Stochastic optimal control guidance law with bounded acceleration

A stochastic optimal control guidance law is derived for a system with nonlinear dynamics having a bounded acceleration command. In the investigated problem of missile guidance, the kinematics are linear and the measurements Gaussian; however, due to the nonlinear dynamics and the bounded controller, the certainty equivalence principle is not valid. Consequently, the optimal guidance law is obtained by numerically solving the Hamilton-Jacobi equation associated with the stochastic optimization problem. The optimal guidance law depends on the conditional probability density function of the estimated states. Moreover, the guidance law is nonlinear in the estimated zero efiort miss distance and in the missile internal states. It is shown that if rate saturation is present or if the acceleration bound is non-symmetric then a non-zero acceleration command is issued even if the zero efiort miss is zero. The aim of this unusual feature of the guidance law is to position the missile as far as possible from saturation limits, thus placing it in as advantageous situation as possible to deal with expected future target maneuvers.

Optimal impact angle control guidance law and its variants for intercepting a maneuvering target are introduced in this paper. The linear quadratic(LQ) optimal control theory is reviewed first to setup framework of guidance law derivation, called the sweep method. As an example, the inversely weighted time-to-go energy optimal control problem to obtain the optimal impact angle control guidance law for a fixed target is solved via the sweep method. Since this optimal guidance law is not applicable for a moving target due to the angle mismatch at the impact instant, the law is modified to three different biased proportional navigation(PN) laws: the flight path angle control law, the line-of-sight(LOS) angle control law, and the relative flight path angle control law. Effectiveness of the guidance laws are verified via numerical simulations.

An optimal cooperative guidance law for multiple vehicles with weak maneuverability against a superior target is studied in this paper. Firstly, the range of zero-effort-miss distance under the relative-frame-based optimal guidance law is derived by considering the acceleration limitation. Secondly, by adopting the concept of a joint maneuver domain that encompasses the escape area, the reachable sets with bias for each vehicle are devised through the establishment of time-varying virtual aiming points. Then, the guidance command for is obtained by minimum energy consumption optimization. Moreover, the relations among the number of vehicles, the coverage area of the target’s large maneuver and the coefficient of guidance law are analyzed by designing a non-overlapping coverage strategy. Finally, simulation results demonstrate the feasibility and performance of the proposed method with bias term, which can realize the perfect interception of the weak multi-vehicles against the strong maneuvering target.

P ROPORTIONAL navigation (PN) is a popular guidance law because of its robustness, simplicity, and ease of implementation. PN guidance law is capable of intercepting both nonmaneuvering and maneuvering targets [1]. PN is also an optimal control effort guidance law in the linearized setting [2]. Modern tactical guidance laws are designed not only to intercept a target but to satisfy other constraints [3] like impact time, impact angle, and other optimality constraints. These guidance laws can often be represented as being comprising of a PN component and other components designed to satisfy other constraints. As these guidance laws require an estimate of the time-to-go, it is important to develop methods to obtain an accurate estimate of the time-to-go for the PNguidance law. An interesting application requiring the estimation of time-to-go is in salvo attack problems [4], in which several missiles, launched at different times, need to achieve interception of the target simultaneously. Using PN guidance law to achieve this task requires an accurate estimation of time-to-go. Impact time, impact angle, and other optimal guidance laws also need information of the time-to-go to compute the guidance command. One of the simplest ways to estimate time-to-go is by computing range over closing velocity (or the negative of the range rate). The range over range rate estimate is accurate only when the closing velocity is constant and the initial heading error is small. Improved methods for time-to-go estimation by solving a linear quadratic control problem is discussed in York and Pastrick [5]. In Tahk et al. [6], a recursive time-to-go method is discussed in which the timeto-go estimates are updated recursively. The time-to-go estimate is updated in every guidance cycle time and a compensationmethod for the time-to-go error is obtained. The time-to-go estimate improves as the engagement proceeds because the guidance law used reduces the heading error. In Sang et al. [7] the commanded guidance history is considered to obtain the time-to-go estimate. The time-to-go estimate in Jeon et al. [4] is obtained as a function of the initial range and heading error. Our time-to-go estimation method also depends on the initial conditions and hence, our results are compared with Jeon et al. [4]. In this Note, an accurate method to obtain a time-to-go estimate of PN guidance law for a stationary target is proposed. The time-to-go for a stationary target is obtained accurately by a simple interpolation. It is shown that from the time-to-go estimate of a particular initial condition, the time-to-go estimate of any other initial condition can be obtained by a simple but elegant time scaling property of the PN guidance law. The time-to-go estimate as a function of the heading error for the particular initial condition is used as the base solution for estimating the time-to-go for other initial conditions. This method also gives estimates for initial conditions with large initial heading errors; even for cases when the missile is launched with a high heading error. Themethod can be used to obtain time-to-go estimates for any initial condition and navigation gain. The time-to-go estimation is extended to constant velocity targets by a simple iterative process. The time-to-go obtained after the initial iteration is exact and need not be computed in every guidance cycle time.

