The proportional navigation dilemma-pure or true?

Two generic classes of proportional navigation (PN) laws are compared in detail. One class consists of pursuer-velocity-referenced systems, which include pure proportional navigation (PPN) and its variants; the second category consists of line-of-sight- (LOS-) referenced systems such as true proportional navigation (TPN), generalized true proportional navigation (GTPN), and generalized guidance laws. The existing closed-form solutions are discussed in detail, and the classical linear and quasilinear analytical solutions are summarized. A critical comparison is then made with regard to the definition, implementation, and analytical aspects of the guidance laws, including the method, the nature of solution, and an appraisal of the behavior of the pursuer motion resulting from the laws. It is established that, in spite of some restricted advantages in the solvability of the equations of motion, the LOS-referenced PN schemes suffer from serious limitations in terms of implementation and trajectory behavior. Among the major drawbacks are forward velocity variation requirement, relatively large control effort requirement, restrictions on initial engagement conditions to ensure intercept, lack of robustness, and possibility of unbounded acceleration. It is concluded that PPN is a better guidance law in a practical sense than TPN and its generalizations     

On the generalization of true proportional navigation

A simple framework to define generalized true proportional navigation (GTPN) guidance laws is presented. It is shown that this framework subsumes many of the generalizations presented in the earlier literature. The capture regions of a number of GTPN guidance laws are obtained through a rigorous qualitative analysis. The method of analysis is simpler and lends itself directly to an easy geometrical interpretation. A considerable amount of misinterpretation in the previous results, arising out of certain basic misconceptions, are corrected here. It is shown that a logical application of the guidance philosophy, through a minor modification of GTPN to take into account the direction of rotation of the line-of-sight (LOS), contributes substantially to the expansion of the capture region in the relative velocity space. In particular, it is shown that the capture region also extends to the negative closing velocity region, thus making the modified GTPN almost comparable, so far as the domain of capturability of guidance laws is concerned, to the pure proportional navigation (PPN) guidance law. A number of new results on the exact bounds on the capture region are derived and illustrated through examples     

Ideal proportional navigation for exoatmospheric interception

Ideal proportional navigation (IPN) is a natural choice for exoatmospheric interception for its mighty capture capability and ease of implementation. The closed-form solution of two- dimensional ideal proportional navigation was conducted in previous public literature, whereas the practical interception happens in the three-dimensional space. A novel set of relative dynamic equations is adopted in this paper, which is with the advantage of decoupling relative motion in the instantaneous rotation plane of the line of sight from the rotation of this plane. The dimension-reduced IPN is constructed in this instantaneous plane, which functions as a three-dimensional guidance law. The trajectory features of dimension-reduced IPN are explored, and the capture regions of dimension-reduced IPN with limited acceleration against nonmaneuvering and maneuvering targets are analyzed by using phase plane method. It is proved that the capture capability of IPN is much stronger than true proportional navigation (TPN), no matter the target maneuvers or not. Finally, simulation results indicate that IPN is more effective than TPN in exoatmospheric interception scenarios.     

3D guidance law modeling

Proportional navigation (PN) guidance laws (GLs) have been widely used and studied in the guidance literature. But most of the guidance literature on PN has concentrated on the evaluation of empirical PNGLs or GLs obtained from very specific optimality considerations. The authors present a novel approach (called guidance laws modeling) to derive new GLs. They consider the basic requirements of capture and define a complete class of GLs that meet these requirements. It is shown that PN is a natural candidate in this class. The main consequence of this modeling process is the definition of two new GLs: one in 2D space and the other in 3D space. These new GLs can be interpreted as new generalizations of the true proportional navigation (TPN) GL. Moreover, it is shown that these generalizations allow the TPNGL to match the capturability performance of the pure proportional navigation (PPN) GL in terms of initial conditions which allow the guided object to reach its target     

飞行器比例导引综述

基于导弹制导和机器人控制等领域，对追踪／截获动目标——导引问题的研究，综观几十年来大量国内外研究成果，系统分析与比较了各种比例导引律的含义、特点与性能。指出纯比例导引及其改进是非常有效的导引方式，可以有效地用于机动目标追踪／截获导引，能为相关领域机动目标导引问题的研究，包括战斗机追踪导引设计提供参考。     

Optimal pure proportional navigation for maneuvering targets

The optimal pure proportional navigation (PPN) guidance law with time-varying navigation gains is considered. Unlike the conventional optimal PPN approach where linearized model was assumed in the optimization process, this work exploits the exact nonlinear formulation of PPN to derive analytically the optimal time trajectory of the navigation gain to minimize a performance index which is a weighted sum of the final time and the integral of the squared acceleration. It is verified that the PPN scheme with constant navigation gain is not only optimal in the vicinity of the interception point, but also optimal for the whole trajectory, if the navigation constant is designed by the methodology proposed here. Based on the optimization results for nonmaneuvering targets, a recursive optimal PPN scheme is proposed for maneuvering targets, wherein the optimal navigation gain and time-to-go are predicted recursively during the interception, and trajectory and performance of the interceptor guided by optimal recursive PPN scheme are evaluated analytically.     

