A Lyapunov-like approach to performance analysis of 3-dimensionalpure PNG laws

The 3-dimensional PPNG (pure proportional navigation guidance) law was proposed about forty years ago, but its performance has been analyzed only on the basis of the linearized trajectory equations, which are valid locally around the collision course. We take into full account the nonlinear dynamic characteristics of pursuit situation in the 3-dimensional space to analyze the performance of the 3-dimensional PPNG law more rigorously. We prove that a missile guided by the 3-dimensional PPNG law can always intercept a target maneuvering randomly in the 3-dimensional space if 1) the target acceleration varies with a known bound, 2) the navigation constant is selected large, and either 3a) the initial heading error is small or 3b) the missile keeps the head toward the target during flight. We also propose a modified PNG law, which seems to exhibit better performance at the final phase of pursuit than the conventional PPNG law. We introduce a Lyapunov-like method that proves to be a very powerful tool in obtaining our results     

Closed-form solution of pure proportional navigation

The closed-form solution of the equations of motion of an ideal missile pursuing a nonmaneuvering target according to the pure proportional navigation law is obtained as a function of the polar coordinates for all real navigation constants N⩾2. The solution is given in the form of a uniformly convergent infinite product which reduces to a product of a finite number of factors if the navigation constant is a rational number. The solution is discussed, and necessary and sufficient conditions are stated for vanishing, bounded, and unbounded missile acceleration in the final phase of pursuit     

Empirical virtual sliding target guidance law design: an aerodynamic approach

We present a novel empirical virtual sliding target (VST) guidance law for the midcourse phase of a long range surface-to-air missile that uses the simplicity of the conventional proportional navigation (PN) guidance law while exploiting the aerodynamic characteristics of a missile's flight through the atmosphere to enable the missile to achieve superior performance than that achieved by conventional PN guidance laws. The missile trajectory emulates the trajectory of an optimal control based guidance law formulated on a realistic aerodynamic model of the missile-target engagement. The trajectory of the missile is controlled by controlling the speed of a virtual target that slides towards a predicted intercept point during the midcourse phase. Several sliding schemes, both linear and nonlinear, are proposed and the effect of the variation of the sliding parameters, which control the sliding speed of the virtual target, on the missile performance, are examined through extensive simulations that take into account the atmospheric characteristics as well as limitations on the missile in terms of the energy available and lateral acceleration limits. Launch envelopes for these sliding schemes for approaching and receding targets are also obtained. These results amply demonstrate the superiority of the proposed guidance law over the conventional PN law.     

Optimal guidance for acceleration constrained missile andmaneuvering target

Explicit formulas of optimal guidance laws for an acceleration-constrained, arbitrary-order missile and maneuvering target are derived. These formulas are given in terms of the transfer function and acceleration constraint of the missile and the transfer function/shaping filter of the target. Optimal full-state feedback guidance law is synthesized against a target performing a barrel roll maneuver, and compared with the performance of proportional navigation (PN) for minimum and nonminimum phase missile. Simulation of a third-order missile shows the relative gain from using the full-order guidance law     

A Qualitative Study of Proportional Navigation

This paper represents a study of the trajectories of an ideal missile homing on a target according to the proportional navigation law. A qualitative study is performed and conditions are determined which enable one to demonstrate that: 1) the missile always reaches the target regardless of the initial conditions at launch; 2) the rotational rate of the line of sight is decreasing at the pursuit end.     

三维制导的几何方法与鲁棒控制方法

 在考虑导弹速度和目标速度均为时变的情况下,将微分几何方法与李雅普诺夫稳定理论结合起来,提出了一种导弹三维导引规律设计新方法。对机动目标,PPNG方法的主要问题是在导引末段,它要求导弹在俯仰和偏航通道上具有较大的加速度,为了解决这一问题,本文应用基于李雅普诺夫稳定理论的鲁棒控制方法,提出了一种三维鲁棒制导算法。这种方法在导引末段不需要过大的加速度命令,不需要知道目标精确的加速度和速度方位信息。     

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     

Biased PNG law for impact with angular constraint

A new homing guidance law is proposed to impact a target with a desired attitude angle. It is a variation of the conventional proportional navigation guidance (PNG) law which includes a supplementary time-varying bias. The proposed guidance law does not require a time-to-go estimation and has a simpler form. Analytic conditions for fulfilling the guidance goal are also provided. Simulation results demonstrate that the proposed guidance law has wider launch envelopes than the previous one and shows a good performance even against a maneuvering target     

Proportional Navigation with a Maneuvering Target

A qualitative analysis of the trajectories of a missile pursuing a maneuvering target according to the proportional navigation law is presented. Conditions for a missile to reach the target from any initial state are determined. The existence of a boundary for the required missile acceleration is demonstrated.     

