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SUMMARY OF THE THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN ELECTRICAL ENGINEERING CONTRIBUTIONS TO THE DEVELOPMENT OF THE ADVANCED ALGORITHMS FOR AIR TARGETS TRACKING Submitted by Ph. D. Student Augustin Sperila, Major, Electrical Engineer Romanian Air Force HQ Thesis under the supervision of Gheorghe Gavriloaia, Colonel (R), Professor, Electrical Engineer Military Technical Academy, Bucharest Bucharest, May 2005 Augustin Sperila – Contributions to the development of the advanced tracking algorithms __________________________________________________________________________________________ CONTENT: Chapter one Introduction Chapter two The actual knowledge on the multisensor data fusion when measurements' origin is uncertain Chapter three Original contributions Chapter four Tracking the air targets when measurements’ origin is uncertain Chapter five Multisensor data fusion Chapter six Simulation set-up and results Chapter seven Overall conclusions Chapter eight Ph. D. Student’s Resume Chapter nine The author’s list of publications KEY WORDS: maneuvering air targets, data association, avoiding tracks coalescence, BLUE data fusion. __________________________________________________________________________________________ Technical Military Academy 2 Augustin Sperila – Contributions to the development of the advanced tracking algorithms __________________________________________________________________________________________ 1. INTRODUCTION Effective track fusion against cluttered background and missed detections has been a major challenge in the target tracking community. Although there have been several algorithms proposed for performing the distributed track fusion when the measurements’ origin is unsure, no general benchmark has been given to date, which would be able to discriminate between their effectiveness against the tracking scenario and system’s parameters. But, no matter the fusion method is used, it has to deal with two major issues. Firstly, it has to prevent the local estimators losing tracks in the presence of missed detections and false measurements; and secondly, it has to account for the local estimates correlation induced by the propagation of the common noise in the cinematic model. Additionally, the problem is made more difficult by the need to search for the best trade-off between the maximum allowed target maneuverability and the system’s capability to maintain the tracks against a heavy clutter density. Supposing that the problem of maintaining the correct tracks identity at the local processors was solved in an appropriate manner, the use of the Best Linear Unbiased Estimate (BLUE) fusion rule appears to be very promising for performing tracks fusion but meanwhile, one should have made provisions for allowing the maximum possible targets maneuverability. The aim of this work is to investigate the effectiveness of a simplified approach in employing the BLUE fusion rule which is avoiding the explicit dependence of the cross-correlation between the local estimates on the history of detections for each track. This approach will be tested together with an altered version of the Jonker-Volgenant-Castanon (JVC) assignment procedure for avoiding tracks coalescence, optimized at the expense of a small reduction in target maneuverability during tracks interaction. The alteration will adapt the JVC procedure to the Probabilistic Data Association Filter (PDAF), in order to balance the need for both targets maneuverability and false alarms handling. __________________________________________________________________________________________ Technical Military Academy 3 Augustin Sperila – Contributions to the development of the advanced tracking algorithms __________________________________________________________________________________________ The herewith IMM-PDAF(altered)-JVC algorithm proposed for avoiding tracks coalescence for maneuvering targets will be validated by extensive simulation trials, on a program built under the MATLAB environment. Then, an assessment of the improvement in the precision of the estimate will demonstrate the power of the BLUE fusion rule when it is employed following the equivalent approach described in chapter five. 2. THE ACTUAL KNOWLEDGE ON THE MULTISENSOR DATA FUSION WHEN MEASUREMENTS' ORIGIN IS UNCERTAIN During the past two decades, among target tracking community there was much work done for refining the known fusion rules, by that making them to effective in a stressful environment. The stressful environment would, at minimum, comprise strong interference between trajectories when those ones cross each other, moderate-to-high miss-detection and false alarms probabilities and, highly maneuvering-capable targets. To the knowledge of the author, none of the works performed to date addressed all the listed issues in a coherent manner. However, by employing the Chong's fusion rule in conjunction with the Joint Probabilistic Data Association Filter (JPDAF), the resulting algorithm should have the capability to appropriately deal with al the listed issues. In one of the