A New Guidance Law for the Defense Missile of Nonmaneuverable Aircraft

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

Design and Analysis of State-Feedback Optimal Strategies for the Differential Game of Active Defense

Abstract: This paper is concerned with a scenario of active target defense modeled as a zero-sum differential game. The differential game theory as developed by Isaacs provides the correct framework for the analysis of pursuit-evasion conflicts and the design of optimal strategies for the players involved in the game. This paper considers an Attacker missile pursuing a Target aircraft protected by a Defender missile which aims at intercepting the Attacker before the latter reaches the Target aircraft. A differential game is formulated where the two opposing players/teams try to minimize/maximize the distance between the Target and the Attacker at the time of interception of the Attacker by the Defender and such time indicates the termination of the game. The Attacker aims to minimize the terminal distance between itself and the Target at the moment of its interception by the Defender. The opposing player/team consists of two cooperating agents: The Target and the Defender. These two agents cooperate in order to accomplish the two objectives: Guarantee interception of the Attacker by the Defender and maximize the terminal Target-Attacker separation. In this paper, we provide a complete, closed form solution of the active target defense differential game; we synthesize closed-loop state feedback optimal strategies for the agents and obtain the Value function of the game. We characterize the Target's escape set and show that the Value function is continuous and continuously differentiable over the Target's escape set, and that it satisfies the Hamilton–Jacobi–Isaacs equation everywhere in this set.

An integrated guidance and control approach in three-dimensional space for hypersonic missile constrained by impact angles

Abstract: A new robust three-dimensional integrated guidance and control (3D-IGC) approach is investigated for sliding-to-turn (STT) hypersonic missile, which encounters high uncertainties and strict impact angle constraints First, a nonlinear state-space model with more generality is established facing to the design of 3D-IGC law With regard to the as-built nonlinear system, a robust dynamic inversion control (RDIC) approach is proposed to overcome the robustness deficiency of traditional DIC, and then it is applied to construct the basic 3D-IGC law combining with backstepping method In order to avoid the problems of "explosion of terms" and high-frequency chattering, an improved 3D-IGC law is further proposed by introducing dynamic surface control and continuous approximation approaches From the computer simulation on a hypersonic missile, the proposed 3D-IGC law not only guarantees the stable flight, but also presents the precise control on terminal locations and impact angles Moreover, it possesses smooth control output and strong robustness

Cooperative Missile Guidance for Active Defense of Air Vehicles

Abstract: In air combat, an active countermeasure against an attacking missile homing into a Target aircraft entails the launch of a defending missile. The Target is protected by the Defender, which aims to intercept the Attacker before the latter reaches the Target aircraft. A differential game is presented where the Target–Defender team strives to maximize the terminal separation between the Target and the Attacker at the time instant where the Attacker is intercepted by the Defender, whereas the Attacker strives to minimize the said separation. This paper discusses the case where the Defender is endowed with a positive capture radius. Optimal strategies for the three agents are derived and simulation examples illustrate the effectiveness of the proposed approach.

Multiple Pursuer Multiple Evader Differential Games

Abstract: In this article an $N$ -pursuer versus $M$ -evader team conflict is studied. This article extends classical differential game theory to simultaneously address weapon assignments and multiplayer pursuit-evasion scenarios. Saddle-point strategies that provide guaranteed performance for each team regardless of the actual strategies implemented by the opponent are devised. The players’ optimal strategies require the codesign of cooperative optimal assignments and optimal guidance laws. A representative measure of performance is employed and the Value function of the attendant game is obtained. It is shown that the Value function is continuously differentiable and that it satisfies the Hamilton–Jacobi–Isaacs equation—the curse of dimensionality is overcome and the optimal strategies are obtained. The cases of $N=M$ and $N>M$ are considered. In the latter case, cooperative guidance strategies are also developed in order for the pursuers to exploit their numerical advantage. This article provides a foundation to formally analyze complex and high-dimensional conflicts between teams of $N$ pursuers and $M$ evaders by means of differential game theory.

