Nonlinear modified bias proportional navigation guidance law against maneuvering targets

To realize the terminal acceleration constraint for a bias proportional navigation guidance law without usage of switching logics, this paper proposes a modified bias term and presents a terminal acceleration constrained bias proportional navigation guidance law against maneuvering targets. First, a so-called virtual planar coordinates whose origin is attached to the point mass of the target is built, so that the original maneuvering target is transformed to a virtual stationary target. On this basis, the common structure of bias proportional navigation guidance law is presented. To realize the terminal acceleration constraint, a modified bias term related to the relative distance between target and missile is used to improve the bias proportional navigation guidance law. With the virtual look angle and the line-of-sight angle constrained, the proposed modified bias proportional navigation guidance law can intercept the maneuvering targets in a desired attack angle. Comparisons with the optimal guidance law in the linear system are carried out, and the proposed law is proved to be near-optimal. The numerical simulation results demonstrate the all-aspect interception capability of the proposed law against maneuvering targets.     

Three-dimensional event-triggered fixed-time cooperative guidance law against maneuvering target with the constraint of relative impact angles

In order to improve the flexibility and reduce the energy consumption of cooperative guidance laws considering the impact angle constraint, this paper proposes a three-dimensional event-triggered fixed-time cooperative guidance law with the constraint of relative impact angles. First, for the purpose of avoiding the precision degradation due to the estimation error of time-to-go especially facing a maneuvering target, the range-to-go and velocity along the line-of-sight (LOS) are taken as the coordination variables for achieving time-cooperative guidance. Secondly, instead of assigning specific desired impact angles for each missile, only the consensus errors of relative impact angles are utilized as the coordination variables for achieving space-cooperative guidance, which can avoid continually maneuvering for maintaining the constant desired impact angles, thus reducing the fuel consumption. Next, the guidance laws along the LOS and perpendicular to the LOS are developed, and the event-triggering mechanisms are designed to reduce the update frequency of cooperative guidance commands, thus further reducing the energy consumption. To guarantee the convergence rate, the fixed-time control theory is adopted and the stability of proposed event-triggered cooperative guidance laws are rigorously proved. In addition, it is also proved that there is no Zeno behavior when implementing the proposed event-triggered cooperative guidance laws. Finally, numerical simulations indicate that the strictly simultaneous attack is achieved and the constraint of relative impact angles is satisfied. Comparative studies demonstrate that the computation burden of cooperative guidance commands is relaxed and the fuel consumption is reduced by the proposed event-triggered cooperative guidance laws with the constraint of relative impact angles.     

Optimal spatial-temporal cooperative guidance against a maneuvering target

This paper proposes an optimal three-dimensional (3-D) spatial-temporal cooperative guidance (STCG) law for intercepting a maneuvering target with impact angle and time constraints. The guidance problem is studied to achieve spatial cooperation for multi-directional attack in the normal channel and temporal cooperation for simultaneous interception in the tangential channel, respectively. Firstly, the 3-D optimal impact-angle-control guidance (OIACG) is introduced to formulate spatial interception geometry. Based on this law, the relative trajectory length is analytically derived and an accurate time-to-go predictor is formulated against maneuvering targets. In the tangential channel, an optimal temporal cooperative guidance is proposed by leveraging high-dimensional Schwarz inequality method. The proposed algorithm is believed to outperform the existing nonlinear cooperative guidance laws due to its optimality with specific performance index for minimizing the control expenditure. The convergence properties of the proposed STCG law are provided to facilitate its practical implementation. Comparison simulations and application under the realistic pursuer model and target estimation are performed to demonstrate the effectiveness and robustness of the proposed cooperative method.     

Distributed cooperative guidance law for multiple missiles with input delay and topology switching

This paper considers the simultaneous attack of a stationary target by multiple missiles. A novel fixed-time distributed guidance law based on the proportional navigation (PN) guidance law is designed by integrating a consistent control technique into the guidance strategy. This guarantees that the time-to-go of the missile becomes consistent. The guidance law adopts a discrete design, and a compensation item driven by normal acceleration is added to tangential acceleration. This eliminates the potential singularity problem when the heading angle is zero before the consistency is obtained, and thus the multiple missile system still converges in fixed time. In addition, the proposed guidance law can be applied to both undirected and directed graphs. Furthermore, two improved guidance laws are proposed to improve the robustness of the system against adverse effects caused by input delays and topology switching failures and to provide a theoretical proof. Finally, a simulation is used to verify the performance of the distributed guidance law and its robustness against the above failures.     