cited papers, a sound mathematical development offers the equations for what is called there the "distributed JPDAF algorithm", but the analysis is confined to the performances of the algorithm in improving the estimation precision on straight trajectories, with a small plant noise covariance. Moreover, despite the heavy processing burden, no indications were given about the rate of lost tracks, and the fusion with incomplete communication rate is approximated only for small noise covariance. On the other hand, there is another newer (1999), more powerful data fusion rule than the Chong’s one, namely the Best __________________________________________________________________________________________ Technical Military Academy 4 Augustin Sperila – Contributions to the development of the advanced tracking algorithms __________________________________________________________________________________________ Linear Unbiased Estimate (BLUE) rule, whose optimality criterion is more restrictive. Indeed, the BLUE fusion rule is using weighting coefficients optimized so that at any time, the Hessian of the estimation error covariance is minimized (for the Chong’s fusion rule, only the trace of the error covariance is imposed to yield the minimum). Starting from its strength, it would be highly desirable to implement the BLUE fusion rule in conjunction with a dedicated algorithm for managing the incertitude concerning the measurements’ origin. The problem which arises here is that for iteratively computing the local estimates’ weighting coefficients, one should have the possibility to rigorously account for the cross-correlation induced in the local estimation errors by the propagation of the common plant noise. But, for any known approach able to deal with the measurements’ origin uncertainty, the cross-covariance of the local estimation errors will become explicitly dependant on the local processors measurements’ validation configuration. This problem is the main drawback in any attempt to perform data fusion on the background of missed detections and false alarms. In a reference paper, that difficulty was overcome by using the Chong’s fusion rule, which is the only one fusion rule which has not to explicitly account for the local estimates’ cross-correlation, but this was done at the expense of using the very complex and computationally demanding JPDAF algorithm. As far as it was widely recognized that the memory requirements for JPDA grow exponentially with both the number of targets in track and false alarm density, its ability to correctly solve the problem of correct data association for interacting targets would make its employment as basic tracking algorithm for BLUE fusion at least questionable in what concerns the worthiness. The reasons which make it unworthy are twofold: on one hand, the previously addressed JPDAF issues will raise too much the local processing complexity, with no guarantee that the (interacting) tracks lost will be completely eliminated and, on the other hand, by using JPDAF the cross- covariance of the local estimates will be over-expanded, asking for the transmission of too much data from the local processors to the fusion center. One possible mid-way solution will be to perform BLUE fusion having the less- __________________________________________________________________________________________ Technical Military Academy 5 Augustin Sperila – Contributions to the development of the advanced tracking algorithms __________________________________________________________________________________________ complex PDAF (the earlier, single-target version of JPDAF) algorithm as the basic local estimator, but in this case the problem of lost tracks caused by trajectories’ interaction will have to be addressed separately. A (partial) approach for dealing with this problem was given in one of the reffered papaers, namely the JVC algorithm. It is a global nearest neighbor optimization technique which is used for avoiding tracks to be lost when they interact, but by taking into account only the missed detections, not also the false alarms. Along this attempt, the author strained to investigate the possibility to adequately adapt the JVC algorithm in order to handle both missed detections and false alarms, on the expense of a small reduction in the targets allowed maneuverability during tracks interaction. The resulting PDAF(altered)-JVC algorithm will then be used as a basic local estimator for performing the BLUE data fusion. 