Guidance laws based on a conventional sliding mode ensures only asymptotic convergence. However, convergence to the desired impact angle within a finite time is important in most practical guidance applications. These finite time convergent guidance laws suffer from singularity leading to control saturation. In this paper, guidance laws to intercept targets at a desired impact angle, from any initial heading angle, without exhibiting any singularity, are presented. The desired impact angle, which is defined in terms of a desired line-of-sight angle, is achieved in finite time by selecting the interceptor's lateral acceleration to enforce nonsingular terminal sliding mode on a switching surface designed using nonlinear engagement dynamics. Numerical simulation results are presented to validate the proposed guidance laws for different initial engagement geometries and impact angles. Although the guidance laws are designed for constant speed interceptors, its robustness against the time-varying speed of interceptors is also evaluated through extensive simulation results.

Nonsingular Terminal Sliding Mode Guidance with Impact Angle Constraints

In recent years, many practical nonlinear optimal control problems have been solved by pseudospectral (PS) methods. In particular, the Legendre PS method offers a Covector Mapping Theorem that blurs the distinction between traditional direct and indirect methods for optimal control. In an effort to better understand the PS approach for solving control problems, we present consistency results for nonlinear optimal control problems with mixed state and control constraints. A set of sufficient conditions is proved under which a solution of the discretized optimal control problem converges to the continuous solution. Convergence of the primal variables does not necessarily imply the convergence of the duals. This leads to a clarification of the Covector Mapping Theorem in its relationship to the convergence properties of PS methods and its connections to constraint qualifications. Conditions for the convergence of the duals are described and illustrated. An application of the ideas to the optimal attitude control of NPSAT1, a highly nonlinear spacecraft, shows that the method performs well for real-world problems.

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Connections between the covector mapping theorem and convergence of pseudospectral methods for optimal control

Recent convergence results with pseudospectral methods are exploited to design a robust, multigrid, spectral algorithm for computing optimal controls. The design of the algorithm is based on using the pseudospectral differentiation matrix to locate switches, kinks, corners, and other discontinuities that are typical when solving practical optimal control problems. The concept of pseudospectral knots and Gaussian quadrature rules are used to generate a natural spectral mesh that is dense near the points of interest. Several stopping criteria are developed based on new error-estimation formulas and Jackson's theorem. The sequence is terminated when all of the convergence criteria are satisfied. Numerical examples demonstrate the key concepts proposed in the design of the spectral algorithm. Although a vast number of theoretical and algorithmic issues still remain open, this paper advances pseudospectral methods along several new directions and outlines the current theoretical pitfalls in computation and control.

https://www.elissarglobal.com/wp-content/uploads/2012/05/Spectral-Algorithm-for-Pseudospectral-Methods-in-Optimal-Control.pdf

Spectral Algorithm for Pseudospectral Methods in Optimal Control

The nonlinear impact angle control problem of a physically constrained pursuer against a stationary ground target is solved through the use of a two-phased proportional navigation guidance scheme. The look angle and acceleration constraints are handled by numerically solving for the required values of the midcourse and terminal navigation gains. Different formulations are presented for three cases. The first one is the unrealistic case of constant velocity included for demonstration purposes. The second one is the more realistic case of constant maximum lift coefficient that is likely to suit subsonic ground-to-ground engagements. The last one is the adaptive update case that is intended for pursuers experiencing wide flight regimes. A simple range observer is also introduced to supplement the last case. The performance of the proposed technique is illustrated with simulations.

