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Differential Game Guidance Law for Interceptor Missiles with a Time

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Differential Game Guidance Law for Interceptor Missiles with a Time Trans. Japan Soc. Aero. Space Sci. Vol. 54, No. 185/186, pp. 189–197, 2011 Differential Game Guidance Law for Interceptor Missiles with a Time-varying Lateral Acceleration Limit By Naiming QI, Yanfang LIU and Xiaolei SUN Department of Aerospace Engineering, Harbin Institute of Technology, Harbin, China (Received May 5th, 2010) A guidance law for an interceptor missile with a time-varying lateral acceleration limit is developed using differ- ential games formulation with bounded controls. If an interceptor has axial acceleration and its lateral acceleration is generated by the aerodynamic lift, its lateral acceleration limit will vary with time due to its time-varying speed and the air density varying with altitude. This time-varying lateral acceleration limit property can also be obtained by designs of the time-varying sum of jets’ thrust when the interceptor is steered by divert reaction jets. The effect of this time-vary- ing limit is investigated in this paper. The results indicate that the time-varying design of this limit leads to cooperation between the mid-course and terminal homing phase, and yields an improvement in homing accuracy. The effectiveness of this guidance law is demonstrated in a realistic ballistic missile defense scenario. It is shown that the proposed guid- ance law guarantees admissible accuracy, and is robust for the target maneuver commands and the delay in the estimation process of the target maneuvers. Furthermore, it provides better homing performance compared to the conventional law if the estimation delay is longer than a certain degree. Key Words: Differential Game, Guidance Law, Interceptor Missile, Time-varying, Game Space Nomenclature P: pursuer (interceptor) E: evader (target) V: velocity x: axial x: state vector r: range 1. Introduction a: realized acceleration amax: acceleration limit The traditional target of an interceptor is an inhabited ac: commanded acceleration aircraft, against which the interceptor has substantial advan- u: normalized acceleration commands tages in speed, maneuverability and agility (i.e., maximum : flight path angle acceleration divided by the first-order time constant). More- ’: angle of climb over, miss distances at the order of a few meters are consid- max: maximum angle of attack ered to be admissible. However, the tactical ballistic missile : time constant (TBM), a new type of target which is able to carry noncon- t: time ventional warheads, is much less vulnerable than an aircraft. y: relative displacement Successful interception of such a target demands a very ÁtSW: target maneuver switch time small miss distance or even a direct hit. Átest: estimation delay A great majority of the advanced missile guidance laws in Ms: guaranteed miss distance current use were developed with two-dimensional models ts: critical time assuming missiles with both constant speeds and lateral z: zero-effort-miss distance acceleration limits.1–4) Actually, the missile-target intercep- : angle of line of sight tion scenarios are characterized by nonlinear kinematics and : pursuer-evader maneuverability ratio time-varying velocities. The results5) derived from linear- : average maneuverability ratio ized guidance theory were compared to the outcome of h: altitude three-dimensional point mass simulations dealing with an : air density interception scenario against a highly maneuverable TBM. CL: lift coefficient This comparison showed clearly that the application of the S: reference area linear guidance theory currently used does not predict the m: mass miss distance well. The discrepancies between the results Subscripts had been attributed to inappropriate assumptions of constant 0: initial velocities and constant lateral acceleration boundaries. f: final In open, accessible technical literature, there are several works addressing the guidance problem of variable-speed Ó 2011 The Japan Society for Aeronautical and Space Sciences 190 Trans. Japan Soc. Aero. Space Sci. Vol. 54, No. 185/186 missiles. A guidance law, based on nonlinear kinematics After the interception scenario is outlined in Section 2, with head-pursuit engagement, was derived using sliding- the DGL/M guidance law is presented in Section 3, with mode control (SMC) methodology,6) and extended to an overview of some other former studied differential guid- head-on and tail-chase engagements.7) The SMC approach ance laws. This is followed by a discussion on DGL/M was also used to derive guidance laws for interceptors with implementation in Section 4, where the focus is on two types constant axial acceleration.8) However, the simulation of time-varying interceptor lateral acceleration limit results indicated that, as the interceptor lateral acceleration profiles. In Section 5, the performance DGL/M is tested limit was not taken into account, these guidance laws using a two-dimensional realistic ballistic missile defense demand high maneuverability. scenario. The paper is concluded in Section 6. As the target maneuvers are controlled independently