature. Estimators based on this assumed behavior are then used to determine the acceleration magnitudes for use by the guidance law. Note that both the optimal guidance law and the augmented proportional navigation guidance law [2] employ this assumption in their derivations. The success of such an estimation methodology depends critically upon how close the hypothesis corresponds to reality. The approach advanced in the present research is to hypothesize the target to be a goal-oriented entity, in the sense that it has a goal it wants to reach, and the evasive maneuvers it employs are to enhance the probability of avoiding the missile. The target s goal orientation and the evasive strategy are then parameterized, and a linear neural network adaptive scheme is used for on-line estimation of the maneuver logic parameters. It will be demonstrated in this paper that the proposed approach will result in a significant improvement in the target acceleration estimates. These two ideas form the central core of the present paper. Target State Estimator Architecture Very low miss distances [3, 4] required in next generation missile systems for defense against tactical ballistic missiles and surface skimming missiles will require the use of advanced guidance laws. While all homing missile guidance systems require target state information in order to achieve target interception, the accuracy of modern homing guidance schemes critically depend on the accuracy of the target state estimates. Classical homing guidance laws such as proportional navigation [2] require only the line-of-sight rate for interception. On the other hand, advanced guidance laws such as augmented proportional navigation [2, 5, 6] and differential game based guidance laws [7, 8] cannot function without accurate target acceleration, velocity and position state estimates. Previous studies have shown that classical guidance laws can out-perform modern guidance laws [9], if the target state estimates are erroneous. Due to the fact that homing guidance is a finite-time task, any information available about the target maneuvers can be used to develop superior guidance tactics. Guidance laws based on differential game theory [7, 8] and reachable sets [1] can use such information to synthesize guidance strategies that can defeat the target maneuvers. Thus, it is critical that accurate and detailed information about the target states be available to the guidance law. The information generally available on-board a missile are the missile body-axis referenced line-of-sight angles and rates from the seeker, missile body acceleration components from accelerometers, and body rate components from rate gyros. For the present research, it will be assumed that the missile incorporates an inertial navigation system and a seeker providing line-of-sight angles and rates, together with range and range rate to the target. Each of the on-board sensor outputs is corrupted by noise. Since the missile incorporates an inertial navigation system (INS), it can be assumed that the noise sources from the rate gyros and the accelerometers have already been accounted for in the INS algorithm. The remaining noise sources to be included in the target state estimation process are primarily from the seeker. The target state estimator uses the data available from the seeker and the inertial navigation system, together with a model of the target behavior to generate the target state vector, which typically includes target position, velocity and acceleration vectors. In order to accomplish this task, the target state estimator relies on a hypothesized model of the target. Typical hypotheses used in the missile industry include nonmaneuvering and circular target maneuver models. Nonmaneuvering target model assumes that the mean value of target acceleration is zero. The entire

2 2 observed behavior is then assumed to be the result of random target acceleration. On the other hand, the circular target maneuver model assumes that the target executes short maneuvers of constant radius of curvature, with the radius of curvature changing as a random walk. Thus, the target jerk components are considered as the integral of zero-mean, Gaussian, white noise. Kalman filtering theory [11, 12] is then used to derive the state estimator. The model used to derive the target state estimator has a major impact on the structure of the estimator and the quality of the state estimates. Linearized models are often used in the derivation. The model nonlinearities are generally included as process noise components. The process noise terms are also expected to capture all the unknown features of the assumed target model, including its maneuver strategy. The general approach for target state estimator is to use as simple a model as possible, leaving the Kalman filter to fix whatever information is missing. Note that in all target