Josef Shinar

Solution Techniques for Realistic Pursuit-Evasion Games

Abstract : Two solution techniques are presented and applied to solve in a closed form realistic air combat problems modelled as perfect information zero-sum differential games. Optimal missile guidance and avoidance solved as a linear differential game with bounded control. Medium range air to air interception is analyzed applying the method of singular perturbations. For more complex problems the combination of the two techniques may be required. (Author)

Time-Varying Linear Pursuit-Evasion Game Models with Bounded Controls

Future end game interception scenarios of autonomous uncrewed e ying vehicles are expected to be characterized by variable velocities and lateral acceleration limits. A time-varying linear pursuit ‐evasion game model with bounded controls is presented that can be used to analyze such scenarios. The usefulness of this model is demonstrated by simulations of a realistic ballistic missile defense scenario, as an example. It is shown that a differential game guidance law derived using this time-varying model provides a signie cant improvement in the homing accuracy compared to a guidance law based on a model with constant velocities and lateral acceleration limits. Moreover, the time-varying linear model provides a much more accurate prediction of the miss distance, cone rming its validity. Also a general review of possible structures of the game space decomposition is presented. Oneofthese structuresimpliesthateven ifthepursuerdoesnot havea maneuverability advantage overtheevader, but has an agility advantage, a zero miss distance can still be achieved for some initial conditions.

Missile guidance laws based on pursuit-evasion game formulations

The increasing maneuverability potential of tactical ballistic missiles motivates the development of improved interceptor guidance laws. In this paper two guidance laws are compared, both based on a pursuit-evasion differential game formulation. The first game is a linear-quadratic one. The second one is formulated with bounded controls and the miss distance as the performance index. The comparison shows that for the same values of physical parameters the capture zone (guaranteeing zero miss distance) of the guidance law using bounded controls, as well as the control effort associated with it, are larger. However, for short duration end games and for large values of interceptor maneuverability advantage the capture zones nearly coincide.

Three-dimensional optimal pursuit and evasion with bounded controls

The missile-aircraft pursuit-evasion problem is formulated by a three-dimensional linearized kinematic model with bounded control. The formulation is valid both for the optimal control (against a known adversary strategy) and the zero sum differential game versions. Assuming perfect information, the linearized kinematic model yields for both versions a solution which can be impletemented in real time for airborne application. The avoidance of a known pursuer by an evader who has no state information is solved by a stochastically optimal periodical maneuver. Other examples of imperfect information are briefly discussed.

Efficient multiple model adaptive estimation in ballistic missile interception scenarios

A novel efe cient algorithm, featuring a highly reduced computational load, is presented for multiple model adaptive estimation in a future real-life ballistic missile defense scenario, where the blind incoming target (having no information on the interceptor’ s state ) performs a bang ‐bang evasive maneuver characterized by a random switching time. The efe ciency of the algorithm derives mainly from its exploitation of the special structure of the hypothesis space in this problem to drastically reduce the number of concurrently active e lters in the bank without incurring any signie cant performance degradation. The proposed algorithm’ s efe ciency allows a substantial increase in the resolution of the discretized hypothesis space, thus enhancing considerably the attainable estimation performance. The effect of the new estimator’ s performance on guidance accuracy is examined. The homing performance of various perfect information guidance laws using this efe cient estimation method is compared, via Monte Carlo simulations, to the use of a Kalman e lter incorporating a shaping e lter representing the random target maneuver. The results demonstrate the superiority and viability of the proposed method.

New Interceptor Guidance Law Integrating Time-Varying and Estimation-Delay Models

The paper presents a synthesis of a new guidance law, derived using differential games concepts, for the interception of highly maneuvering targets. The synthesis is based on the integration of two recently demonstrated improvementfeatures:namely,theuseofa time-varying linearkinematicsgamemodelandthecompensationofthe inherent estimation delay of thetarget acceleration. The new guidance law is implemented in a genericyet realistic noise-corruptedthree-dimensionalnonlinearballisticmissiledefensescenariobyusingasuitablethree-dimensional estimator.Thetestagainstworst-casetargetmaneuversdemonstratesasignie cantimprovementcomparedto other known guidance laws indicating a potential breakthrough in interceptor guidance. I. Introduction H ISTORICALLY, the typical target of interceptor missiles has been a manned aircraft, against which the missile had substantial advantage in speed, maneuverability, and agility. Moreover, miss distances of a few meters, compatible with the lethal radius of the missile warhead, were considered admissible. Because of these facts, even a simple guidance law such as proportional navigation (PN) could guarantee the target destruction. The Gulf War introduced the tactical ballistic missile (TBM), able to carry nonconventional warheads, as a new type of target. Successful interception of a TBM, much less vulnerable than an aircraft, requires a very small miss distance or even a direct hit. Several ballistic missile defense systems are currently in development. Because of advances in technology, these systems (such as ARROW and PAC-3) succeeded to demonstrate, using conventionalguidance concepts, excellent homing accuracy against such nonmaneuvering targets. 1;2 Although known TBMs were not designed to maneuver because of their high reentry speed, they have a substantial maneuverability potential in the atmosphere. Moreover, this potential can be made applicable by a modest technical effort. The same is true for future high-speed antiship or cruise missiles.Paradoxically, the successful current development of ballistic missile defense systems can serve to motivate the future development of maneuverable antisurface missiles. Against such threats interceptor missiles will have only a marginal maneuverability advantage. Hence, the required small miss distances are not achievable by using conventional guidance laws even in a noise-free environment, as it was demonstrated by recent simulation studies. 3;4 Most missile guidance laws used at the present, including PN, were derived using a linear quadratic optimal control formulation assuming perfect information. 5 Such a formulation requires an assumption on the future evolution of the target maneuver. If this