True proportional navigation with maneuvering target

We analytically obtain the capture regions for the true proportional navigation (TPN) missile guidance law against an intelligently maneuvering target. Two versions of TPN are considered. The first is the original TPN which assumes the commanded lateral acceleration to be directly proportional to the line-of-sight (LOS) rate only, the proportionality factor being an arbitrary constant or dependent only on the initial closing velocity. The other, known as RTPN (realistic TPN), assumes the commanded lateral acceleration to be directly proportional to the LOS rate and also the current closing velocity. The target is assumed to maneuver in such a way as to increase the LOS rate and thus directly oppose the proportional navigation (PN) philosophy of annulling the LOS rate. A necessary and sufficient condition for capture is derived for the original TPN, and using it, the exact capture region is obtained. A sufficient condition for capture is derived for RTPN and is used to obtain its capture region partially. Some necessary conditions for capture are also derived for RTPN and are used to obtain an upper bound on its complete capture region. Using these conditions some important results on the existence of capture regions and a comparative study of capturability of TPN laws are also presented     

某型战机追踪导引研究

基于对广泛应用与研究的各种比例导引律的比较与分析，提出应用纯比例导引实现战机追踪导引的方法，并通过追踪以不同机动方式逃逸的飞行目标的仿真，说明了它的正确性与可行性。     

基于纯比例导引的拦截碰撞角约束制导策略

拦截碰撞角约束制导是当前导弹制导研究的关键问题之一。首先基于理想比例导引（IPN）律拦截非机动目标的解析解,推导了纯比例导引律（PPN）拦截固定目标的解析解,得到了弹目相对距离、制导指令加速度和导弹前置角的显示表达式,并进一步得到了拦截碰撞角与弹目相对运动状态和比例导引系数之间的解析表达式。其次,基于该解析表达式,提出了基于PPN的拦截碰撞角约束制导策略（PPNIACG）,并探讨了在铅垂面内进行落角约束打击和水平面内进行拦截碰撞角约束打击的2种实现方式。最后,以弹道成型制导律（TSG）和最优碰撞角约束制导律（OIACG）为参考,通过数值仿真算例,对PPNIAC的拦截性能进行了对比分析,验证了所提出制导策略的有效性和正确性。     

径向基神经网络在优化导引律中的应用

应用神经网络的非线性的特点,在纯比例导引律的导引下,分析战机的运动轨迹,通过采样取得的一系列载机的控制输入数据,根据追踪导引控制的内在要求(目标视线角速度的绝对值减小并趋于零),设计载机导引控制器的相应的离散输出值,离线训练一个径向基神经网络模块,嵌入到载机控制回路中,作为战机导引的控制输入,实现载机在纯比例导引下的性能优化;仿真结果表明,在节能及省时方面,能达优化纯比例导引律的目的。     

一种攻击大机动目标的变参数组合导引律

针对目标作大机动时的导引问题,提出了一种既简单又有效的比例加变结构组合导引律,给出了其设计机理,并在理论上对变参数进行了优化设计。通过仿真结果比较,验证了此组合导引律可以在固定时间内明显地减少追踪者的脱靶量。     

空间拦截修正比例导引律研究

基于追踪导引律和比例导引律,提出了一种新的空间拦截末制导律。通过调整两种导引法的系数决定其在导引法中的权重。考虑空间末段拦截的具体情况,在相对运动方程进行线性化处理的基础上可以得出运动方程的解析解。将非线性方程得出的数值解和这种导引律的解析解相比,仿真表明这种方法的解析解与数值解一致,且避免了最优制导复杂的解析结果。     

异构多导弹系统自适应分布式协同制导

针对异构多导弹系统的分布式协同制导问题,在传统比例导引律的基础上设计出一类领航跟随分布式协同制导律。运用代数图论、分布式网络同步原理以及非线性系统一致性理论,领弹采用基于固定系数的比例导引律,从弹采用基于可变系数的参数自适应比例导引律。该领航-跟随分布式自适应协同制导律可使领弹和从弹实现对目标的同时攻击,且各相邻导弹间仅传输各自的可测状态信息,算法具有较低的通信代价和较好的可扩展性。最后给出了相关数值仿真算例,仿真结果验证了控制算法的有效性。     

Design of Time-constrained Guidance Laws via Virtual Leader Approach

Guidance problems with flight time constraints are considered in this article. A new virtual leader scheme is used for design of guidance laws with time constraints. The core idea of this scheme is to adopt a virtual leader for real missiles to convert a guidance problem with time constraints to a nonlinear tracking problem, thereby making it possible to settle the problem with a variety of control methods. A novel time-constrained guidance (TCG) law, which can control the flight time of missiles to a prescribed time, is designed by using the virtual leader scheme and stability method. The TCG law is a combination of the well-known proportional navigation guidance(PNG) law and the feedback of flight time error. What's more, this law is free of singularities and hence yields better performances in comparison with optimal guidance laws with time constraints. Nonlinear simulations demonstrate the effectiveness of the proposed law.     