models discussed in the literature, the target is assumed to be maneuvering in an open-loop fashion, without any explicit goal or strategy. This assumption leads to purely reactive target state estimators, often producing sluggish state estimator response in the presence of agile targets. After the initial transient, the estimator gains become constant, making the estimator dynamics time-invariant. From this point onwards, the filter may be unable to follow aggressive target maneuvers. An approach advocated in the literature to avoid this difficulty is to periodically restart the filter to ensure rapid response. Another approach to preserving the agility of the target state estimator without sacrificing its accuracy is to assume that that the target can be modeled as consisting of a set of models and a switching logic. The switching logic is then assumed to select any one of these models at any time instant. Thus, to an observer, at each time instant, the target model appears to act like any one of these models. The resulting target state estimator consists of a bank of Kalman filters that is switched or blended using a hypothesis-testing algorithm. Methods that employ such approaches are called Interacting Multiple Model (IMM) estimation techniques [13-19]. These methods will not be discussed in this work. In the present research, the target model is assumed to consist of three chains of three integrators driven by an adaptive maneuver strategy. The target equations of motion are assumed to be of the form: &&& x = j, &&& && z& = j 1 y = j2, The variables x, y, z are the position components of the target in an inertial frame, and j 1, j 2, j 3 are the jerk components obtained from the target maneuver logic. The physical model of the target is assumed to be known, but the exact maneuver strategy is assumed to be unknown. The maneuver strategy is parameterized using a neural network and the parameters of the network are then determined on-line, together with the target states. Since the target maneuvering logic is adaptively determined, the resulting estimation scheme can be expected to have agile response to any changes in the target behavior. A block diagram of the adaptive target state estimation scheme is given in Figure 1. Present research will employ linear neural networks [2, 21] to realize the adaptive maneuver logic. These networks have found extensive applications in speech prediction, speech recognition, adaptive noise cancellation, long distance telephone echo cancellation, high accuracy phased-array tracking radars, 3 sonar signal processing and a myriad of other signal processing applications. Note that the present work represents one of the first applications of adaptive neural networks for target state estimation. Judging from the success it has had in other applications, the adaptive neural network can be expected to result in significant improvements in the target maneuver logic modeling problem. The use of nonlinear neural [39, 41] networkbased target state estimators will be future interest. Missile State Estimates Linear/Nonlinear FIR Filter Adaptive Maneuvering Logic Maneuver Logic Parameter Estimates LOS Angle, (LOS Rate, Range) Measurements Target Model Robust Target State and Parameter Estimator Target Model States Target State Estimates Fig. 1. Conceptual Structure of the Target State Estimator The function of the neural network adaptive maneuvering logic is to generate jerk inputs to the target equations of motion based on an assumed goal of the target, and the difference between missile and the target states. For present research, the target goal is assumed to be the missile launch point. While the exact form of the adaptive maneuvering logic is unimportant, the chief requirement is that it be rich enough to capture most of the expected behavior of the target. For instance, if the target is a tactical ballistic missile, the adaptive maneuvering logic must be able to generate maneuvers such as variable frequency/variable amplitude spiraling that has been observed in such missiles [38], in addition to the nominal ballistic trajectory. The maneuver model must also include the possibility that the target may act in such a way as to cause the worst case perturbations in the missile guidance law, even though it may just be on a ballistic trajectory with the missile launch point the final destination. On the other hand, if the target was a high performance aircraft or advanced cruise missile, it may employ tactics for actively evading the missile. Since the goal of the target can be approximately determined before missile launch, this information can be used in the adaptive maneuver logic. Since the target maneuver logic is unknown, the coefficients of the linear neural network have to be determined from the measurements. Thus, the target maneuver logic must be implemented using an on-line adaptive neural