3. ORIGINAL CONTRIBUTIONS The main scope of this work is to adapt the JVC algorithm in order to handle both missed detections and false alarms in conjunction with PDAF, meanwhile making provisions for allowing to the maximum possible extent the targets maneuverability. This leads to a variant of the what could be called the “IMM-PDAF(altered)-JVC” algorithm. This algorithm will be used as a basic estimator for the BLUE fuser, in order to avoid tracks coalescence when they cross each other and meanwhile allowing for the maximum targets maneuverability. Further, an efficient approach to compute the cross-covariance of the local estimation errors of the BLUE fuser is proposed. This approach eliminates the dependence of the cross- covariance on the local measurements’ validation configurations, by using an equivalent Kalman filter to perform the transition between the actual predicted and updated local estimation errors. __________________________________________________________________________________________ Technical Military Academy 6 Augustin Sperila – Contributions to the development of the advanced tracking algorithms __________________________________________________________________________________________ The equivalence is established via the Information Filter equations, by considering the information reduction at the local processors due to the measurements’ origin uncertainty is actually caused by the update made with a less-precise sensor, when the measurements origin is not questionable. The effectiveness of the resulting BLUE/IMM-PDAF-JVC algorithm will be extensively trialed by simulation, in order to define both its capability to avoid the tracks coalescence and to improve the fused estimate’s precision, allowing for important conclusions about its worthiness for real life applications. 4. TRACKING THE AIR TARGETS WHEN THE MEASUREMENTS ORIGIN IS UNCERTAIN For managing missed detections, clutter, and maneuvering targets the employment of the IMM-PDAF(altered)-JVC instead of IMM-NN-JVC is needed, because it allows both tracking maneuvering targets against heavy clutter and making the JVC technique effective by its very precise estimates. IMM-PDAF is a combination between the IMM and PDAF algorithms, but it fails to track closely spaced targets. That’s why the JVC algorithm is used to yield a unique optimal pairing between the tracks and the measurements, based on a cost function. The cost function weights are the probabilities that the normalized innovations squared exceed the actual values: cij P 2 Dij 1 P 2 Dij (1) The assignment becomes a linear optimization problem, where all feasible track-to-measurement pairs for the current validation configuration are first mapped, than added together into the cost function, via weighting coefficients. Thgere are sought the variables x ij so that: n m n m x ij arg min x ijcij x ij 1 ; x ij 1 ; 0 xij 1 i, j (2) i 1 j1 i i i 1, n ; j1, m __________________________________________________________________________________________ Technical Military Academy 7 Augustin Sperila – Contributions to the development of the advanced tracking algorithms __________________________________________________________________________________________ IMM-PDAF replaces the Kalman filters in the IMM module with PDAF filters. The difference is that the modes probabilities are formed using the PDA “likelihood” function: j k p Zk / M j k , Zk 1 PD m k (3) Vk mk 1 PD Vk1 m k N z t k ; Hx jk / k 1 , Sk ˆ m k t 1 At a first glance, the use of the IMM-JVC assignment procedure in the PDAF context seems to be trivial. Indeed, the JVC algorithm would assign the common validated measurements to the tracks such as, all but the measurement which do not belong according to JVC decision to the track under consideration will be used for updating the track by the IMM-PDAF algorithm. By trials performed using the JVC-driven assignment, a quite high track lost rate was obtained. The main reason for that was the bias introduced by the wrong assignment decided in the JVC algorithm when the number of the common validated measurements was equal to one, more important at low clutter densities. In order to eliminate the possibility the tracks being misled by incorrect decisions made in JVC when there was only one commonly validated measurement, in those situations the measurements were discarded for both tracks in the current update iteration. If there were any other measurements validated for the two tracks, they were used for the current update. If the common validated measurement was the only measurement validated for any of the two tracks, that estimate was predicted for the track, but the measurement would update the associated covariance, in order to avoid an artificial increase in the size of the following validation gate. At high clutter densities, the combined effect of the clutter drift and tracks interaction bias was an important generator of tracks lost. That effect was minimized by shrinking the size of the validation gates, by computing the “innovation spread” in the covariance update only for the measurement which is the closest to the prediction: __________________________________________________________________________________________ Technical Military Academy 8 Augustin Sperila – Contributions to the development of the advanced tracking algorithms __________________________________________________________________________________________ q k arg min i k i k T (4) i 1, m k k q k 1 q k q k Tk q (5) Pk / k I 1 0 I K k H k Pk / k 1 K k k K T k (6) Those above alterations in the basic PDAF were the main reasons for the degradation in the allowed target maneuverability mentioned in the first section. 