Missile Guidance Problem Research Articles

Linear Pseudospectral Method with Chebyshev Collocation for Optimal Control Problems with Unspecified Terminal Time

In this paper, a linear Chebyshev pseudospectral method (LCPM) is proposed to solve the nonlinear optimal control problems (OCPs) with hard terminal constraints and unspecified final time, which uses Chebyshev collocation scheme and quasi-linearization. First, Taylor expansion around the nonlinear differential equations of the system is used to obtain a set of linear perturbation equations. Second, the first-order necessary conditions for OCPs with these linear equations and unspecified terminal time are derived, which provide the successive correction formulas of control and terminal time. Traditionally, these formulas are linear time varying and cannot be solved in an analytical manner. Third, Lagrange interpolation, whose supporting points are orthogonal Chebyshev–Gauss–Lobatto (CGL), is employed to discretize the resulting problem. Therefore, a series of analytical correction formulas are successfully derived in approximating polynomial space. It should be noted that Chebyshev approximation is close to the best polynomial approximation, and CGL points can be solved in closed form. Finally, LCPM is applied to the air-to-ground missile guidance problem. The simulation results show that it has high computational efficiency and convergence rate. A comparison with the other typical OCP solvers is provided to verify the optimality of the proposed algorithm. In addition, the results of Monte Carlo simulations are presented, which show that the proposed algorithm has strong robustness and stability. Therefore, the proposed method has potential to be onboard application.

Optimal guidance method based on conjugate gradient method and pseudospectral method

This paper aims to propose a method to solve the nonlinear optimal guidance problems with various performance indexes. First, the conjugate gradient method is derived based on the performance indexes of general form and specified form, and the execution steps of the optimal guidance method are obtained. Secondly, the pseudospectral collocation scheme is used to solve some intermediate variables in the algorithm, and the differential equations are transformed into algebraic equations, so as to improve the computational efficiency. Finally, the proposed method is applied to the air-to-ground missile guidance problem. By comparing with the simulation results of GPOPS-II, the optimality of the proposed method is verified. Additionally, the high computational efficiency of the algorithm is also verified.

Optimal Trajectory Design Accounting for Robust Stability of Path-Following Controller

This study presents an optimal control framework that designs an optimal trajectory while additionally accounting for the stability of a path-deviation error controller that is dependent on the states provided by the optimal trajectory. Thus, instead of designing the controller independent of the trajectory, the stability is enforced as part of the optimization constraints in connection with the optimal trajectory it should follow. This allows to trade off controller and trajectory optimally and thus permits an improved exploitation of the system capabilities. To achieve this goal, the classic trajectory optimization problem formulation is augmented by stability constraints and feedback gains, with the latter being used as optimization parameters. These stabilize the path-following error dynamics by means of gain scheduling and enforce desired response characteristics. Here, the response is shaped, for once, by enforcing limits on the position of the real and imaginary parts of the eigenvalues as well as by limiting their sensitivity. Especially the latter is a way of robustifying the feedback controller in the time domain. The applicability of the proposed method is shown in an aviation-related scenario by solving a missile guidance problem.

Parallel-Structured Newton-Type Guidance by Using Modified Chebyshev–Picard Iteration

This paper presents a parallel-structured Newton-type guidance law for the missile guidance problems. In the proposed approach, the modified Chebyshev–Picard iteration is used to approximate the dynamic equations, by which the continuous-time guidance problem is transcribed into a sequence of discrete-time subproblems, and both serial and parallel computing can be supported. The subproblems can then be iteratively solved by the Newton-type methods. Numerical demonstrations in a 3-D air-to-surface missile guidance problem with both impact-time and impact-angle constraints are provided to illustrate the effectiveness of the proposed guidance law. The simulation results show that the proposed guidance is superior to the state-of-the-art adaptive damped Newton-type guidance and quasi-spectral model predictive static programming in both computational efficiency and accuracy, even before parallel implementation.

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 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 in the performance index on the trajectory and command is demonstrated by numerical simulations.

A novel finite-time disturbance observer-based partial control design: A guidance application

This paper presents a novel technique to design a robust finite-time partial stabilizer, based on the non-singular terminal sliding mode method and a disturbance observer for the missile guidance problem. In the proposed method, the finite-time stability is desired for only a part of the state variables in the guidance system. Accordingly, the guidance system is divided into two subsystems where the finite-time stability is desired only for the first subsystem. Then, a partial diffeomorphism transformation is used to convert the first subsystem into the normal form. Finally, the components of the input vector appearing in the transformed form of the first subsystem are designed to achieve finite-time stability based on a partial disturbance observer. In the proposed finite-time disturbance observer, the disturbances are estimated in a finite time without any knowledge about their upper bounds. Simulation results demonstrate the effectiveness of the designed guidance law to intercept maneuvering and non-maneuvering targets compared to the existing methods.