network [2, 39]. A single input, single output adaptive linear neural network is shown in Figure 2. The network output consists of a weighted linear combination of the current and previous n samples of the input. Note the variable z is the z-transform variable. The weights p, p 1...p n are all adjusted by an adaptation algorithm such as the LMS algorithm [2, 39]. Alternately, these weights can be included as additional states to be estimated. This latter approach is adopted for the present research. Given sufficient number of elements, the adaptive linear neural network can emulate a large class systems, including ones that contain pure time-delays. Note that the adaptive neural network includes only zeros and it can never go unstable in the classical sense. This property makes it eminently suitable for use in on-line adaptive estimation schemes. Although the exact structure of the adaptive maneuvering logic will depend upon the type of targets, three 5 stage linear

3 3 neural networks were used for the present research. The benefits in using additional neural elements will be explored during future research. The target state estimation problem including the on-line determination of the adaptive linear neural network weights can be formulated as an extended Kalman filter. In order to formulate the problem this way, difference equations for individual parameters are first defined as: Input z-1 z-2 z z-n p = pi + n p p1 p2 p3 pn Output Adaptation Adaptation Signal Algorithm Fig. 2. Adaptive Linear Neural Network The noise components n in the parameter differential equations are assumed to be zero-mean, Gaussian white noise. These difference equations are then appended to the target difference equations to form the dynamical model for the Kalman filter. In order to simplify the target state estimator formulation, the Kalman filter is formulated in terms of six pseudomeasurements. The pseudo-measurements are the three components of the target position vector with respect to the missile R x, R y, and R z, and three components of the target velocity vector with respect to the missile R x, R y and R z. The position and velocity position components are resolved along the inertial reference frame. The pseudo-measurements are constructed using the six actual measurements: range R, pitch line-of-sight angle λ y, yaw line-of-sight angle λ z, range rate R, pitch line-of-sight rate λ y and yaw line-of-sight rate λ z. An extended Kalman filter is then designed using well known approaches discussed in the literature [11]. Since these simulations involve stochastic phenomenon, comparisons will be presented using Monte-Carlo simulations. The engagement scenario considered here is that of a radially incoming target trajectory, at an altitude of 1 meters, and with a speed of Mach 2.5. Along its path, the target exhibits a random weave (sinusoidal wave) in horizontal plane, with a magnitude 5g s and a period of 4 seconds. The lateral acceleration component estimated using conventional and adaptive target state estimators are given in Figure 3. Identical process and measurement noise covariance matrices are used in both cases. It may be observed that the adaptive estimator tracks the actual acceleration much more closely than the conventional state estimator. Increased noise in the adaptive estimator output arises from the adaptation process. In order to further illustrate the benefits of using adaptive target state estimator, Monte-Carlo simulations were carried out for this engagement scenario. The results from these Monte-Carlo runs are given in Figures 4 through 7. These results indicate that the use of the adaptive filter decreases of both mean and standard deviation of the target state estimates. Y Acceleration (ft/s2) Fig. 3. Y-Component of the Target Acceleration Estimate Using Conventional and Adaptive Filters Solid Line: Adaptive Filter, Dashed Line: Actual, Dash- Dot Line: Conventional Filter In this engagement scenario, the mean and standard deviation for the miss distance for the conventional filter are 7.21 feet and 7.52 feet respectively. The mean miss distance for the adaptive filter is 3.69 feet and the standard deviation is 1.35 feet. These figures suggest that in this engagement scenario, the miss distance can be reduced by almost 5% through the use of the adaptive target state estimator. Adaptive Estimator Performance Evaluation This section will illustrate the performance of the adaptive target state estimator in one engagement scenario. In order to serve as a baseline for comparisons, the performance of the conventional estimator will also be presented. The difference between the adaptive estimator and the conventional estimator is that the latter does not incorporate a neural network maneuver subsystem. An optimal guidance law is used in every case, and the simulations are continued until closing rate becomes positive.