5. MULTISENSOR DATA FUSION The BLUE fusion rule seeks its fused estimate as: T x k Wk X k ˆ (7) X k [ x (1) ,...,x (1) ]T ˆk ˆk (8) ( ( Wk [Wk1) ,...,Wk1) ]T (9) The weighting coefficients are: 1 m W r m k C1 Cij1 rj (10) j1 i, j1 with k ˆ k ˆk T Cij E x i x k x j x k (11) With PDAF, the current cross-covariance of local estimation errors will be obtained dependent on the current measurements validation configuration, which is computationally demanding and communication bandwidth consuming. __________________________________________________________________________________________ Technical Military Academy 9 Augustin Sperila – Contributions to the development of the advanced tracking algorithms __________________________________________________________________________________________ This issue could be overcome by considering that at each step, the update was made on an equivalent classical Kalman Filter, in the absence of incertitude concerning the measurements’ origin. The Kalman Filter should have the covariance of measurements errors set such as, it would be capable to make the transition between the actual (PDAF-like obtained) predicted and updated estimates and covariance matrices. Thus, the recursion of the cross-covariance between the local estimates is obtained via Information Filter formulae, as: ( 1 Cij1 Pki/)k FPki1 / k 1FT Q k () (12) 1 T FCij FT Q Pk j/k FPk j1 / k 1FT Q k The above formula is no more dependent on the current validation configuration and can be easily used with complete/incomplete communication rate and without or with feedback. 6. SIMULATION SET-UP AND RESULTS In order to validate the proposed method, a 2D scenario was simulated in the MATLAB™ environment. Two identical radars having the measurements errors variances of 40 meters in range and 0.2 degrees in bearing are used for updating the tracks of two targets moving with 300 m/s each, whose trajectories cross at a difference angle of 15 degrees in the middle of the simulation period. The simulation lasts for 120 seconds. The radar scan period is 2 seconds. The radar detection probability was varied between 0.8 and 1 and the clutter density was trialed in the range from 0.001 to 0.3 returns per square km. The above parameters reconstruct the basic constraints of two very demanding scenarios found in literarture. The basic IMM-PDAF algorithm was used when the validation gates do not overlap, and the IMM-PDAF(altered)- __________________________________________________________________________________________ Technical Military Academy 10 Augustin Sperila – Contributions to the development of the advanced tracking algorithms __________________________________________________________________________________________ JVC when tracks interact each other. Each IMM-PDAF has two “nearly- constant velocity” models: a benign one with the noise variance of 3 m/s 2 and a “noisy” one, tuned for allowing the tracking of a target which maneuvers at 7g at very low clutter density (0.001 returns per sq. km). The average maximum g-factor of the considered target plotted via trials against the clutter density is shown in the next figure: Target max. g-factor 10 8 6 4 2 0 0 0.1 0.2 0.3 Clutter density, ret. / sq. km Pd=1 Pd=0.9 Pd=0.8 Figure 1 The noise variance for the IMM maneuvering model was optimized by trials depending on the clutter density. A successful sample run of the simulation, plotted for PD=0.9 and clutter density equal to 0.1 returns/km2 is shown in Fig. 2: Figure 2 __________________________________________________________________________________________ Technical Military Academy 11 Augustin Sperila – Contributions to the development of the advanced tracking algorithms __________________________________________________________________________________________ The output of the simulation runs for assessing the IMM-PDAF-JVC capability to avoid track coalescence, averaged over 100 runs, are summarized in the table 1 below: Table 1 IMM-PDAF-JVC Track Lost Percentage (%, 100 Runs) PD Clutter density (ret/km2) 0.001 0.1 0.2 0.3 1 1 1 1 1 0.95 2 2 2 2 0.9 4 4 4 3 0.8 13 11 9 - First of all, one should note that due to the JVC hard assignment procedure, tracks are no more lost by their convergence, as in the standard IMM-PDAF, but by shifting the tracking from one trajectory to another. Another fact is that, when the tracking scenario is symmetrical (the targets approach each other with the same speed), the JVC assignment decisions are confused by the symmetrical disposal of the measurement errors, and the JVC-driven assignment will fail in 4 percent of the trials when PD=0.9. If the targets speeds are slightly different, the discrimination by the JVC becomes much more efficient. The percentage of tracks lost will decrease to below one when targets speed difference equals 15 m/s. Also, the unacceptable decrease in the JVC assignment performance when PD falls below 0.8 excludes this probability of detection from the instrumented range of the application. Extensive trials were performed in order to determine how much the alteration proposed for the basic IMM-PDAF affects the maximum target maneuverability under the most stressful conditions. The trials output is shown in Fig. 3, and Fig. 4 is a successful