多导弹时间协同制导:一种领弹-被领弹策略

首先,假设各枚导弹的速度为相同常值,领弹采用经典比例导引（PNG）,被领弹采用经典比例导引和机动控制相结合的方式,推导出采用领弹-被领弹策略的多导弹时间协同控制设计模型,该模型实际上描述的是一个非线性的弹目相对运动状态跟踪控制系统。在此系统中,领弹的弹目距离与导弹前置角作为两个参考状态量,被领弹的弹目距离与导弹前置角作为两个待控制的状态量。针对这一弹目相对运动状态跟踪控制系统,采用时标分离的方法设计了期望的慢子系统和快子系统。对这两个子系统分别进行动态逆控制设计,得到了被领弹的机动控制指令。该机动控制指令用于调整被领弹相对目标的运动状态,来逼近领弹相对目标的运动状态,这就保证了所有的导弹能够同时攻击目标。然后,通过为每枚被领弹引入一个与其速度相同的虚拟领弹,将上述方法推广到各枚导弹速度可为不同常值的情况。仿真结果验证了本文方法的有效性。     

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。     

Optimal and Suboptimal Guidance for a Short-Range Homing Missile

Optimal and suboptimal guidance laws for short-range homing missiles are developed and compared to the commonly mechanized quidance law of proportional navigation. The optimal controller is derived as an optimal feedback regulator; the suboptimal controller is an approximation of the optimal regulator and consists of timevarying proportional navigation plus a time-varying gain term times a calculated target acceleration. Monte Carlo studies of the three controllers show that the optimal and suboptimal controllers are much superior to proportional navigation for the case of combined constant target acceleration, line-of-sight rate noise, and missile acceleration saturation.     

Capturability of the 3-dimensional pure PNG law

Although a lot of effort has been recently devoted to the capturability analysis of the pure proportional navigation guidance (PPNG) law, there are still significant discrepancies between the analysis results and the actual capturability of the PPNG law. In this paper, a Lyapunov-like approach along with a vector analysis method based on the so-called LOS (line of sight) plane is introduced to analyze the capturability of the 3-dimensional PPNG law against a target maneuvering arbitrarily with time-varying normal acceleration. Thereby, our analysis result can explain the high capturability of the PPNG law theoretically much better than those presented in the prior literature. In our capturability analysis, the nonlinear dynamics of 3-dimensional pursuit situations have been taken into full account. Our approach is also expected to be useful for the design and analysis of other PPNG-like missile guidance laws     

带有不确定因素的最优拦截制导律

本文从微分对策观点出发,利用极小极大原理讨论了带有不确定因素的最优拦截制导律。在选择特定的过渡时间时,最优拦截制导律具有综合形式。仍由二部分组成:一部分是变系数比例导航,另一部分是与导弹加速度有关的修正项。     

基于H∞ 控制的非线性末制导律设计

 针对三维目标拦截问题,提出一种新的具有强鲁棒性的非线性H_∞末制导律。基于三维弹目相对运动学的非线性关系,将目标机动作为系统扰动,建立了弹目相对运动的数学模型。同时,基于零化弹目视线角速率的思想,提出一种全局非线性H_∞稳定控制策略,得到了连续的非线性末制导律。该方法利用Lyapunov稳定性理论严格证明了制导系统的全局渐近稳定性,并且无需求解哈密尔顿-雅可比-艾萨克斯（HJI）偏微分方程,同时也无需控制弹目相对运动速度。数字仿真表明,和比例导引律相比,这种制导律对高速大机动目标具有很强的鲁棒性和适应性,并能获得良好的制导精度。     

Discrete optimal guidance for high-order missile and maneuvering target

An explicit closed-form formula of a discrete guidance law for a linear time-invariant, acceleration commanded, arbitrary-order missile and a linear, time-invariant arbitrary-order, maneuvering target is derived. The formula is given in terms of the missile and target transfer functions and responses to initial conditions. Examples are explicitly solved for zero-lag and a first-order missile. As an example, it is shown that the discrete proportional navigation ratio is higher than 3 and time varying in contrast to the continuous case.<>     

Saturated super-twisting sliding mode missile guidance

This paper concentrates on developing a missile terminal guidance law against a highly maneuvering target whose maneuvering acceleration is very close to that of the missile or even exceeds the missile normal acceleration in a finite period of time. A new saturated super-twisting algorithm is proposed and applied to the design of missile guidance law. The proposed algorithm has the advantages of simple structure, easy parameter tuning rules and a full utilization of the limit control input. The designed saturated super-twisting sliding mode guidance law is then employed in a missile guidance system. Simulation and its superior performance against strong maneuvering targets is demonstrated.     

用于被动寻的导弹的带末端落角约束的变结构导引律

使被动寻的导弹在较低的平飞弹道情况下，以较小的俯冲过载和最佳攻击姿态精确命中目标；采用带落角约束的变结构导引律，对弹目距离及距离变化率进行估计，克服被动寻的导弹不能测距的约束，同时使切换开关增益随时间自适应变化以减小俯冲过载；利用某型在研被动寻的导弹的气动参数，对此变结构导引律进行了数学仿真，仿真结果表明，在较低的平飞弹道约束条件下及过载要求范围内，该导引律能以期望落角命中静止与机动的坦克目标，对弹目距离变化率及距离的较大估计误差具有较强的鲁棒性。     

捷联寻的制导系统弹体追踪导引方法研究

对捷联寻的系统的简易制导方法—弹体追踪法进行全面剖析 ,解析分析了弹体追踪法对导弹速度、目标速度的约束 ,以及弹体阻尼、滞后、舵偏到攻角之间的传递系数对弹体追踪导引性能的影响。通过数学推导及应用控制理论分析与仿真 ,弹体追踪法较比例导引和速度追踪法在导引特性和打击机动目标时性能较差 ,但其成本较低 ,是一种降低制导武器成本的简易制导方法。但采用弹体追踪法时 ,导弹和目标的速度不能太大 ,当制导武器系统在形成导引误差信号存在固有滞后时 ,弹体稳定回路需要有一定阻尼 ,且舵偏到攻角之间的传递系数不能太大 ,即飞行过程中攻角要较小。     

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.