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     

Capturability of 3D RTPN guidance law against true-arbitrarily maneuvering target with maneuverability limitation

The capturability of the Three-Dimensional(3D) Realistic True Proportional Navigation(RTPN) guidance law is thoroughly analyzed. The true-arbitrarily maneuvering target is considered, which maneuvers along an arbitrary direction in 3D space with an arbitrary but upperbounded acceleration. The whole nonlinear relative kinematics between the interceptor and target is taken into account. First, the upper-bound of commanded acceleration of 3D RTPN is deduced,using a novel Lyapunov-like approach. Sec...     

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     

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     

Pure proportional navigation against time-varying target manoeuvres

Capturability of the pure proportional navigation (PPN) guidance law against a target executing bounded piecewise continuous time-varying manoeuvres is investigated. A qualitative analysis is carried out to obtain a set of sufficient conditions for capture defined on the engagement parameters and initial conditions. These conditions are significantly less restrictive than the ones obtained previously by others using the Lyapunov method. It is shown that the actual capture region for time-varying target manoeuvres, obtained by using the conditions derived, is much larger than that obtained from the Lyapunov technique. We also show that though a bounded tine varying target manoeuvre does change the constant target manoeuvre capture region to some extent, it does not reduce it drastically. Further, we show that the worst case capture region is obtained when the target executes a constant manoeuvre equal to the bound on the manoeuvre level. Some bounds on the missile lateral acceleration are also obtained for certain regions in the engagement plane. These results are generalizations and extensions of existing results on the capturability of the PPN guidance law against targets executing constant or time-varying target manoeuvres     

考虑运动耦合的BTT导弹三维制导律设计

针对倾斜转弯(BTT)导弹制导过程中的双通道耦合问题,设计了一种新型三维(3D)制导律。首先,通过矢量描述的方式,将导弹运动学模型分解成俯冲、转弯和耦合3项,俯冲和转弯两项在笛卡儿坐标系下描述,耦合项则在SO(3)空间中描述。然后,针对无终端约束和有终端约束情况,分别采用比例控制和SO(3)群上的广义比例-微分(PD)控制方式进行了相应的制导律设计。所得制导律既克服了李群方法的结构复杂性,又避免了解耦方法的信息损失。仿真表明,所设计的制导律能够满足BTT导弹精确制导的要求。     

Approach and landing guidance design for reusable launch vehicle using multiple sliding surfaces technique

An autonomous approach and landing (A&L) guidance law is presented in this paper for landing an unpowered reusable launch vehicle (RLV) at the designated runway touchdown.Considering the full nonlinear point-mass dynamics,a guidance scheme is developed in threedimensional space.In order to guarantee a successful A&L movement,the multiple sliding surfaces guidance (MSSG) technique is applied to derive the closed-loop guidance law,which stems from higher order sliding mode control theory and has advantage in the finite time reaching property.The global stability of the proposed guidance approach is proved by the Lyapunov-based method.The designed guidance law can generate new trajectories on-line without any specific requirement on off-line analysis except for the information on the boundary conditions of the A&L phase and instantaneous states of the RLV.Therefore,the designed guidance law is flexible enough to target different touchdown points on the runway and is capable of dealing with large initial condition errors resulted from the previous flight phase.Finally,simulation results show the effectiveness of the proposed guidance law in different scenarios.     

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。     

Adaptive nonlinear guidance law considering control loop dynamics

A new adaptive nonlinear guidance law is proposed here. The fourth order state equation for integrated guidance and control loop is formulated taking into consideration the target uncertainties and control loop dynamics. The state equation is further changed into the normal form by nonlinear coordinate transformation. Using the normal form of state equation, an adaptive nonlinear guidance law is proposed to compensate for the uncertainties in both target acceleration and control loop dynamics. The proposed law adopts the sliding mode control approach with adaptation for unknown bound of uncertainties. The present approach can effectively solve the existing guidance problem against target maneuver and the limited performance of control loop. We have provided the stability analyses and performed simulations comparing favorably our approach to the state of the art.     

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     

A finite-time 3D guidance law based on fixed-time convergence disturbance observer

In order to achieve accurate interception of high-speed maneuvering targets, this paper presents a relative Line-of-Sight (LOS) velocity based finite-time three-dimensional guidance law design framework, and discusses the application of fixed-time convergence disturbance observer in this framework. Firstly, a simple Lyapunov function is provided to show that the coupled terms in the relative kinematics can be ignored in the proposed guidance law design framework. Secondly, the realizations of several classical guidance laws are analyzed with the proposed framework, including TPN guidance law, finite-time Input-to-State Stability (ISS) guidance law, and sliding mode guidance law. Thirdly, fixed-time convergence disturbance observers are introduced to design the composite finite-time 3D guidance law, and Lyapunov method is employed to show the stability of the guidance system. Numerical simulations with different scenarios show that the proposed generalized guidance law performs high interception accuracy.     

LQG 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 missile's acceleration saturation.     

