Nonlinear optimal missile guidance for stationary target interception with pendulum motion perspective

This study outlines the set of equations constituting the necessary conditions that should be solved to determine the optimal guidance command for a missile to intercept a stationary target along a desired impact direction at a prespecified final time. Unlike the earlier studies on nonlinear optimal guidance problems, the present study formalises the optimal control problem with both final time and final state fixed. The pure control effort quadratic norm is considered as the performance index to be minimised. A noticeable finding from the study of the necessary conditions is that the flight path angle of the optimal trajectory obeys the simple pendulum dynamics. Full characterisation of the exact optimal solution requires numerically solving a set of four nonlinear algebraic equations with respect to four unknowns

Three-dimensional guidance method with course modification for altitude shaping in endoatmospheric interception

This study presents a three-dimensional guidance law for the interception of an endoatmospheric target. The proposed method takes an empirical design approach which first specifies the structure of the lateral acceleration command as that of a linear optimal guidance law for zero-effort-miss nullifcation. Then, the direction of pursuit and the guidance gain are designed in accordance with the physical understandings of the motion characteristics of an aerodynamically-controlled interceptor. More specifically, the proposed method induces an intentional increase in the flight altitude around the initial phase while respecting the maximum altitude constraint, all of which are realised through modification of the desired flight path angle in the vertical plane. The proposed guidance method does not rely on explicit definition of design elements such as engagement planes, guidance phases, complicated time-to-go estimation, and waypoints. Moreover, the proposed design approach of modifying the desired course based on the collision courses naturally facilitates smooth handover to the terminal phase near the collision condition. Numerical simulation shows that the proposed guidance method is effective in intercepting a nonmanoeuvring target over a wide range of engagement conditions to the target in comparison to the existing guidance laws developed for homing and midcourse flight

Generalized formulation of linear nonquadratic weighted optimal error shaping guidance laws

This study presents a novel extension to the theory of optimal guidance laws represented by the nontraditional class of performance indices: nonquadratic-type signal Lp" role="presentation">Lp norm for the input weighted by an arbitrary positive function. Various missile guidance problems are generally formulated into a scalar terminal control problem based on the understanding of the predictor–corrector nature. Then, a new approach to derive the optimal feedback law, minimizing the nonquadratic performance index, is proposed by using the Hölderian inequality. The proposed extension allows a more general family of formulations for the design of closed-form feedback solutions to various guidance problems to be treated in a unified framework. The equivalence between the proposed approach and other design methodologies is investigated. In general, the type of input norm mainly determines the variability of input during the engagement while trading off against the rate of error convergence. The analytic solution derived in this study is verified by comparison with the solution from numerical optimization, and the effect of the exponent p" role="presentation">p in the performance index on the trajectory and command is demonstrated by numerical simulations

Inverse optimality of pure proportional navigation guidance for stationary targets

The main contribution of this study is the optimality analysis of the PPNG performed in full generality. The new theoretical findings can explain the result of the former analysis in which the PPNG is derived as the minimum effort solution [5] and also describe a comprehensive design framework including the observability-enhanced guidance laws developed for the dual homing guidance problem. Furthermore, this study provides several examples illustrating how the PPNG with various navigation gain functions can be understood as optimal control solutions

Look-angle-constrained control of arrival time with exact knowledge of time-to-go

The capability to control the time of arrival at a goal position as desired endows a single vehicle or a coalition of many of them with the strategic advantage to perform time-critical missions. Arrival time coordination can be used as an element to solve multi-agent, multidepot routing and task planning problems in cooperative unmanned aerial robots. The tactic known as Salvo, which either designates or synchronizes the impact times across multiple missiles to enhance their collective survivability as well as attack effectiveness, strongly depends on control of arrival time. In principle, control of arrival time is essentially adjustment of the arc length of the vehicle’s flight path through manipulation of the curvature, provided that most vehicles flying in the atmosphere often prefer not to change their speeds excessively. On the other hand, the capability to take measurements of the target with onboard sensors provides a higher degree of autonomy to the vehicle and hence allows a more intelligent behavior. Modern autonomous vehicles acquire information about the designated destination or the surrounding environment with imaging sensors, in particular. An onboard sensor that collects emission or reflection from the target is usually not likely to be omni-directional yet possesses only a finite field-of-regard. The requirement to ensure continuous acquisition of target-originated signals necessitates a measure to keep the information source inside the sensor’s field of view that spans over a solid angle of limited range. That is, a box constraint is imposed on the look angle

Approximation of achievable robustness limit based on sensitivity inversion

Introduction: The sensitivity function, defined as the closed-loop transfer function from the exogenous input to the tracking error, is central to the multi-objective design and analysis of a feedback control system. Its frequency response determines many performance characteristics of the closed-loop system, such as disturbance attenuation, reference tracking, and robustness against uncertainties and noise. It is well known that the nominal sensitivity peak, i.e., the H∞ -norm of the sensitivity function, is a direct measure of stability robustness, because the sensitivity magnitude quantifies both the attenuation of the effect of external disturbances on the closed-loop output and the variations of the closed-loop system with respect to the plant perturbations

Composite model reference adaptive control with parameter convergence under finite excitation

A new parameter estimation method is proposed in the framework of composite model reference adaptive control for improved parameter convergence without persistent excitation. The regressor filtering scheme is adopted to perform the parameter estimation with signals that can be obtained easily. A new framework for residual signal construction is proposed. The incoming data is first accumulated to build the information matrix, and then its quality is evaluated with respect to a chosen measure to select and store the best one. The information matrix is built to have full rank after sufficient but not persistent excitation. In this way, the exponential convergence of both tracking error and parameter estimation error can be guaranteed without persistent oscillation in the external command which drives the system. Numerical simulations are performed to verify the theoretical findings and to demonstrate the advantages of the proposed adaptation law over the standard direct adaptation law

Analysis of guidance laws with non-monotonic line-of-sight rate convergence

This study presents analyses of guidance laws that involve non-monotonic convergence in heading error from a new perspective based on an advanced stability concept. Pure proportional navigation with range-varying navigation gain is considered, and the gain condition to guarantee asymptotic convergence to the collision course is investigated while allowing the heading error to exhibit patterns that involve intermediate diversion. The extended stability criterion considered in this study allows local increase of the function in some finite intervals, which is less conservative than the standard stability theorem. The existing guidance laws involving intentional modulation of the heading error as well as the design of the navigation gain are discussed with respect to the new stability criterion

Analytic approach to impact time guidance with look angle constraint using exact time-to-go solution

This paper proposes an analytic approach for impact time control guidance laws against stationary targets using biased proportional navigation. The proposed guidance scheme realizes the impact time control in two different ways: the first approach directly uses the exact time-to-go error to satisfy both the impact time control and the field-of-view constraint, while the second approach adopts a look angle tracking law to indirectly control the impact time, with the reference profile of the look angle generated using the exact time-to-go solution. The stability properties of the proposed guidance laws are discussed, and numerical simulations are carried out to evaluate their performance in terms of accuracy and efficiency

Data-efficient active weighting algorithm for composite adaptive control systems

We propose an active weighting algorithm for composite adaptive control to reduce the state and estimate errors while maintaining the estimation quality. Unlike previous studies that construct the composite term by simply stacking, removing, and pausing observed data, the proposed method efficiently utilizes the data by providing a theoretical set of weights for observations that can actively manipulate the composite term to have desired characteristics. As an example, a convex optimization formulation is provided, which maximizes the minimum eigenvalue while keeping other constraints, and an illustrative numerical simulation is also presented