Haim Weiss

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

Kalman filtering for matrix estimation

A general discrete-time Kalman filter (KF) for state matrix estimation using matrix measurements is presented. The new algorithm evaluates the state matrix estimate and the estimation error covariance matrix in terms of the original system matrices. The proposed algorithm naturally fits systems which are most conveniently described by matrix process and measurement equations. Its formulation uses a compact notation for aiding both intuition and mathematical manipulation. It is a straightforward extension of the classical KF, and includes as special cases other matrix filters that were developed in the past. Beyond the analytical value of the matrix filter, it is shown through various examples arising in engineering problems that this filter can be computationally more efficient than its vectorized version.

LQC guidance law with bounded acceleration command

A novel missile guidance law that is dependent on the conditional probability density function of the estimated states is presented. The guidance law is derived by analyzing an interception scenario in the framework of an linear quadratic Gaussian (LQG) terminal control problem with bounded acceleration command. The nonlinear saturation function is represented by the equivalent random input describing function. Since for the investigated problem the certainty equivalence property is not valid, the resulting controller depends on the measurement noise level and on the saturation limit. In comparison to the classical optimal guidance law (OGL), the maximal value of the effective navigation gain is achieved during the engagement instead of near the terminal time. Thus, the saturation limit is reached earlier so as to have enough time to reduce the guidance errors. Using Monte-Carlo simulations, the superiority of the new guidance law over the classical OGL is shown. This validates the new approach of designing an estimation statistics dependent guidance law by using a random input describing function to approximate the missiles acceleration saturation.

Derivation of Formation-Flight Guidance Laws for Unmanned Air Vehicles

A guidance-based approach to formation flight of unmanned air vehicles is presented. Most of the works related to formation flight deal with the control of the relative positions between the vehicles in the formation. In the proposed approach a pursuit law replaces the control law. A new rule of pursuit denoted as detective deviated pursuit is then developed based on known guidance rules. The proposed guidance law regulates the distances between vehicles and the heading angles with respect to the lines of sight. We show that the detective deviated pursuit rule can be applied to formation flight, where the formation geometry determines the guidance-law parameters. The work discusses the conditions for a stable formation flight in the case of a maneuvering leader and proposes a control scheme for implementation. The performance of the new guidance-based formation flight is evaluated via simulations. Nomenclature C D = drag coefficient CL = lift coefficient D = drag g = acceleration caused by gravity K = drag polar constant K xI = integral controller for the x state K xP = proportional controller for the x state n = load factor R = radius r = leader‐follower range T = thrust t = time V =v elocity W = weight X = states vector x, y = inertial coordinates

Guidance Laws in Target—Missile—Defender Scenario with an Aggressive Defender

Abstract An encounter among a target, an intercepting missile and a defending missile is studied in a linear quadratic game setting. The purpose of the defending missile is to destroy the intercepting missile, before the latter reaches the target. The limiting values of the three participants optimal strategies is studied as the quadratic weight on the defending missiles acceleration command tends to zero. It is shown that in the limit the intercepting missiles and the targets optimal strategies are identical in form to that obtained in the game without the defending missile.

Improving the performance of existing missile autopilot using simple adaptive control

Abstract A simple add-on adaptive control algorithm is presented. It is demonstrated via example that the performance of existing missile autopilot can be improved. The algorithm involves the synthesis of parallel feedforward which guarantees that the controlled plant is almost strictly positive real (ASPR). It is proved in the paper that such a parallel feedforward always exists. The proof is based on the parameterization of a set of stabilizing controllers. This parameterization enables straight-forward design and implementation of the add-on simple adaptive control (SAC) algorithm.

Missile guidance with constrained terminal body angle

A novel missile guidance law that simultaneously minimizes the miss distance and imposes a predetermined terminal body angle is presented. The guidance law is derived by analyzing an interception scenario within the framework of a linear quadratic (LQ) terminal control problem. The derivation is for first order linear missile dynamics and a non-maneuvering target. The performance of the proposed guidance law is demonstrated in a planar interception scenario.

Trajectory shaping in linear-quadratic pursuit-evasion games

of the American Control Conference, American Automatic Control Council, Evanston, IL, 2003, pp. 402–407. 13Manchester, I. R., and Savkin, A. V., “Circular Navigation Missile Guidance with Incomplete Information and Uncertain Autopilot Model,” AIAA Paper 2003-5448, Aug. 2003. 14Rusnak, I., “Optimal Guidance Laws with Uncertain Time-of-Flight,” IEEE Transactions on Aerospace and Electronics Systems, Vol. 36, No. 2, 2000, pp. 721–725. 15Khalil, H. K., Nonlinear Systems, second ed., Prentice–Hall, Upper Saddle River, NJ, 1996, pp. 211–216. 16Chilali, M., and Gahinet, P., “H∞ Design with Pole Placement Constraints: An LMI Approach,” IEEE Transactions on Automatic Control, Vol. 41, Vol. 3, 1996, pp. 358–367.

TRAJECTORY SHAPING AND TERMINAL GUIDANCE USING LINEAR QUADRATIC DIFFERENTIAL GAMES

The linear quadratic optimization theory is applied to the missile guidance problem including a running cost on the state vector. This inclusion enables to develop a new and effective way for trajectory shaping. For this case, optimal control signal decomposition is presented and the control signal components are established. Then a new optimal guidance control decomposition strategy is developed whereby the terminal guidance phase is separated from the trajectory-shaping phase, thus producing a sub-optimal control. A new formulation of linear quadratic differential game with a penalty on the target estimation error is then proposed for the noisecorrupted environment. This approach, together with the inclusion of a running cost on the state vector, enables to develop a new effective guidance law against smart targets. Numerical comparisons of the new guidance law with some widely used representative guidance laws are performed. *†‡

TARGET MANEUVER ADAPTIVE GUIDANCE LAW FOR A BOUNDED ACCELERATION MISSILE

A new estimator-guidance law is presented, for a bounded acceleration interceptor pursuing a target performing a sudden step maneuver. The linearized end-game scenario is investigated. The missile’s acceleration saturation is represented by the equivalent random input describing function. The estimator proposed for this problem features a mechanism constructed to e‐ciently identify a sudden target maneuver. Due to the bound on the missile acceleration, the certainty equivalence property is not valid, and so, the resulting controller (guidance law) depends on the measurement noise level, statistics of the target maneuver, and the saturation limit. The derivation of this guidance law proceeds in two stages. In the flrst simplifled stage, it is assumed that the instance that the target issues a step command in its acceleration is known. However, the size of the target maneuver jump must still be estimated. Simulation results show signiflcant improvement over the deterministic optimal guidance law when the jump times vary from 10 to 3 missile time constants prior to the intercept time. If the jump occurs earlier or later than this duration no improvement is gained. In the second stage, a suboptimal guidance law is derived using the multiple model adaptive control formulation for a target performing a single step maneuver near the intercept time. Note that this sub-optimal guidance law does not require a-priory knowledge of the time of the jump in the target acceleration command. Performance improvement of the same order of magnitude as was observed for the known jump time scheme (that also serves as a performance bound) is achieved, verifying the validity of this approach. Sensitivity analysis to various noise levels and expected target maneuvers was carried out emphasizing the robustness and advantages of this scheme.