助推段防御比例导引法研究

基于助推段防御和比例导引法,分析了拦截弹与目标导弹的相对运动关系并给出了比例导引方程;采用差分法分析了拦截弹与目标导弹的位置关系、构建了数学模型并推导了拦截弹的差分方程;以拦截时间和拦截弹道为研究对象,分析了拦截弹的速度和比例系数对拦截时间和拦截弹道的影响,并对比例导引法的拦截弹道进行了三维仿真。仿真结果表明,助推段防御使用变系数比例导引法,可有效缓解拦截时间与拦截弹机动性之间的矛盾。     

带有双闭环滤波器的有限时间稳定变结构制导律

 在末制导过程中,为了取得较高的命中精度,制导律必须使视线(LOS)角速率较快收敛。为了提高制导律的性能,在假设未知不确定时变项有界的前提下,对带有双闭环滤波器的变结构制导律进行了研究。提出的变结构制导律能够使视线角速率及其一阶导数在有限时间内收敛至滑动模态域内,得到更加精确的等效控制并有效克服干扰的影响。通过引入高阶滑动模态,对有关定理进行了重新证明,简化了证明过程,加强了定理的结论,弱化了定理成立的条件。最后利用数值仿真验证了所研究方法的有效性。     

带有引诱角色的多飞行器协同最优制导方法

考虑我方高价值飞行器面临对方拦截器拦截时,发射两枚防御器对拦截器进行拦截的情形。针对对方拦截器采用扩展比例导引律,在4个飞行器均具有一阶线性动力学特性假设下,基于最优控制理论设计了能在拦截末端施加相对拦截角的显式协同制导律。显式协同制导律将高价值飞行器和两枚防御器三者的协同考虑在内,给出了三者最优控制输入的解析解。仿真结果表明,设计的制导律能使两防御器成功拦截敌方拦截器,且在拦截末端施加一个相对拦截角。通过与只考虑两防御器协同的隐式协同制导律进行比较,可发现显式的协同制导律在控制要求和能量消耗上,要优于隐式的协同制导律。此外,还验证了在不同发射条件下的协同制导律的稳定性。     

具有碰撞角约束的三维圆轨迹制导律

 针对再入飞行器带碰撞角约束的导引问题,设计了一种新型三维(3D)制导律。改进并扩展了圆轨迹导引算法,定义了2个圆轨迹跟踪误差变量。通过对导引任务的分析,提出闭环修正导引方法。在此基础上,对再入飞行器制导过程的动力学方程进行解析推导,设计出能适应再入飞行器速度大小变化的三维闭环圆轨迹制导律(3CCGL)。数学仿真结果表明:此制导律能导引再入飞行器沿终端约束方向精确命中目标;同已有算法相比,该制导算法优势明显,其导引的飞行路径短,终端碰撞速度大,并能实现大角度转向攻击,大幅提高再入飞行器的末段机动能力。     

Three-dimensional finite-time cooperative guidance for multiple missiles without radial velocity measurements

In order to simultaneously attack a target with impact angle constraint in threedimensional(3-D) space, a novel distributed cooperative guidance law for multiple missiles under directed communication topologies is proposed without radial velocity measurements. First, based on missiles-target 3-D relative motion equations, the multiple missiles cooperative guidance model with impact angle constraint is constructed. Then, in Line-of-Sight(LOS) direction, based on multiagent system cooperative control theory, one guidance law with directed topologies is designed with strict proof, which can guarantee finite time consensus of multiple missiles' impact times. Next, in elevation direction and azimuth direction of LOS, based on homogeneous system stability theory and integral sliding mode control theory, two guidance laws are proposed respectively with strict proof, which can guarantee LOS angles converge to desired values and LOS angular rates converge to zero in finite time. Finally, the effectiveness of the designed cooperative guidance law is demonstrated through simulation results.     

基于普通分离原理的制导/估计综合设计方法

现有的先进制导律研究常常分离设计最优估计器和最优制导律,但是这种分离设计没有被证明是有效和最优的设计方法.本文基于普通分离原理(GST),提出一种制导/估计综合设计方法.首先,将拦截器反导作战模型考虑为一类非线性、非高斯追逃拦截问题,建立了噪声环境中追逐-逃逸对策模型;其次,引入追逃未达集概念,利用几何方法将估计器设计...     

A novel biased proportional navigation guidance law for close approach phase

A novel biased proportional navigation guidance (BPNG) law is proposed for the close approach phase, which aims to make the spacecraft rendezvous with the target in specific relative range and direction. Firstly, in order to describe the special guidance requirements, the concept of zero effort miss vector is proposed and the dangerous area where there exists collision risk for safety consideration is defined. Secondly, the BPNG, which decouples the range control and direc-tion control, is designed in the line-of-sight (LOS) rotation coordinate system. The theoretical anal-ysis proves that BPNG meets guidance requirements quite well. Thirdly, for the consideration of fuel consumption, the optimal biased proportional navigation guidance (OBPNG) law is derived by solving the Schwartz inequality. Finally, simulation results show that BPNG is effective for the close approach with the ability of evading the dangerous area and OBPNG consumes less fuel compared with BPNG.     

有向拓扑条件下针对机动目标的分布式协同制导律设计

研究了在有向拓扑条件下针对机动目标的分布式协同制导律的设计问题。首先，根据飞行器与目标之间追击过程的几何关系建立制导过程系统模型，针对该模型的非线性问题采用动态反馈线性化方法进行处理。将目标未知机动视为干扰，通过扩张状态观测器进行观测，同时将该估计值用于制导律的设计中，通过直接补偿的方式剔除目标机动对飞行器剩余飞行时间的影响。然后，将所设计的制导律代入到制导模型中，利用一致性分析方法将多飞行器协同制导问题转化为一致性问题，利用极点分析的方法对非一致性子空间的收敛性进行分析，得到协同制导律收敛的充要条件。最后，通过仿真分析的方式对所设计的协同制导律以及制导律的参数选取方法进行分析。