Research on the Dynamic Evolution Behavior of Group Loitering Air Vehicles

In this paper we focus on integrated Reconnaissance/Strike LAV, so LAV can be in one of four possible situations: search, attack, BDA or removed. LAV is always being searching if it has not acquired a real target until it was destroy. While LAV discovers a target it attempts to identify if it is a real target. If the target is an unreal target, the LAV moves on with its search. If the target is a real target, The LAV attacks it. Once a LAV enters an attack stage, it cannot go back to the any other stage. A LAV may fail with enemy’s air defense or technical failure or accident [1].

The combat mission constraints and rules of conduct of their behaviour are as follows:

(1) Judge the physical constraints of LAV [20], if the LAV meets physical characteristics and communication capability, it will form the Group LAVs, and perform a search mission;

(2) When obstacles, threats of enemy air defence weapons appear, The LAVs select obstacle avoidance, hedge bypass;

(3) When the target enters the visual field of the LAV, using information fusion and processing technology on the target judgment, can perform strike missions against the enemy targets;

(4) When perform strike missions against the enemy targets, can perform damage assessment against the enemy targets;

(5) When finish performing damage assessment task against the enemy targets , can perform the next round of decision.

Let vector δ→k(t):k=1,2,…,K$\begin{array}{} \displaystyle {{\vec \delta _k}(t):k = 1,2, \ldots ,K} \end{array}$ denote the task status of Group LAVs at time t. In order to obtain the probability of the state transition, the configuration equation of group LAVs system is derived, which is described as follows:

N→=∑k=1KNkδ→k(t),$$\begin{array}{} \displaystyle \vec N = \sum\limits_{k = 1}^K {{N_k}{{\vec \delta }_k}(t)}, \end{array}$$

where N_k is the number of LAVs at the state k and N→$\begin{array}{} \displaystyle \vec N \end{array}$ is the distribution in each state of Group LAVs system.

Suppose that the probability distribution of group LAVs system state is P(n,t) at time t. Then at time t + τ, the probability distribution is P(n,t + τ). Define W_t(η, n) as the transition probability density from state η to state n in unit time during time interval [t,t + τ]. So the transition probability from state η to state n during time interval [t,t + τ] is τW_t(η, n). According to [20], the primary equation model as follows:

dP(n,t)dt=∫{W(η,n)P(η,t)−W(n,η)P(n,t)}dη.$$\begin{array}{} \displaystyle \frac{{dP(n,t)}}{{dt}} = \int {\left\{ {W(\eta ,n)P(\eta ,t) - W(n,\eta )P(n,t)} \right\}d} \eta. \end{array}$$

In order to obtain evolution equation under the statistical average, we can get the state time evolution equation as follows:

∂∂t〈n→〉=∂n→∂t∫n→P(n→,t)dn→=∫n→∫{W(n→′,n→)P(n→′,t)−W(n→,n→′)P(n→,t)}dn→′dn→=∫∫(n→′−n→)W(n→′,n→)P(n→,t)dn→′dn→=〈∫n→′W(n→′,n→)−n→W(n→′,n→)dn→′〉.$$\begin{array}{} \displaystyle \begin{split} \frac{\partial }{{\partial t}}\left\langle {\vec n} \right\rangle &= \frac{{\partial \vec n}}{{\partial t}}\int {\vec n{\rm{ }}P(\vec n,t)} {\rm{ }}d\vec n\\ &= \int {\vec n\int {\left\{ {W\left( {\vec n',\vec n} \right){\rm{ }}P(\vec n',t) - W\left( {\vec n,\vec n'} \right){\rm{ }}P(\vec n,t)} \right\}} } {\rm{ }}d\vec n'd\vec n\\ &= \int {\int {\left( {\vec n' - \vec n} \right)} } {\rm{ }}W\left( {\vec n',\vec n} \right){\rm{ }}P(\vec n,t){\rm{ }}d\vec n'd\vec n\\ &= \left\langle {\int {\vec n'W\left( {\vec n',\vec n} \right) - \vec nW\left( {\vec n',\vec n} \right)d\vec n'} } \right\rangle. \end{split} \end{array}$$

So it easy to verify that process of the state transition is a stability Markov chain. And it has some notation as following:

λ_ij = 0–the state i is a transient state, once the process enters this state immediately leave.

λ_ij = ∞–the state i is an absorbing state; once the process enters this state will never leave.

Obviously, the last state of the LAV is an absorbing state.