Switched-Gain Guidance for Impact Angle Control Under Physical Constraints

Impact time control guidance law with field of view constraint

F OR decades, many advanced guidance laws with terminalimpact-angle constraints have been devised to maximize the warhead effect of antiship or antitank missiles and to ensure a high kill probability. The proposed suboptimal guidance with a terminal body-attitude constraint for reentry vehicles in [1] appears to be the first attempt to design an impact-angle-constrained guidance. In [2], an energy-optimal impact-angle control lawwas proposed by solving the linear quadratic optimal control problem with arbitrary missile dynamics. As an extension of this study, the authors also proposed an optimal impact-angle controller that can minimize the time-to-go weighted energy-cost function [3]. Using the Schwartz inequality and differential game theory, terminal-impact-angle-constrained guidance laws for maneuvering targets were developed in [4,5], respectively. To intercept a stationary target with zero terminal acceleration as well as a specified impact angle, the guidance law called the time-to-go polynomial guidance law was suggested in [6], where the acceleration command was assumed initially as a polynomial function of the time-to-go with two unknown coefficients. In [7], a modified proportional navigation (PN) guidancewith a time-varying bias, which is a function of relative range, was proposed using a nonlinear planar engagement and Lyapunov-like function. Although various guidance laws to control the impact angle have been developed so far, most of these laws are difficult to implement, especially for a passive homing missile equipped with an infrared seeker, because an accurate time-to-go estimation or range information is required. The authors of [8–10] proposed two-phase guidance schemes with terminal-angle constraints on the basis of the conventional PN guidance. The guidance scheme suggested in [8] comprises PN guidance with a navigation gainN < 2 for covering all impact angles from 0 to −π and PN with N 2 for intercepting stationary targets with a desired impact angle in surface-to-surface engagements. This guidance scheme was further extended to the case of moving targets in [9]. Using the biased PN (BPN) guidance, the authors of [10] developed a similar two-phase scheme in which the missile follows BPN with a constant bias for the initial homing phase and then switches to PN (i.e., BPN with zero bias) when the integral value of the bias satisfies a certain value calculated from initial engagement conditions and desired impact angle. Because these two-phase guidance schemes only use the line-of-sight (LOS) rate information for the impact-angle control, they can be applied to passive homingmissile systems. However, these guidance schemes have some drawbacks. 1) If the limitation of missile acceleration capability exists, a large miss distance or impact-angle error is generated. 2) A higher look angle, which may result in seeker lock-on failure or instability, is produced in the first-phase guidance. To overcome the drawbacks resulting from the look angle and acceleration limits, we first propose a bias-shaping method based on the two-phase BPNguidance scheme of [10], which can achieve both the terminal-angle constraint and look-angle limitation to maintain the seeker lock-on condition. Next, we investigate analytically the guidance performance of BPN with the proposed bias-shaping method in consideration of the limited acceleration capability. The proposed bias-shaping method consists of two time-varying biases and switching logic similar to the proposed logic of [11] and only requires the LOS rate information to generate the guidance command, thereby easily implementing the proposed law in practical passive homing missiles.

Bias-Shaping Method for Biased Proportional Navigation with Terminal-Angle Constraint

A new optimal guidance problem with impact angle constraint and seeker’s field-of-view limits is investigated for a missile with a strapdown seeker. Impact angle control to satisfy the terminal fli...

Optimal impact angle control guidance law considering the seeker’s field-of-view limits:

In this paper, an impact angle control guidance law, which considers simultaneously the impact angle and seeker's look angle constraints, is proposed for a constant speed missile against a stationary target. An optimal control theory with state variable inequality constraint is used to design the guidance law, for which a control energy performance index with the weighting function of the range-to-go is minimized. Various forms of guidance and trajectory shaping are possible by selecting a proper gain of the weighting function. To handle the seeker's look angle limits when the missile trajectory is highly curved by controlling the impact angle, the proposed guidance law generates three types of acceleration commands as the guidance phases: the first acceleration command for an initial guidance phase makes an initial seeker's look angle reach the maximum look angle; the second one for a midguidance phase maintains the maximum look angle; the final one for a terminal guidance phase intercepts the target with the desired impact angle. The performance of the proposed guidance law was investigated with nonlinear simulations for various engagement conditions.

Range-to-go weighted optimal guidance with impact angle constraint and seeker's look angle limits

A biased proportional navigation guidance law, considering the seeker's look angle and acceleration capability limits, for impact-angle-control against nonmaneuvering targets is proposed. The proposed law consists of two time-varying biases: One is to achieve the desired impact angle within the command limit; the other is to keep the look angle constant for not exceeding the boundary value. Under physical constraints, the maximum achievable impact angle is obtained for practical implementation. The performance of the proposed law is demonstrated through nonlinear simulations.