sample run for clutter density equal to 0.2. __________________________________________________________________________________________ Technical Military Academy 12 Augustin Sperila – Contributions to the development of the advanced tracking algorithms __________________________________________________________________________________________ Target max. g-factor (PD=0.9) 8 6 4 2 0 0 0.1 0.2 0.3 Clutter density, ret. / sq. km IMM-PDA IMM-PDA-JVC Figure 3 Figure 4 The trials run for assessing of the effectiveness of the proposed variant for BLUE fusion rule provided the following averaged results over 100 runs: Table 2 Errors in BLUE fusion rule, no feedback, communication rate 1/3 (meters) Clutter density (returns/km2) PD 0.001 0.1 0.2 0.3 Local Fus Inc Local Fus Inc Local Fus Inc Local Fus Inc 1 81.2 56.8 67.9 80.6 55.9 64.9 78.2 56.1 64.3 77.2 54.3 63.3 0.95 79.7 56.6 66.2 78.6 55.4 63.9 78.9 56.3 64.7 80.4 56.8 64.2 0.9 80.6 57.3 66.2 80.7 57.0 65.8 80.5 55.4 64.8 81.3 56.9 64.9 Local - local estimate; Fus: fused estimate, complete communication rate; Inc – fused estimate, incomplete communication rate. __________________________________________________________________________________________ Technical Military Academy 13 Augustin Sperila – Contributions to the development of the advanced tracking algorithms __________________________________________________________________________________________ Table 3 Errors in BLUE fusion rule, feedback (meters) Clutter density (returns/km2) PD 0.001 0.1 0.2 0.3 Local Fus Local Fus Local Fus Local Fus 1 61.3 48.9 60.9 48.4 59.5 48.3 59.0 48.6 0.95 61.6 49.7 61.6 49.6 60.2 48.4 59.8 47.8 0.9 63.8 50.5 63.2 50.8 62.7 51.5 61.4 51.0 Local - local estimate; Fus: fused estimate, feedback. From the above tables, one could observe that the errors are constant over the whole clutter range and they grow very slowly as a function of the decrease in detection probability. With complete communication rate, the increase in precision for the fused estimate over the local one is roughly 30 percent, equal to that one in which Chong’s fusion rule and JPDAF were used; in the current approach, the advantage over the previously mentioned one is the linear dependency of the computational burden on the number of tracks (JPDAF made that dependency exponential). The precision improvement in the fused estimate falls from 22 percent to 12 percent, when communication rate decreases from 1/2 to 1/5. When feedback is used, the precision improvement is about 40 percent over the non-feedback local estimates (the 30 percent improvement reported with Chong-JPDAF was obtained with feedback too). 7. OVERALL CONCLUSIONS An altered version of the IMM-PDAF-JVC algorithm was optimized via computer simulations for avoiding tracks convergence for closely spaced trajectories. The simulations were carried out in stressful conditions, which considered both missed detections and moderate-to-heavy clutter. __________________________________________________________________________________________ Technical Military Academy 14 Augustin Sperila – Contributions to the development of the advanced tracking algorithms __________________________________________________________________________________________ The algorithm was proved as being effective in avoiding tracks convergence, on the expense of a small reduction in the allowed target maneuverability during tracks interaction. The proposed IMM-PDA-JVC algorithm was then used as the basis for of a simplified implementation of the BLUE fusion rule. The flexibility of that one allowed its implementation with complete/incomplete communication rate and feedback. The overall improvement in the precision of the fused estimates over the local ones was better than one reported with the Chong-JPDAF due to the more demanding optimality criterion defining BLUE fusion rule than one used in the Chong’s one. __________________________________________________________________________________________ Technical Military Academy 15 Augustin Sperila – Contributions to the development of the advanced tracking algorithms __________________________________________________________________________________________ AUGUSTIN SPERILĂ Major, Electrical Engineer, Ph. D. Student OBJECTIVE Obtaining the Ph. D. degree in Electrical Engineering EXPERIENCE September 1993 – June 1999: Commissioned as 2nd Lt. (EE) with Romanian Air Force, GBAD Unit (promoted as Capt., August 1995) Deputy Commander, SAM Battalion Conducting logistics management Planning and supervising scheduled maintenance Participating in SAM live firings Searching for improved maintenance procedures in order to increase equipment availability June 1999 – March 2002: Air Force Staff, Technical Development Branch, Logistics Division Staff Officer Providing technical expertise in GBAD and Radar fields Drawing operational requirements for equipment related to major procurement programs Acting as consultant for Air Force Technical Standardization Board Keeping track of and promoting Air Force Technical Library Publishing articles in engineering international symposia March 2002 – November 2002: Air Force