A modified cooperative proportional navigation guidance law

In Jeon et al. (2010), a cooperative PNG (Proportional Navigation Guidance) law was proposed to achieve the salvo attack using multiple missiles. When the leading angle is near zero, singularity would occur, and the cooperation may fail. In this paper, a modified cooperative guidance law is presented to avoid such singularity existing in the guidance law in Jeon et al. (2010). The time-to-go control efficiency under small leading angle is also improved in this paper. The convergence of times-to-go to the same value using the proposed guidance law is guaranteed by the Lyapunov stability analysis. Simulation results show the effectiveness of the modified guidance law and the improved performance.     

Time-varying sliding mode guidance scheme for maneuvering target interception with impact angle constraint

In this paper, a guidance scheme for impact angle control against maneuvering targets with unknown target acceleration is proposed. In this scheme, the unknown target acceleration is estimated via a linear extended state observer; a novel time-varying global slide mode control technique is presented to eliminate the reaching phase and enforce a desired impact angle exactly at the time of interception with finite-time convergence, good robustness, high precision and smooth guidance command. Moreover, feasible guidance logics are developed to achieve all-aspect interception with the tolerance of large initial heading errors. Numerical simulations in various scenarios are performed to verify the performance of the proposed guidance scheme.     

Distributed observer-based cooperative guidance with appointed impact time and collision avoidance

The problem of cooperative guidance is considered with appointed impact time and collision avoidance for the leader-following flight vehicles, which consist of one leader with the target seeker and the other seeker-less followers. A fixed-time convergent guidance law is presented for the leader to achieve appointed impact time. To guarantee the simultaneous arrival of all the flight vehicles, a cooperative guidance law is proposed to make the follower-leader ranges keep proportional consensus with the range-to-go of leader. A distributed observer is put forward for the followers to estimate the range-to-go of leader. Moreover, the collision avoidance can be reliably fulfilled by the collaborative action of the direction-based and distance-based means.     

Three-dimensional predefined-time impact angle control guidance law with field-of-view limit

In this paper, an impact angle control guidance (IACG) law with predefined convergence time and seeker’s field-of-view (FOV) limit is proposed in three-dimensional (3D) scenario. First, a predefined-time error dynamic is developed whose significance is revealed by comparison with conventional methods. Second, based on coupled engagement dynamics, a 3D predefined-time IACG law is derived by applying the proposed error dynamic. To tackle the FOV limit, two auxiliary functions are introduced into the IACG law. The robustness against disturbances and uncertainties is further improved by utilizing the terminal sliding mode technique. With the proposed guidance law, the impact-angle error can converge to zero exactly at a tunable predefined time. Finally, the effectiveness and performance of the proposed IACG law are shown by several simulations with comparative study.     

Three-dimensional vector guidance law with impact time and angle constraints

The three-dimensional (3D) impact time and angle guidance problem is of great practical significance but remains open because of the coupling nonlinearity and multiple constraints. To solve this problem, a 3D vector guidance law is proposed in this paper to intercept a non-maneuvering target at the desired impact conditions. First, a 3D vector impact angle constrained guidance law with explicit time-to-go estimation is developed by extending the planar one into the 3D space. Then, the intercepting component of the above guidance law is augmented by a time-to-go feedback term, which leads to the proposed 3D vector impact time and angle guidance law. Stability analysis and parameter selection criteria are presented to show the advantageous features of the proposed design. In particular, the proposed guidance law does not require the switch logic, numerical algorithms, or decoupling strategy, which outperforms similar existing results in terms of continuous command and convenient implementation. Finally, several numerical simulations are performed to validate the theoretical findings.     

Terminal angle constraint finite-time guidance law with input saturation and autopilot dynamics

This paper proposes a finite-time command filtered backstepping guidance law (FCFBGL) with the terminal angle constraint while accounting for the input saturation and the autopilot dynamics. To eliminate the adverse effect induced by the filtering errors and the acceleration saturation, a new finite-time error compensation mechanism is integrated in the guidance law design. The proposed FCFBGL not only guarantees the the line-of-sight (LOS) angle error to converge to a small neighborhood of the origin in finite time but also achieves the continuity of the input signal. in finite time. Moreover, with the aid of the fractional power extended state observer (FPESO), the proposed FCFBGL requires no information on the target acceleration and the acceleration derivative of the missile, which is preferable in the practical application. The finite-time stability of the proposed guidance law is derived with the Lyapunov methodology. Simulation results illustrate the effectiveness and superiority of the proposed guidance law.     