4 Mean Y Position Error (ft) 2 2 S.D. in Y Acceleration Error (ft/s/s) Fig. 4. Mean Error in the Estimate of the Y- Target Position Component Using Conventional and Adaptive Filters Solid: Adaptive Filter, Dash-Dot: Conventional Filter Fig. 7. Standard Deviation of the Y - Target Acceleration Component Estimation Error Using Conventional and Adaptive Filters Solid Line: Adaptive Filter, Dash-Dot Line: Conventional Filter Mean Y Velocity Error (ft/s) Fig. 5. Mean Error in the Estimate of the Y - Target Velocity Component Using Conventional and Adaptive Filters Solid : Adaptive Filter, Dash-Dot: Conventional Filter Mean Y Acceleration Error (ft/s/s) Conclusions This paper discussed the development of an adaptive target state estimator. The estimator is based on a nine-state dynamical model and three 5-tap linear neural network serving as the maneuvering logic. Kalman filter theory was used to derive the estimator. The adaptive estimator performance has been assessed in several engagement scenarios. Monte-Carlo simulations indicate that the adaptive filter can provide nearly 5% improvement in the miss distances. The following observations can be made on the basis of the present research. 1. Adaptive estimator consistently provides better state estimates than a conventional filter. The adaptive estimator is especially effective in estimating acceleration of a maneuvering target. In certain cases, however, a better performance of adaptive filter is achieved at the expense of noisier estimates. 2. Adaptive filter does not provide any improvement in estimating acceleration components of a non-maneuvering target. In fact, the estimates are generally much more noisier than those provided by the conventional filter due to the higher order of the adaptive filter. Additional improvements may be realizable using larger nonlinear neural networks for modeling the target maneuver logic, and by employing guidance laws that exploit the more detailed information about the target acceleration components provided by the adaptive maneuvering logic. These and other related issues will be of future research interest Fig. 6. Mean Error in the Estimate of the Y - Target Acceleration Using Conventional and Adaptive Filters Solid: Adaptive Filter, Dash-Dot: Conventional Filter Acknowledgement This research was supported under Navy Contract N C- 132, with Mr. John E. Bibel of NSWC as the technical monitor. References 1. Basar, T., and Bernhard, P., H -Optimal Control and Related Minimax Design Problems, Birkhauser, Boston, MA, 1991.

5 5 2. Zarchan, P., Tactical and Strategic Missile Guidance, American Institute of Aeronautics and Astronautics, Reston, VA, Ohlmeyer, E. J., Root-Mean-Square Miss Distance of Proportional Navigation Missile Against Sinusoidal Target, Journal of Guidance, Control, and Dynamics, Vol. 19, May-June 1996, pp Bibel, J. E., Malyevac, D. S., and Ohlmeyer, E. J., Robust Flight Control for Surface Launched Tactical Missiles, Naval Surface Warfare Center Dahlgren Division Technical Digest, September Bryson, A. E., and Ho, Y. C., Applied Optimal Control, Hemisphere, New York, NY, Vergez, P. L., and Liefer, R. K., Target Acceleration Modeling for Tactical Missile Guidance, Journal of Guidance, Control, and Dynamics, Vol. 7, No. 3, May-June 1984, pp Isaacs, R., Differential Games, Robert E. Krieger, New York, Anderson, G. M., A Comparison of Air-to-Air Missile Guidance Laws Based on Optimal Control and Differential Game Theory, AIAA Guidance and Control Conference, Boulder, CO, August 6-8, Zarchan, P., Proportional Navigation and Weaving Targets, Journal of Guidance, Control, and Dynamics, Vol. 18, No. 5, 1995, pp Gupta, N. K., Reachable Set Methods, in Control and Dynamic Systems, Vol. 12, C. Leondes (editor), Academic Press, New York, NY Gelb, A. (Editor), Applied Optimal Estimation, MIT Press, Cambridge, Jazwinski, A. H., Stochastic Processes and Filtering Theory, Academic Press, New York, NY, Blom, H. and Bar-Shalom, Y., The Interacting Multiple Model Algorithm for Systems with Markovian Switching Coefficients, IEEE Transactions on Automatic Control, Vol. 23, Aug. 1988, pp Watson, G. and Blair, W., Interacting Acceleration Compensation Algorithm for Tracking Maneuvering Targets, IEEE Transactions on Aerospace and Electronic