Smooth Interpolation-Based Fixed-Final-Time Command Generation

This paper proposes a command generation approach based on the smooth interpolation without twist constraints and the input–output relation developed by the generalized model predictive static programming. Commands are efficiently generated to achieve the desired terminal condition at a specified final time in an interpolative manner featuring a noniterative nature. The effectiveness of this approach is demonstrated through three different simulation studies including a double integrator benchmark, a soft lunar landing problem, and a cluster missile guidance problem.

Model Predictive Convex Programming for Constrained Vehicle Guidance

A new model predictive convex programming is proposed in this paper for state and input constrained vehicle guidance design. The proposed method defines a convex optimization framework considering a flexibly designed cost function subject to inequality constraints and a sensitivity relation between state increments and input corrections. This formulated convex optimization problem can be solved in a computationally efficient manner. Simulation studies of nonlinear missile and aircraft landing guidance problems demonstrate the effectiveness of the proposed approach.

Missile Mathematical Model and System Design

Recently, aerospace (flight) engineers, having more solid mathematical backgrounds, have become familiar with the newest results in control theory and are able not only to formulate control problems but also solve them without attracting the attention of control experts. Moreover, they try to apply new results in control theory to specific aerospace and missile guidance problems. As a result, control theory is losing its biggest supplier and, to a certain degree, this slows down its progress.

The design of particle swarm optimization guidance using a line-of-sight evaluation method

In this paper, an improved particle swarm optimization guidance (IPSOG) is proposed. The particle swarm optimization guidance (PSOG) is presented to solve nonlinear and dynamic missile guidance problems. However, the miss distance (MD) tends to be large. The objective function of the relative distance in the PSOG leads the missile to the current position of a target. Therefore, the PSOG is similar to pursuit guidance. In the IPSOG, a new objective function for the PSOG is introduced to improve the guidance performance. The line-of-sight (LOS) rate is taken as the objective function. The fitness function is then evaluated according to the defined objective function. Numerical simulation results show that the guidance performance of the IPSOG is better than the PSOG.

A Predictor Observer for Seeker Delay in the Missile Homing Loop

In this work, we employ a classical predictor observer for linear feedback systems to stabilize a well-known missile guidance law degraded by an optical seeker measurement delay in the near-collision course of a missile-target intercept engagement. In the framework of a canonical, planar intercept missile guidance problem, coupled with a delay in the measurement of the missile-to-target line-of-sight angle, we tailor the general formula for the predictor observer to the guidance law. We then use numerical simulations to evaluate the performance of the delaycompensated guidance law. The results indicate that for the scenario considered, the predictor observer is able to stabilize the guidance kinematics for realistic values of the measurement delay, thus encouraging further feasibility studies for using this approach in practice.

State Constrained Model Predictive Static Programming: A Slack Variable Approach

Abstract An incorporation of state variable inequality constraint for well established Model Predictive Static Programming(MPSP) is proposed in this work. A transformation technique by using slack variables to transform the State Variable Inequality Constraints into an equality constraints, and thereby converting the constraint control problem into an unconstraint one of higher dimension, is incorporated into MPSP. MPSP is a suboptimal control design technique which is applicable for finite horizon nonlinear problems with terminal constraints for class of optimal nonlinear control problems. MPSP is basically a combination of model predictive control and approximate dynamic programming. MPSP is inherently silent on handling the state variable inequality constraint along the trajectory and no augmentation has been proposed till date to handle the same. In this work, we propose an augmentation to handle the state variable inequality constraint in MPSP framework by using slack variable transformation technique. Control problem with state variable inequality constraint is first transformed into an unconstrained problem, then it is formulated in MPSP framework, with q th time derivative of slack variables as control variable. The proposed philosophy, for verification purpose, is tried on air to air missile guidance problem through simulations. State variable inequality constraint on missile is to keep its altitude within certain limit. It is observed from the extensive simulations that State Constrained MPSP works better as compared to MPSP in terms of keeping the altitude of the missile within constraint.