Staff, Standardization Branch Specialist Officer Participating in the Drafting Board for 14 National Technical Military Standards Keeping track of Air Force Standardization Library Drawing policy and monitoring NATO technical STANAGs implementation November 2002 – February 2005: Air Force Staff, International Military Cooperation Office (promoted as Maj., June 2004) Staff Officer Registering as Ph. D. student and participating in the doctoral training program (undergoing all due examinations and partial dissertations) __________________________________________________________________________________________ Technical Military Academy 16 Augustin Sperila – Contributions to the development of the advanced tracking algorithms __________________________________________________________________________________________ Preparing working, administrative and protocol arrangements for international activities Participating in international operational seminars, working parties and exercises Getting english language proficiency certification SLP 3.3.3.3 NATO STANAG 6001 Publishing articles in technical national journals and participating with papers in engineering international symposia February 2005 - present: Air Force Staff, Technical Supply Section, Logistics Division Staff Officer Keeping track of and facilitating excess equipment disposal Providing technical expertise in GBAD and Radar matters Preparing the full Ph. D. Thesis and Dissertation Publishing articles in technical national journals and participating with papers in engineering international symposia EDUCATION 1988–1993 Military Technical Academy, Bucharest, Romania B.S., Electronics for Air Defence Systems, graduated second in file. June 2004 Radar System Design Course, Military Technical Academy Bucharest, Romania (Co-sponsored by Technical University Delft and Thales Netherlands) INTERESTS History, computers, traveling, bodybuilding. __________________________________________________________________________________________ Technical Military Academy 17 Augustin Sperila – Contributions to the development of the advanced tracking algorithms __________________________________________________________________________________________ THE AUTHOR’S LIST OF PUBLICATIONS Papers in international conferences and symposia: 1. Gheorghe Gavriloaia, Adrian Stoica, Augustin Sperilă, „An ad-hoc method for avoiding tracks coalescence in PDAF for tracks fusion” sent to „Telsiks 2005”, Niş, Yugoslavia, September 2005; 2. Gheorghe Gavriloaia, Augustin Sperilă, “BLUE data fusion rule implemented on an altered version of IMM-JVC-PDAF”, accepted for publication in NAV-MAR Conference, Constanta, June 2005; 3. Gheorghe Gavriloaia, Augustin Sperilă, “The BLUE sensor fusion rule. A benchmark”, Technical Military Academy Symposium “Modern technologies in the XXIst century”, Bucharest, November 2003; 4. Gheorghe Gavriloaia, Augustin Sperilă, “Matching a m out of n track initiation algorithm to system’s parameters”, XXXIVth International Scientific Symposium of METRA, Bucharest, May 2003; 5. Augustin Sperilă, “Discrimination gain for sensor resource allocation”, Communications 2002 IEEE International Conference, Bucharest, May 2002; 6. Augustin Sperilă, “Tracking maneuvering targets. Interacting multiple models or jump/pruning models for Kalman filter banks?”, XXXIIIrd International Scientific Symposium of METRA, Bucharest, May 2002; 7. Augustin Sperilă, Ioan Burduşel, “An approach for enhancing the positional estimate of an air target within a radar data fusion system”, Communications 2002 IEEE International Conference, Bucureşti, decembrie 2002; 8. Ioan Burduşel, Constantin Căliman, Augustin Sperilă, “Considerations concerning the employment of the strategic wargames”, XXXIst International Scientific Symposium of METRA, Bucharest, September 1999. __________________________________________________________________________________________ Technical Military Academy 18 Augustin Sperila – Contributions to the development of the advanced tracking algorithms __________________________________________________________________________________________ Articles in Romanian Armaments Directorate’s Journal “Tehnica Militara” : 1. Gheorghe Gavriloaia, Augustin Sperilă, “Aplicarea eficientă a regulii de fuziune a celui mai bun estimat liniar nedeplasat pentru datele radar”, Revista Academiei Tehnice Militare, nr. 1/2005; 2. Gheorghe Gavriloaia, Augustin Sperilă, “Evitarea pierderii însoţirii ţintelor aeriene la intersectarea traiectelor pe fondul clutterului”, Tehnica Militară, Supliment Ştiinţific, nr. 2/2004; 3. Gheorghe Gavriloaia, Augustin Sperilă, “Însoţirea ţintelor aeriene manevriere cu modelul lui Singer pentru filtrul Kalman”, Tehnica Militară, Supliment Ştiinţific, nr. 1/2003; 4. Augustin Sperilă, “Detalii privind alegerea modelelor şi iniţializarea filtrelor Kalman, Tehnica Militară, nr. 3/2002; 5. Augustin Sperilă, “Un model de filtru Kalman extins, pretabil pentru aplicaţii de control al traficului aerian”, Tehnica Militară, nr. 1/2002 __________________________________________________________________________________________ Technical Military Academy 19