Finite-time cooperative circumnavigation control of multiple second-order agents with desired formations

This paper investigates the finite-time cooperative circumnavigation control of multiple second-order agents, in which the agents should surround a moving target with desired formation and circular velocity based on local information. Firstly, the controller design is transformed into design control parameters such that the error system, including distance error, speed error and angle error, is finite-time consensus. The error system is viewed as a cascaded system containing two second-order subsystems, and then a distributed finite-time controller composed of two parts is delivered. The finite-time stability of the entire system is given by employing cascaded control theory. One significant advantage of the proposed controller is that it allows the agents to converge to desired trajectory in a finite time instead of asymptotically. Another merit is that the desired formation is an extensive case and unlimited, including different tracking radii and angular spacing. Furthermore, the proposed controller can be implemented by each agent in its local frame, utilizing only local information. These properties significantly extend the application scope of cooperative circumnavigation. Finally, simulations are carried out to validate the effectiveness of the proposed method.     

Cooperative localization and circumnavigation of multiple targets with bearing-only measurements

This paper studies the multi-target localization and circumnavigation problem for a networked multi-agent system using bearing-only measurements. A more general case that only some of the agents are responsible for measuring the bearing angles with respect to the targets is considered. First, a novel estimator is developed for the agents to locate the targets collaboratively, based on which the geometric center of multi-target is reconstructed by each agent. Then, an estimator-based distributed controller is proposed to steer the agents, such that they can enclose the targets along different circles centered at the geometric center of multi-target with any desired angular spacing. By using Lyapunov stability theory, graph theory and consensus algorithm, global exponential stability of the overall system is analyzed rigorously. Besides, it is proved that bounded angular velocity of each agent and collision avoidance between the target and agent can be guaranteed in the whole movement process. Finally, numerical simulations are given to corroborate the theoretical results.     

Fixed-time leader-following consensus of multiple uncertain nonholonomic systems: An adaptive distributed observer approach

This paper discusses the fixed-time leader-following consensus problem for multiple uncertain nonholonomic systems, which are widely used in engineering models. According to our literature review, either the system is assumed to be known, or the uncertainty only contains state information, which does not meet the actual requirements. For this reason, this paper investigates more general nonholonomic systems with uncertainties driven by inputs and states. First, a fixed-time adaptive distributed observer is proposed to estimate the leader’s state and structural parameters, which ensures that the estimation errors converge to zero within a fixed time. Second, two regulator equations based on the idea of cooperative output regulation are constructed, and a novel observer-based distributed switching control law is proposed. This control law overcomes the nonholonomic constraints and appropriately relaxes the assumptions of uncertain functions in the existing references. Finally, the simulation results verify the effectiveness of the proposed control scheme.     

Fixed-time stability of dynamical systems with impulsive effects

The present article is concerned with the fixed-time stability(FxTS) analysis of the nonlinear dynamical systems with impulsive effects. The novel criteria have been derived to achieve stability of the non-autonomous dynamical system in fixed-time under the effects of stabilizing and destabilizing impulses. The fixed time stability analysis due to the presence of destabilizing impulses in dynamical system, that leads to behavior of perturbing the systems’ stability, have not been addressed much in the existing literature. Therefore, two theorems are constructed here, for stabilizing and destabilizing impulses separately, to estimate the fixed-time convergence precisely by using the concept of Lyapunov functional and average impulsive interval. The theoretical derivation shows that the estimated fixed-time in this study is less conservative and more accurate as compared to the existing FxTS theorems. Further, the theoretical results are applied to the impulsive control of general neural network systems. Finally, two numerical examples are given to validate the effectiveness of the theoretical results.     