Integrated Estimation/Guidance Design Approach for Improved Homing Against Randomly Maneuvering Targets

Interceptor missiles, designed against aircraft, have substantial speed and maneuverability advantage over their targets. Thus, by exploiting the technological progress, even simple guidance concepts yielded satisfactory performance. For the interception of antisurface missiles, higher guidance precision is required. Using conventional guidance and estimation concepts, existing missile defense systems have demonstrated hit-to-kill accuracy against nonmaneuvering targets. Guaranteeing a similar performance against maneuvering targets can be achieved only if the estimation errors against such targets are minimized. This paper introduces a new, logic-based estimation/ guidance algorithm, that explicitly uses the time-to-go in the estimation process and modifies the guidance law to reduce the consequence of estimation errors. The successful outcome of the new approach is illustrated by an extensive Monte Carlo simulation study.

SOLUTION OF A DELAYED INFORMATION LINEAR PURSUIT-EVASION GAME WITH BOUNDED CONTROLS

A class of linear pursuit-evasion games with first-order acceleration dynamics and bounded controls is considered, where the evader has perfect information and the pursuer has delayed information on the lateral acceleration of the evader. The other state variables are perfectly known to the pursuer. This game can be transformed to a perfect information delayed control game with a single state variable, the centre of the uncertainty domain created by the information delay. The delayed dynamics of the game is transformed to a linear first-order partial differential equation coupled with an integral-differential equation, both without delay. These equations are approximated by a set of K + 1 ordinary differential equations of first order, creating an auxiliary game. The necessary conditions of optimality derived for the auxiliary game lead to the solution of the delayed control game by a limit process as K → + ∞. The solution has the same structure as the other, already solved, perfect information linear pursuit-evasion games with bounded controls and indicates that the value of the delayed information pursuit-evasion game is never zero. Asymptotic expressions of the value of the game for small and large values of the information delay are derived.

ON THE VALIDITY OF LINEARIZED ANALYSIS IN THE INTERCEPTION OF REENTRY VEHICLES

The paper compares previous results derived from linearized guidance theory with the outcomes of three dimensional point mass simulations of interception scenarios against a highly maneuvering tactical ballistic missile. Due to the differences between the realistic (nonlinear, variable speed) simulation scenario and the simplified mathematical model used in linearized guidance theory, the comparison of the results has not been a straight forward task. Nevertheless, the comparison allowed to confirm that the major conclusions of earlier papers, based on the simplified linear model, are valid. In particular, it was demonstrated again that currently used guidance laws and estimation techniques cannot guarantee a hit-to-kill accuracy in the interception of highly maneuvering targets, such as reentering tactical ballistic missiles, to be expected in the future. The sensitivity trends of the guaranteed miss distance with respect to the parameters of the interception scenario were found to be similar (but not identical) in both models. The main reasons for the differences in the results have been identified.

Using a Multiple-Model Adaptive Estimator in a Random Evasion Missile/Aircraft Encounter

The terminal phase (end game)of an encounter between an air-to-air missileequipped with an activemonopulse radar seeker and an evading e ghter aircraft, possibly employing electronic countermeasures in the form of electronic jinking, is addressed. The missile uses a guidance law derived from linear differential game theory, which is implemented by using a multiple-model adaptiveestimator (MMAE). TheMMAE identie es theevasion strategy of theaircraft, which consists of thecombination of evasion maneuverand electronic jinking. An extensivenumerical study is used to demonstrate the viability of the concept. In comparison with a previously proposed mixed strategy guidance methodology, the new MMAE-based approach leads to a substantial improvement in the guaranteed single-shot kill probability for the missile. ODERN air-to-air missiles, designed to intercept highly maneuverable aircraft equipped with electronic countermeasures (ECM), have to operate in a highly uncertain environment. This paper is concerned with the terminal phase (end game) of an encounter between a missile equipped with an active monopulse radar seeker and an evading e ghter aircraft having the option to use ECM. The terminal homing phase of the interception starts when the active seeker of the missile locks on its target, generating also a warning signal (including some threat identie cation ) in the aircraft. This warning is the only information the pilot has. At the moment when the warning is received, the pilot starts to execute a sequence ofperiodicalevasivemaneuvers.Simultaneously,toenhanceitssur