面向协同standoff跟踪问题的无人机制导律

针对多无人机（UAV）协同standoff跟踪问题,提出了UAV的横侧向和纵向制导律。对参考点制导（RPG）进行改进,作为UAV的横侧向制导律。然后,采用一组非线性微分方程对UAV和目标相对距离的调节过程进行建模,在此基础上证明了改进RPG的渐近稳定性,并推导了RPG参数与系统性能的关系,为RPG参数的选取提供了依据。最后,给出了UAV的纵向制导律,并分析了其渐近稳定性。仿真结果表明,改进RPG的跟踪误差和时间乘以误差绝对值积分（ITAE）指标均优于Lyapunov向量场制导（LVFG）和模型预测控制（MPC）,故改进RPG具有更快的响应速度和更高的稳态精度。     

Generalized guidance law for homing missiles

The concept of a generalized guidance law is presented, and the closed-form solution for a homing missile pursuing a maneuvering target according to generalized guidance laws is given. It is shown that the guidance laws appearing in the literature are merely special cases of the one proposed by the authors. The derived generalized forms of capture area, missile acceleration, and homing time duration that are derived provide insight into the performance of the guidance laws being considered and lead to the discovery of new ones. The problem of finding a nonlinear optimal guidance law for a homing missile with controlled acceleration, applied so as to capture a maneuvering target with a predetermined trajectory while minimizing a weighted linear combination of time of capture and energy expenditures, is solved in closed form. The derived optimal control law is equal to the LOS (line of sight) rate multiplied by a trigonometric function of the aspect angle. Numerical simulation shows that the resulting guidance law appears to yield a significant advantage over true proportional navigation     

Impact time control using biased proportional navigation for missiles with varying velocity

A feasible guidance scheme with impact time constraint is proposed for attacking a stationary target by missiles with time-varying velocity. The main idea is to replace the constant velocity with the future mean velocity; therefore, the existing time-to-go estimation algorithm of the proportional navigation guidance law can be improved to adapt to varying conditions. In order to obtain the prediction of the velocity profile, the velocity differential equation to the downrange is derived, which can be numerically integrated between the current downrange and the target position by the on-board computer. Then, a third-order polynomial is introduced to fit the velocity profile in order to calculate the future mean velocity. At the beginning of each guidance loop, the future mean velocity is predicted and the time-to-go information is updated, based on which a novel biased proportional navigation guidance law is established to achieve the impact time constraint. Finally, numerical simulation results verified the effectiveness of the time-to-go estimation algorithm and the proposed law.     

Guidance law with impact time and impact angle constraints

A novel closed-form guidance law with impact time and impact angle constraints is pro- posed for salvo attack of anti-ship missiles, which employs missile’s normal acceleration (not jerk) as the control command directly. Firstly, the impact time control problem is formulated as tracking the designated time-to-go (the difference between the designated impact time and the current flight time) for the actual time-to-go of missile, and the impact angle control problem is formulated as tracking the designated heading angle for the actual heading angle of missile. Secondly, a biased proportional navigation guidance (BPNG) law with designated heading angle constraint is constructed, and the actual time-to-go estimation for this BPNG is derived analytically by solving the system differential equations. Thirdly, by adding a feedback control to this constructed BPNG to eliminate the time-to-go errorthe difference between the standard time-to-go and the actual time-to-go, a guidance law with adjustable coefficients to control the impact time and impact angle simultaneously is developed. Finally, simulation results demonstrate the performance and feasibility of the proposed approach.     

Three-dimensional Large Landing Angle Guidance Based on Two-dimensional Guidance Laws

 Both the design process and form of the three-dimensional (3D) suboptimal guidance law (3DSGL) are very complex. Therefore, we propose the use of two-dimensional (2D) guidance laws to meet the guidance requirements of 3D space. By analyzing the relationship between the flight-path angle and its projections on OXY and OXZ planes, we obtain the ideal design requirements of the guidance laws. Based on the requirements, we design a 2D suboptimal guidance law used in the horizontal plane; combining the 2D vertical suboptimal guidance law, we create a whole ballistic simulation of six degree-of-freedom. The results are compared with those using other three guidance modes in the case of large windage of the initial azimuth angle. When the proportional navigation guidance (PNG) law is used in the horizontal planes, the landing angle will obviously decrease. With the proposed guidance mode, the large landing angle can be realized and meet the guidance precision requirements. Moreover, the required overload can decrease to meet the control requirement. The effects of the proposed guidance mode are close to that of 3DSGL despite its very simple form.     

变结构控制在导弹制导中的应用综述

简要介绍了变结构控制理论和导弹制导律，对基于变结构控制的导弹制导律进行了比较详细的研究。重点介绍了各种制导律中滑模面的选择和制导律的设计，并给出了一些仿真结果。研究表明，由于变结构控制的应用．使得制导律具有很强的鲁棒性，比传统的比例导引律有更多的优越性。