Biased PNG With Terminal-Angle Constraint for Intercepting Nonmaneuvering Targets Under Physical Constraints

I N ADVANCED guidance law designs, impact angle control has been widely required to maximize the effect of thewarhead and to achieve a high kill probability. Since the first paper for impact angle control [1] was published, to the best of our knowledge, many guidance laws with impact angle constraint have been studied. Ryoo et al. [2] proposed a pure energy optimal guidance law with impact angle constraint. As an extension of this work, Ryoo et al. [3] proposed an optimal impact angle control guidance law minimizing the time-to-go weighted energy cost function. Lu et al. [4] developed three-dimensional guidance laws, which were based on proportional navigation (PN) with adaptive guidance parameters, to achieve impact angle requirements. Ratnoo and Ghose proposed a two-phase guidance law for capturing all possible impact angles against a stationary target [5] and a nonstationary nonmaneuvering target [6]. The proposed guidance law used PN with N < 2 for the initial phase to cover impact angles from zero to −π through an orientation trajectory and PNwithN ≥ 2 for the final phase to intercept the target with the desired impact angle. Erer andMerttopcuoglu [7] proposed a similar two-phase guidance law switching from biased PN to PN when the integral value of the bias met a certain value determined by engagement conditions. Most of research mentioned has focused on the impact angle as well as a zero terminal miss distance. The impact angle control, however, made the missile trajectory highly curved, which might have then missed the target within the seeker’s lookangle limit. Since this leads to the mission failure, it is very important to consider the missile’s physical constraints, such as the seeker’s look-angle and maximum acceleration limits. Recently, studies considering the look-angle limits as well as the impact angle have been carried out. Park et al. [8] proposed a composite guidance law comprising PN with N 1 to maintain the constant look angle and PNwithN ≥ 2 to intercept the target with the desired impact angle. In [9], the error feedback loop of the look angle was included in PN with N 1 for robustness and converging from the arbitrary look angle to the desired one. In [10], a pure energy optimal guidance lawwith an impact angle constraint and the seeker’s look-angle limit was developed using optimal control theory with state variable inequality constraint. Kim et al. [11] proposed a biasshaping method, based on the work in [7], to consider the look-angle and acceleration limits. Tekin and Erer [12] presented a two-phase guidance law with a numerical process for calculating navigation gains to handle the look-angle limits and acceleration constraints. Also, Erer et al. [13] proposed another two-phase guidance scheme to address the look-angle constraint problem, which can select an initial phase guidance lawbetween PNandbiasedPN.Ratnoo [14] dealt with a similar problem to [8] and showed analytic guarantees for achieving all impact angles with any finite field-of-view (FOV) limit. These works have been studied against stationary targets, so the impact angle errors may appear at the instant of interception if the guidance laws presented in [8–14] are applied to the case of moving targets. In this Note, a composite guidance scheme, studied in [8,9], is extended to the case of a nonmaneuvering moving target. The proposed guidance scheme is composed of modified deviated pure pursuit (DPP) with the error feedback loop of the look angle for the initial guidance phase and PN with N ≥ 3 for the final guidance phase: the first phase is to maintain the constant look angle of the seeker, and the second is to intercept themoving targetwith a terminal angle constraint. The switching of guidance phases occurs when satisfying a specific line-of-sight (LOS) angle determined by engagement conditions. Guidelines on the gain tuning of modified DPP and calculation of themaximumachievable impact angle, which is calculated by taking into account the seeker’s look-angle and maximum acceleration limits, are also investigated for guidance designers.

Bong-Gyun Park

Papers

Bias-Shaping Method for Biased Proportional Navigation with Terminal-Angle Constraint

Optimal impact angle control guidance law considering the seeker’s field-of-view limits:

Range-to-go weighted optimal guidance with impact angle constraint and seeker's look angle limits

Biased PNG With Terminal-Angle Constraint for Intercepting Nonmaneuvering Targets Under Physical Constraints

Composite Guidance Scheme for Impact Angle Control Against a Nonmaneuvering Moving Target