of the interceptor, the optimal guidance law1) has an inherent 2. Problem Statement deficiency. Thus, the scenario of intercepting a maneuvera- ble target has to be formulated as a zero-sum pursuit-evasion This paper deals with the terminal homing phase for the game. Linear quadratic game formulation with interceptor endoatmospheric interception of a maneuverable TBM. In ideal dynamics model (i.e., where control commands are this section, the interception scenario is formulated as a realized without errors and time delay) requires proportional linear zero-sum, pursuit-evasion game with limited controls. navigation9) as an optimal guidance law and analysis consid- The deterministic game analysis is based on the following ering boundary controls and first-order interceptor missile set of simplified assumptions:3,4,13) lateral dynamics; thereby yielding an optimal guidance 1) The engagement between the interceptor and the TBM law (DGL/0)10,11) that is robust to the type of target maneu- takes place on a plane. ver but does not guarantee a zero-miss distance. An im- 2) The trajectories of both missiles can be linearized proved guidance law12) (DGL/1), where the dynamics of around the initial line-of-sight (LOS). the target maneuvers are approximated by first-order lags 3) The velocity profiles of both missiles on the designed to their command signals, guarantees a zero-miss distance trajectories are known and can be expressed as functions in a perfect information structure, if the agility of the inter- of time. ceptor is superior to that of the target. A comparison study5) 4) The lateral acceleration limit of each missile is known showed that these differential guidance laws provide im- as a function of time. proved homing accuracy. Based on the time-varying linear 5) The information structure is perfect and contains two pursuit-evasion game model, a guidance law13) (DGL/E) parts: a) both missiles’ designers have perfect knowledge was proposed against targets with known speed and lateral of the engagement parameters, and b) both missiles can acceleration limit profiles. accurately measure all of the state variables. However, in realistic implementation, as there are distur- Though in reality the TBM has no information about the bances and measurement noise, the optimal game guidance interceptor’s state variables, it can carry out a very close laws produce very disappointing results. The miss distance realization of the optimal interception avoidance strategy is never zero and highly sensitive to target maneuvers.14) if it maneuvers randomly. Thus, the second part of assump- Taking the estimation delay into account, a comparison4) tion 5 represents the worst case for the interceptor. was made between DGL/0 and DGL/1. The results showed The typical profiles for both missiles were obtained by that, if the estimation delay is too long, it is better not to three-dimensional point mass simulations in other stud- incorporate the evader’s acceleration in the guidance law ies.5,13) In this paper, these profiles (as shown in Fig. 1) at all; that is, to use DGL/0. Considering the estimator are generated using engineering estimates, and the details inherent delay, a compensated differential guidance law4) are given in the Appendix. It can be seen that, during the (DGL/C) demonstrates robustness with respect to the timing of the evader’s maneuvers. In this paper, a novel guidance law, denoted as DGL/M, for an interceptor missile with time-varying lateral accelera- tion limit is proposed. The guidance law is derived using dif- ferential game formulations with boundary controls under the assumption that the target maneuver dynamics are ideal. DGL/M is more than just a simplification of DGL/E. As it does not incorporate the target’s acceleration, it gains robust- ness with respect to the target’s maneuver commands and estimated delay of the target’s acceleration. Further discus- sion is focused on the effect of two types of time-varying purser’s lateral acceleration limit profiles. It is shown that a properly designed acceleration limit profile leads to coop- eration between the mid-course and terminal homing phase, Fig. 1. End-game velocity/altitude, velocity and maneuverability pro- as well as an improvement in homing accuracy. files. Nov. 2011 N. QI et al.: Differential Game Guidance Law for Interceptor Missiles 191 end game, the speed of the TBM remains almost unchanged :¼fAE; bE; cE; dEg is the TBM realization, with transfer À1 while the altitude decreases. Its lateral acceleration limit is function GEðsÞ¼cEðsI À AEÞ bE þ dE. Here uP and uE monotonically increasing due to the larger air density of are the normalized controls of the interceptor and TBM, the lower altitudes. It is also shown that the velocity and respectively, altitude of the interceptor are monotonically increasing in u ¼ ac=amaxðtÞ; ju j1; i 2fP; Eg: ð3Þ the end game. If the interceptor’s lateral acceleration is pro- i i i i duced by aerodynamic lift, its lateral acceleration limit is Based on assumptions 2 and 3, the final time of intercep- proportional to the product of the square of its velocity tion can be computed for any given initial range X0 of the and the air density.
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