This paper proposed an optimal time-varying proportional navigation guidance law based on sequential convex programming. The guidance law can achieve the desired impact angle and impact time with look angle and lateral acceleration constraints. By treating the multiconstraints’ guidance problem as an optimization problem and changing the independent variable to linearize the problem and constraints, the original nonlinear and nonconvex problem is transformed into a series of convex optimization problem so that it can be quickly solved by sequential convex programming. Numerical simulations compared to nonlinear programming and traditional analytical guidance law demonstrate the effectiveness and efficiency of the proposed algorithm. Finally, the proposed guidance law is verified to satisfy different impact time periods and impact angle constraints.

An optimal pulsed guidance law with a time-varying weighted quadratic cost function that enables imposing a predetermined intercept angle is presented. Due to the characteristic of impulse force, admissible variance of control is redefined. The optimal pulsed guidance law is deduced via extended maximum principle. The optimal pulsed guidance law is eventually transformed to solve the two-point boundary value problem. To decide a shooting point, an efficient algorithm is proposed by combining particle swarm optimization and Kriging surrogate model method. The optimal pulsed guidance law is implemented in several representative engagements. From simulation results, it can be seen that the proposed guidance law can achieve small miss distance with terminal impact angle constraint under different conditions. Moreover, the performance of the proposed guidance law is satisfactory with the comparison of sliding-mode pulsed guidance law.

The guidance laws are commonly designed to yield as small a miss distance as possible, harmonious with the missile’s acceleration capability. In recent decades, the concept of optimized guidance law is well understood in applications where information concerning the target range and line-of-sight angle is available. Researchers' efforts have been continually made to apply modern control theory to conventional and adaptive autopilot designs, even though the classical theory is still applicable to autopilots. It can be noted that it is desirable to perform a detailed computer-aided feasibility study within the context of a realistic missile-target engagement model. Development and evaluation of guidance and control laws for simplified missile-target engagement scenarios are extended and adapted to the air-to-air missile situation and implemented in a complete three-dimensional engagement model. Thus, this study proposed a computational method for constructing an optimal midcourse guidance law, which is based on the optimal control theory and initial boundary conditions. This proposed guidance law is derived from an optimal control theory with the boundary conditions such as allowed relative distance between missile and target at the final time, low line-of-sight rate. A numerical simulation verifies the performance of this guidance law with the impact of harmonic wind. The simulation results demonstrate that the quality of effectiveness as well as the applicability of this proposed algorithm.

A novel guidance law based on extended quadratic cost function is proposed to avoid losing performance index, which is commonly appeared when using cost function with quadratic form. Stochastic optimal guidance laws under target maneuver and target non-maneuver conditions are deducted and are validated by simulating in turn. The simulation results show that guidance trajectories based on extended quadratic performance index are smoother than those of proportional guidance la w and optimal guidance law based on quadratic performance index. Another advantage of this guidance law can be seen from the comparison of normal overloads of different guidance laws is that the proposed guidance law possesses a better process control and requires lower overload at the end of the attack.