Systems, Vol. 31, No. 3, Jul. 1995, pp Blom, H.A. P., An Efficient Filter for Abruptly Changing Systems, IEEE Conference on Decision and Control, Las Vegas, NV, Dec. 1984, pp Helmick, R. E., Blair, W. D. and Hoffman, S. A., One-Step Fixed-Lag Smoothers for Markovian Switching Systems, IEEE Transactions on Automatic Control, vol. 41, no. 7, July 1996, pp Hoffman, S. A. and Blair, W. D., Guidance, Tracking and Radar Resource management for self defense, Proceedings of the 1995 IEEE Conference on Decision and Control, IEEE, Piscataway, NJ, pp Watson, G. A. and Blair, W. D., Solution to second benchmark problems for tracking maneuvering targets in the presence of false alarms and ECM, Proceedings of the conference on Signal and Data Processing of Small Targets, San Diego, CA, 1995, pp Watson, G. A., and Blair, W. D., Interacting acceleration compensation algorithm for tracking maneuvering targets, IEEE National Radar Conference, Lynnfield, MA, 1993, pp Widrow, B. and Stearns, S. D., Adaptive Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, Miller, W., T., Sutton, R. S., and Werbos, P. J. (Editors), Neural Networks for Control, The MIT Press, Cambridge, MA, Hassibi, B. and Kailath, T., On Nonlinear Filters for Mixed H 2 /H Estimation, Proceedings of the American Control Conference, Albuquerque, NM, June 1997, pp Bibel, J. E., SBIR N98-71 Problem Definition, NSWC Electronic Mail, Dated October 14, Urban, T. J., RF Semi-Active Seeker Model and Cruise Missile/TBM Target Generator, Johns Hopkins University- Applied Physics Laboratory Memo, June 11, Menon, P. K., and Iragavarapu, V. R., Integrated Design of Agile Missile Guidance and Control Systems, Optimal Synthesis Report No. 8, Prepared Under NSWCDD Phase I SBIR Contract, October Menon, P. K., and Iragavarapu, V. R., Fuzzy Logic Based Integrated Guidance and Control Systems for Missiles, Optimal Synthesis Report No. 7, Prepared Under NSWCDD Phase I SBIR Contract, July Menon, P. K., and Iragavarapu, V. R., Robust Nonlinear Control Technology for High-Agility Missile Interceptors, Optimal Synthesis Report No. 5, Prepared Under NSWCDD Phase I SBIR Contract, July Ohlmeyer, E. J., Pepitone, T. R., Miller, B. L., Malyevac, D. S., Bibel, J. E., and Evans, A. G., Appliation of GPS/INS to Extended-Range Guided Munitions and Tactical Ballistic Missile Interceptors, Naval Surface Warfare Center Dahlgren Division Technical Digest, 1996, pp Cloutier, J. R., Evers, J. H., and Feeley, J. J., Assessment of Air-to-Air Missile Guidance and Control Technology, IEEE Control Systems Magazine, October 1989, pp Boberg, B., et. al., Estimation of Target Lateral Acceleration for Augmented Proportional Navigation, AIAA Conference on Guidance, Navigation and Control, Aug , 1997, New Orleans, LA, pp Papoulis, A., Probability, Random Variables and Stochastic Processes, McGraw-Hill, New York, NY, Nagpal,K.M., and Khargonekar, P.P., "Filtering and Smoothing in an H Setting," IEEE Transactions on Automatic Control, V 36, N 2, February, Shaked, U., and Theodor, Y., "Game Theory Approach to Optimal Linear Estmation in the Minimum H Norm Sense," Proceedings of the28th IEEE Conference on Decision and Control, Tampa, FL, December, 1989, pp Shaked, U., and Theodor, Y., "H Optimal Estmation: A Tutorial," Proceedings of the 31st IEEE Conference on Decision and Control, Tucson, AZ, December, 1992, pp Yao, Y. X., Darouach, M., and Schaefers, J., "A Robust H Estimator Design for Linear Uncertain System," Proceedings of the American Control Conference, Baltimore, MD, June, 1994, pp Appleby, B. D., Dowdle, J. R., and VanderVelde, W., "Robust Estimator Design Using m Synthesis, Proceedings of the 3th IEEE Conference on Decision and Control, Brighton, England, December, 1991, pp Bryson, A. E., and Ho, Y. C., Applied Optimal Control, Hemisphere, Washington, Chadwick, W. R., Reentry Flight Dynamics of a Non- Separating Tactical Ballistic Missile, Proceedings of AIAA/BMDO Interceptor Technology Conference, San Diego, CA, 1994.

6 6 39. Widrow, B., and Walach, E., Adaptive Inverse Control, Prentice Hall, Upper Saddle River, NJ, Ogata, K., Discrete Time Control Systems, Prentice-Hall, Englewood Cliffs, NJ, Anderson, J. A., and Rosenfeld, E. (Editors), Neurocomputing: Foundations of Research, The MIT Press, Cambridge, MA, 1989.