Integrated guidance and control with partial state constraints and actuator faults

In this paper, a novel adaptive integrated guidance and control (IGC) scheme is proposed for skid-to-turn (STT) missile with partial state constraints and actuator faults. Considering the strict-feedback form of the IGC model, the dynamic surface control (DSC) approach is adopted to design the IGC scheme. To prevent the attack angle, sideslip angle and velocity deflection angle from violating the constraints, the barrier Lyapunov function (BLF) and modified saturation function are employed in the IGC design procedure. Moreover, an auxiliary system is constructed to remove the adverse effects that caused by the modified saturation function. The adaptive laws are constructed to estimate the actuation effectiveness of actuators and the upper bounds of lumped uncertainties in the IGC model. It is theoretically shown that all signals in the closed-loop system are bounded while the state constraints are not violated in presence of actuator faults and uncertainties. Numerical simulation results are presented to verify the effectiveness and robustness of the proposed IGC scheme.     

Fixed-time scaled consensus of multi-agent systems with input delay

The design of fixed-time scaled consensus protocol for multi-agent systems with input delay is developed in this article. First, by virtue of Artstein model reduction method, the time-delay system is converted into a delay-free one. Then, two novel controllers are designed such that the fixed-time scaled consensus of multi-agent systems can be realized for the undirected and directed topology, respectively. Sufficient conditions are derived to guarantee that all agents converge to the assigned ratios instead of the same value under any bounded input delay. Besides, an explicit estimate can be given for the uniform convergence time independent of the initial conditions. Moreover, it is proved that the convergence value of the system is not affected by the initial states of agents any more, but only related to initial states of the virtual agents set in advance. Finally, numerical simulations are given to demonstrate the feasibility of the proposed algorithms.     

Fixed-time stabilization of general linear systems with input delay

In this paper, the fixed-time stabilization control problem for general linear systems with input delay is addressed. In addition to the Artstein–Kwon–Pearson reduction transformation, a pre-compensation control structure is established first to convert the original system into a single input delay-free linear system. Then, we show that the origin of the transformed system is fixed-time stabilizable by an additional homogeneous control design if the original system is controllable. Finally, an example is used to validate the proposed method via simulation results.     

Fault tolerance of cooperative interception using multiple flight vehicles

Cooperative interception of a moving target by multiple vehicles is studied. The main contributions of research work presented in this paper include: (1) cooperative interception is achieved for multiple vehicles to reach the target simultaneously at a finite time, by proposing and solving for a novel finite-time consensus problem and (2) in addition, the cooperative interception is investigated with tolerance of actuator or network failures, where novel fault tolerant consensus protocols are proposed to address actuator failures (or loss of effectiveness) and network failures, respectively. The maximum fault tolerance against network failures can be estimated. Simulations of a three-against-one interception case are presented to demonstrate the effectiveness of the proposed design approaches.     

Echo state network with multiple delayed outputs for multiple delayed time series prediction

In this paper, considering the influence of multiple delayed output items on the prediction accuracy of echo state network, a novel echo state network with multiple delayed outputs (MDO-ESN) is proposed for time series prediction with multiple delayed outputs. Firstly, for a given learning task, through studying the autocorrelation of output signal, its delayed characteristics can be determined, and then the corresponding delayed item of output equation of the MDO-ESN can be adjusted adaptively. Secondly, in order to improve the adaptability of the MDO-ESN in different learning tasks, a sufficient condition is given to satisfy the stability of the MDO-ESN. Thirdly, a parameter optimization method is given to reduce the dependence of the prediction accuracy of the MDO-ESN on the reservoir parameters of the MDO-ESN. Finally, two numerical simulation examples and one actual simulation example are used for verifying the effectiveness of the MDO-ESN.     

Backstepping-based adaptive dynamic programming for missile-target guidance systems with state and input constraints

In this paper, a novel backstepping-based adaptive dynamic programming (ADP) method is developed to solve the problem of intercepting a maneuver target in the presence of full-state and input constraints. To address state constraints, a barrier Lyapunov function is introduced to every backstepping procedure. An auxiliary design system is employed to compensate the input constraints. Then, an adaptive backstepping feedforward control strategy is designed, by which the tracking problem for strict-feedback systems can be reduced to an equivalence optimal regulation problem for affine nonlinear systems. Secondly, an adaptive optimal controller is developed by using ADP technique, in which a critic network is constructed to approximate the solution of the associated Hamilton–Jacobi–Bellman (HJB) equation. Therefore, the whole control scheme consists of an adaptive feedforward controller and an optimal feedback controller. By utilizing Lyapunov's direct method, all signals in the closed-loop system are guaranteed to be uniformly ultimately bounded (UUB). Finally, the effectiveness of the proposed strategy is demonstrated by using a simple nonlinear system and a nonlinear two-dimensional missile-target interception system.