The purpose of this paper is to develop an optimal guidance law that takes into account both the missile acceleration limit and a maximum value allowed for the guidance gain. The maximum value of the guidance gain is dictated mainly by the possible coupling between the missile body motions and the measurement of the inertial target line-of-sight angle rate. The tools for this work are the linear quadratic stochastic Gaussian optimal control theory coupled with the use of random-input describing functions. Although the use of the random-input describing function introduces an approximation in the work, it has been shown in the past that the use of random-input describing functions in miss distance calculations results in good approximations for the predicted miss distance. The paper includes the derivation of the optimal guidance law that accounts for both the missile acceleration limit and the constraint on the guidance gain and a simple one-dimensional Monte Carlo simulation to examine the performance of the guidance law.

Optimal guidance law design for guided munitions based on segmental linear functions

The three-body engagement where a target aircraft protects itself by using a cooperative defender missile to intercept an attacking missile is investigated. It is formulated as a constrained linear quadratic optimal problem. Two different optimal cooperative guidance laws for the target and defender are proposed in two cooperation schemes. Since any control effort to reduce the miss distance to smaller than missile’s lethal radius is wasted, the guidance laws are derived to achieve an upper bound on the missile–defender miss distance. In the two-way cooperation scheme, the target and the defender act as a team. How the target makes a trade-off between aiding the defender and evading the missile is investigated by considering both the missile–target zero-effort miss distance and the control effort into the cost function. Without the penalty weight on the missile–target zero-effort miss distance, the two-way minimum control effort guidance laws are available. In the one-way cooperation scheme, the target uses a known evasion strategy independently. The optimal cooperative guidance law is derived for minimizing the control effort of the defender. Simulation results show that these proposed guidance laws can provide a specified missile–defender miss distance and save the control effort compared with the zero-miss-distance guidance law. Two-way cooperation scheme outperforms one-way cooperation scheme.

Optimal Guidance Law for Impact Angle Control

Simulated navigations of an autonomous underwater vehicle (AUV) achieved by the minimum time guidance within undersea areas of current disturbances are presented. When an AUV has to transit to a given destination within an area of current disturbance, ingenious guidance enables the minimum time navigation. In this study, a numerical solution procedure for an optimal heading guidance law is developed. As the optimal heading reference, the solution of the optimal guidance law is fed to the heading controller. Simulated optimal navigations are realized on the basis of the dynamics of an AUV 'r2D4'. The r2D4 is a deep-ocean-exploring AUV, developed by the Institute of Industrial Science, University of Tokyo. The developed procedure never fails to derive the optimal heading sequence within a finite computational time, provided that the current velocity in the navigation region is known a priori. As a fail-safe strategy in achieving the optimal navigation, the concept of quasi-optimal navigation is presented. The quasi-optimal navigation is implemented by on-site updates of optimal guidances followed by the heading tracking control.

In this paper, two composite inverse optimal guidance laws against maneuvering targets have been proposed and the stability analysis is also presented. The first guidance law is constructed through a combination of Sontag’s formula and first order sliding mode control. In order to mitigating the undesired chattering caused by the discontinuous sign function, another robust guidance law is then synthesized using adaptive super-twisting algorithm. By virtue of the proposed adaptive law, the second guidance law only requires the time derivatives of the lumped uncertainties are bounded, but their upper bounds can be unknown. An important feature of the proposed adaptive algorithm is in non-overestimating the values of the control gains, which is achieved by introducing a ‘detector’ term in the updating law. Finally, some numerical simulations are carried out to demonstrate the effectiveness of the proposed formulations.

Optimal predictive sliding-mode guidance law for intercepting near-space hypersonic maneuvering target