Robust partial control design for non-linear control systems: a guidance application

In this paper, a general approach for robust partial stabilization of uncertain non-linear systems is presented. In this approach, the non-linear dynamic system is divided into two subsystems, called the first and the second subsystems. This division is done based on the required stability properties of the system’s states. The reduced input vector (the vector that includes components of the input vector appearing in the first subsystem) is designed to asymptotically stabilize the first subsystem. The proposed scheme is then applied for designing a guidance law as a potential application. Indeed, the paper presents a new approach to the missile guidance problem and shows that asymptotic stability behaviour is not realistic for all states of the guidance system. The effectiveness of the proposed guidance law in interception of manoeuvring targets is demonstrated analytically and through computer simulations.

Proportional navigation-based collision avoidance for UAVs

A collision avoidance algorithm for unmanned aerial vehicles (UAVs) based on the conventional proportional navigation (PN) guidance law is investigated. The proportional navigation guidance law being applied to a wide range of missile guidance problems is tailored to the collision avoidance of UAVs. This can be accomplished by guiding the relative velocity vector of the aircraft to a vector connecting the current aircraft position to the safety boundary of the target aircraft. Stability of the proposed algorithm is also studied using the circle criterion. The stability condition can be established by choosing the navigation coefficient within a certain bound. The guidance law is extended to 3-dimensional maneuver problems. Inherent simplicity and robustness of the PN guidance law provides satisfactory collision avoidance performance with different initial conditions.

Precision missile guidance using radar/multiple-video sensor fusion via communication channels with bit-rate constraints

We consider a precision missile guidance problem in which the objective is twofold: to come as close as possible to hitting the target, and also to do so from a particular direction. The effectiveness of a guidance law is strongly dependent on the quality of information available to it. In this work we construct a precision guidance law that combines information from an on-board video camera with data transmitted from ground-based radars and video cameras mounted on unmanned aerial vehicles. The communication channels are bit-rate limited, and recent results in control and estimation over finite-data-rate channels are used to construct a nonlinear state estimator. Simulations show the performance improvements which are possible compared to the use of the on-board sensor alone.

Current Trends in Tactical Missile Guidance

The problem of tactical missile guidance is very challenging and has been treated using several basic metlfodologies in the past four decades. Major techniques can be grouped under classical guidance laws, modern guidance laws, l'aws for manoeuvring targets, predictive guidance for endgame scenario, and guidance laws based on intelligent control methods. Each technique has some advantages and disadvantages while implementing in a practical system. Guidance law selection is dictated by nature of flight profile like boost, midcourse, terminal homing, etc, and also miss-distance and a single-shot kill probability. This paper presents a brief survey of the existing techniques and current trends in tactical missile guidance.

Multimodel LQ Pursuer-Evader Game for Missile Guidance

We study a multimodel pursuit evasion LQ differential game. The multimodel concept allows to improve robustness of the designed strategies in the case of presence of some parametric uncertainty. The game solution corresponds to a Nash-equilibrium point of this game. This technique permits to transform the initial game problem, given in Banach space, to a static game given in finite dimensional space. The corresponding numerical procedure is discussed. The obtained weights participate in an extended Coupled Riccati Differential Equation. The effectiveness of the designed controllers is illustrated by the implementation to a two-dimensional Missile Guidance problem.

비례항법을 이용한 무인 항공기의 최적 충돌 회피 기동

Optimal collision avoidance algorithm for unmanned aerial vehicles based on proportional navigation guidance law is investigated this paper. Although proportional navigation guidance law is widely used in missile guidance problems, it can be used in collision avoidance problem by guiding the relative velocity vector to collision avoidance vector. The optimal navigation coefficient can be obtained if an obstacle if an obstacle moves at constant velocity vector. The stability of the proposed algorithm is also investigated. The stability can be obtained by choosing a proper navigation coefficient.

3D Nonlinear H2/H∞ Adaptive Guidance Law for Homing Missiles

This paper explores the effectiveness of nonlinear H2/H∞ optimal guidance design applied to a realistic missile guidance problem with target models. Extended the concept of augmented proportional navigation (APN), a newly-developed technique is applied and solutions are obtained for guidance designs of missile system combine the traditional H2 quadratic performance criterion for noise rejection with the H∞ performance criterion for the maximum robustness against the error of maneuvering target estimated.

Problem of precision missile guidance: LQR and H//sup∞/ control frameworks

Addressed here is the precision missile guidance problem where the successful intercept criterion has been defined in terms of both minimizing the miss distance and controlling the missile body attitude with respect to the target at the terminal point. We show that the H/sup /spl infin// control theory, when suitably modified, provides an effective framework for the precision missile guidance problem. The existence of feedback controllers (guidance laws) is investigated for the case of finite horizon and non-zero initial conditions. Both state feedback and output feedback implementations are explored.