In order to conduct saturation attacks on a static target, the cooperative guidance problem of multimissile system is researched. A three-dimensional guidance model is built using vector calculation and the classic proportional navigation guidance (PNG) law is extended to three dimensions. Based on this guidance law, a distributed cooperative guidance strategy is proposed and a consensus protocol is designed to coordinate the time-to-go commands of all missiles. Then an expert system, which contains two extreme learning machines (ELM), is developed to regulate the local proportional coefficient of each missile according to the command. All missiles can arrive at the target simultaneously under the assumption that the multimissile network is connected. A simulation scenario is given to demonstrate the validity of the proposed method.

Saturation attack, which involves simultaneous attack from different missiles in a communication network, is an important combat manner to penetrate the missile defence system. In fact, a group of well-organized missiles of low cost and poor performance may yield better results than a single excellent one. The key to cooperative guidance of multimissile system is that all missiles reach the target at the same time in the case of saturation attacking.

Cooperative control theories have been researched broadly with respect to different agents, such as unmanned aerial vehicle [

Different from the unmanned aerial vehicle or satellite, the maneuver flight of missile is mainly based on aerodynamic force and its guidance does not include task assignment, cooperative path planning, and path tracking. This makes it particularity hard to design a cooperative strategy for the multimissile system. In the case of a group of missiles intercepting a single maneuver target, an optimal cooperative guidance law [

Many classical literatures [

In order to achieve the simultaneous attack of missiles, the consensus of time-to-go (remaining flight time, i.e., arriving time) is considered in this paper. From the previous literature, it is known that the time-to-go of missiles can be changed by adjusting proportional coefficient

In this paper, we design a cooperative guidance strategy to achieve simultaneous attack based on expert system using ELM, which just requires that the communication network is connected. Considering that the communication between missiles might be incomplete on account of the disturbances from defense system and other environment factors, the centralized cooperative guidance strategy is liable to fail and the distributed strategy is more effective. Via a distributed protocol through the connected network, which aims at asymptotical consensus of time-to-go commands, the

The remainder of this paper is organized as follows. Section

Consider the cooperative guidance of

Set

The relative movement of target and missile.

Let

In the relative movement coordinate system, the command acceleration vector

From the three-dimensional model, it is easy to know that the LOS angular velocity vector

Suppose that the mass of the missile is

Additionally, the moment of inertia to the origin can be expressed as

From expression (

Since the magnitude of

Compared to expression (

On account of the fact that the direction of

Considering that the cooperative guidance problem is to make a series of missiles attack the target simultaneously, the distance between missile and target is decreasing and the symbol of

Substitute (

Next, the cooperative guidance problem of multiple missiles is introduced. Almost all of the variables including position and velocity vectors are time-variant and the subscript

How multimissiles cooperatively attack the ground static target is shown in Figure

The concept map of multimissiles cooperative saturation attack.

Then the objective of cooperative guidance is to find the appropriate acceleration command

In order to achieve simultaneous attack, a distributed cooperative guidance strategy is adopted as shown in Figure

The framework of cooperative guidance strategy.

Denote

Similarly, when the required time-to-go

Figure

The mapping relation between proportional guidance coefficient and time-to-go.

ELM is a simple learning algorithm for single-hidden layer feedforward neural networks (SLFNs) which achieves fast learning through increasing the number of hidden nodes and obtains good generalization performance [

Compared with BP, ELM only needs to adjust the number of hidden nodes and avoids the multiple iterations; hence, it avoids the problem of local minimum or infinite training iteration and can reach the minimum training error. Owing to the strong generalization ability of ELM, the precision of training results can be guaranteed as long as the range of sample data is appropriate.

In order to approximate

Expert system with two SLFNs.

The time-variant undirected graph [

If there is no cooperation among missiles, the original

In order to make

The purpose of

From Figure

System (

Let

Define energy function

By substituting (

Because

Usually, we should choose big

In this section, a scenario is given to illustrate the proposed cooperative guidance strategy [

The topological structure of the communication network among them is shown in Figure

Topological structure of the network.

With the algorithm of ELM and Neural Network Toolbox of Matlab, these two SLFNs can be built and trained conveniently. Each SLFN includes 250 neurons in the hidden layer and adopts activation functions “tansig” and “purelin” for the hidden layer and output layer, respectively. In order to generate enough samples, the guidance of a single missile is simulated with the fixed proportional guidance coefficient iteratively, with the initial conditions

The training results of the SLFN with 250 hidden neurons using ELM are quite good. The error of mean square is 0.0069, which is close to the result of BP. However, the training time is only 4.3 seconds, which is much less than BP of 51.1 seconds. When increasing the number of hidden neurons to 500, the training time with ELM still needs only 9.5 seconds but the training results improve significantly. In conclusion, the training speed and fitting precision of ELM both have a great advantage over other ANN algorithms.

With the two trained SLFNs, setting the parameters

The trajectories of missiles.

The trajectories of proportional coefficients.

The converging process of time-to-go.

In this paper, the three-dimensional cooperative guidance for simultaneous attack on a static target of multimissile system is investigated. The classic PNG law is extended to three dimensions using vector operation; besides, an indirect and distributed cooperative proportional guidance strategy, based on expert systems and first-order

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported in part by the National Nature Science Foundation of China (nos. 61473124, 61203081 and 61174079), Doctoral Fund of Ministry of Education of China (no. 20120142120091), Fundamental Research Funds for the Central Universities of HUST (no. 2013054), and Precision Manufacturing Technology and Equipment for Metal Parts (no. 2012DFG70640).