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

Optimal Cooperative Pursuit and Evasion Strategies Against a Homing Missile

Abstract: Optimal-control-based cooperative evasion and pursuit strategies are derived for an aircraft and its defending missile. The aircraft-defending missile team cooperates in actively protecting the aircraft from a homing missile. The cooperative strategies are derived assuming that the incoming homing missile is using a known linear guidance law. Linearized kinematics, arbitrary-order linear adversaries' dynamics, and perfect information are also assumed. Specific limiting cases are analyzed in which the attacking missile uses proportional navigation, augmented proportional navigation, or optimal guidance. The optimal one-on-one, noncooperative, aircraft evasion strategies from a missile using such guidance laws are also derived. For adversaries with first-order dynamics it is shown that depending on the initial conditions, and in contrast to the optimal one-on-one evasion strategy, the optimal cooperative target maneuver is either constant or arbitrary. These types of maneuvers are also the optimal ones for the defender missile. Simulation results confirm the usefulness and advantages of cooperation. Specifically, it is shown how the target can lure in the attacker, allowing its defender to intercept the attacking missile even in scenarios in which the defender's maneuverability is at a disadvantage compared with the attacking missile.

Cooperative Differential Games Strategies for Active Aircraft Protection from a Homing Missile

Abstract: Cooperative pursuit―evasion strategies are derived for a team composed of two agents. The specific problem of interest is that of protecting a target aircraft from a homing missile. The target aircraft performs evasive maneuvers and launches a defending missile to intercept the homing missile. The problem is analyzed using a linear quadratic differential game formulation for arbitrary-order linear players' dynamics in the continuous and discrete domains. Perfect information is assumed. The analytic continuous and numeric discrete solutions are presented for zero-lag adversaries' dynamics. The solution of the game provides 1) the optimal cooperative evasion strategy for the target aircraft, 2) the optimal cooperative pursuit strategy for the defending missile, and 3) the optimal strategy of the homing missile for pursuing the target aircraft and for evading the defender missile. The obtained guidance laws are dependent on the zero-effort miss distances of two pursuer―evader pairs: homing missile with target aircraft and defender missile with homing missile. Conditions for the existence of a saddle-point solution are derived and the navigation gains are analyzed for various limiting cases. Nonlinear two-dimensional simulation results are used to validate the theoretical analysis. The advantages of cooperation are shown. Compared with a conventional one-on-one guidance law, cooperation significantly reduces the maneuverability requirements from the defending missile.

Cooperative Multiple-Model Adaptive Guidance for an Aircraft Defending Missile

Abstract: A cooperative guidance law, for a defender missile protecting an aerial target from an incoming homing missile, is presented. The filter used is a nonlinear adaptation of a multiple model adaptive estimator, in which each model represents a possible guidance law and guidance parameters of the incoming homing missile. Fusion of measurements from both the defender missile and protected aircraft is performed. A matched defender’s missile guidance law is optimized to the identified homing missile guidance law. It utilizes cooperation between the aerial target and the defender missile. The cooperation stems from the fact that the defender knows the future evasive maneuvers to be performed by the protected target and thus can anticipate the maneuvers it will induce on the incoming homing missile. Moreover, the target performs a maneuver that minimizes the control effort requirements from the defender. The estimator and guidance law are combined in a multiple model adaptive control configuration. Simulation results show that combining the estimations with the proposed optimal guidance law, that utilizes cooperation between the defending missile and protected target, yields hit-to-kill closed loop performance with very low control effort. This facilitates the use of relatively small defending missiles to protect aircrafts from homing missiles.

Line-of-Sight Interceptor Guidance for Defending an Aircraft

Abstract: = missile evasive maneuver normal to defendermissile line of sight amp = missile evasive maneuver component normal to target-missile line of sight at, ad, am = target, defender, and missile lateral accelerations, respectively at max, ad max, am max = target, defender, and missile maximum lateral accelerations, respectively atplos, adplos, amplos = target, defender, and missile accelerations normal to the line of sight, respectively Rd, Rm, Rdm = target-defender, target-missile, and defendermissile closing ranges, respectively tf = defender-missile interception time vt, vd, vm = target, defender, and missile speeds, respectively vtlos, vdlos, vmlos = target, defender, and missile speeds along the line of sight, respectively vtplos, vdplos, vmplos