Pij(t+Δt)−Pij(t)=∑k∈IPik(Δt)Pkj(t)−Pij(t)=Pii(Δt)Pij(t)−Pij(t)+∑k∈IPik(Δt)Pkj(t)$$\begin{array}{} \displaystyle {P_{ij}}(t + \Delta t) - {P_{ij}}(t) = \sum\limits_{k \in I} {{P_{ik}}(\Delta t)} {P_{kj}}(t) - {P_{ij}}(t) = {P_{ii}}(\Delta t){P_{ij}}(t) - {P_{ij}}(t) + \sum\limits_{k \in I} {{P_{ik}}(\Delta t)} {P_{kj}}(t) \end{array}$$

Thus we can obtain,

(Pii(Δt)−1)≤Pij(t+Δt)−Pij(t)=(Pii(Δt)−1)Pij(t)+∑k∈IPik(Δt)Pkj(t)≤(1−Pii(Δt))$$\begin{array}{} \displaystyle \left( {{P_{ii}}(\Delta t) - 1} \right) \le {P_{ij}}(t + \Delta t) - {P_{ij}}(t) = \left( {{P_{ii}}(\Delta t) - 1} \right){P_{ij}}(t) + \sum\limits_{k \in I} {{P_{ik}}(\Delta t)} {P_{kj}}(t) \le \left( {1 - {P_{ii}}(\Delta t)} \right) \end{array}$$

Thus,

|Pij(t+Δt)−Pij(t)|≤(1−Pii(Δt))$$\begin{array}{} \displaystyle \left| {{P_{ij}}(t + \Delta t) - {P_{ij}}(t)} \right| \le \left( {1 - {P_{ii}}(\Delta t)} \right) \end{array}$$

Because the state is not a transient state. Therefore, the transition probability satisfies regularity conditions:

limt→0Pij(t)={1,i=j0,i≠j$$\begin{array}{} \displaystyle \mathop {\lim }\limits_{t \to 0} {P_{ij}}(t) = \left\{ \begin{array}{l} 1,i = j\\ 0,i \ne j \end{array} \right. \end{array}$$

Then we have the conclusion as follows:

limΔt→0|Pij(t+Δt)−Pij(t)|=0.$$\begin{array}{} \displaystyle \mathop {\lim }\limits_{\Delta t \to 0} \left| {{P_{ij}}\left( {t + \Delta t} \right) - {P_{ij}}\left( t \right)} \right| = 0. \end{array}$$

Accordingly, p_ij is a consistent continuous function at time t.

To the best knowledge of the authors, the finite state machine model was established under the assumptions stated in the following.

(1) LAVs are two-way information exchange via data link;

(2) The time of translation state form one task to another task is exponential distribution;

(3) The parameters of encountered obstacles during combat threats are random variable; Differential equation model was established using finite state machine automatically.

{dNS(t)dt=1λAνNAS(t)+1λBDANBDA(t)−αsNS(t)NT(t)−αAsNS(t)(NS(t)+C).dNAν(t)dt=−1λAνNAν(t)+αAANA(t)(NA(t)+C)+αAsNS(t)(NS(t)+C)+αABNBDA(t)(NBDA(t)+C).dNA(t)dt=αsNS(t)NT(t)−1λANA(t).dNBDA(t)dt=αABNAB(t)−1λBDANBDA(t)+αABNBDA(t)(NBDA(t)+C).dNT(t)dt=−αsNS(t)NT(t).$$\begin{array}{} \displaystyle \left\{ \begin{array}{*{20}{c}} {\frac{{d{N_S}(t)}}{{dt}} = \frac{1}{{{\lambda _{A\nu }}}}{N_{AS}}(t) + \frac{1}{{{\lambda _{BDA}}}}{N_{BDA}}(t) - {\alpha _s}{N_S}(t){N_T}(t) - {\alpha _{As}}{N_S}(t)({N_S}(t) + C).} \hfill \\ {\frac{{d{N_{A\nu }}(t)}}{{dt}} = - \frac{1}{{{\lambda _{A\nu }}}}{N_{A\nu }}(t) + {\alpha _{AA}}{N_A}(t)({N_A}(t) + C) + {\alpha _{As}}{N_S}(t)({N_S}(t) + C) + {\alpha _{AB}}{N_{BDA}}(t)({N_{BDA}}(t) + C).} \hfill \\ {\frac{{d{N_A}(t)}}{{dt}} = {\alpha _s}{N_S}(t){N_T}(t) - \frac{1}{{{\lambda _A}}}{N_A}(t).} \hfill \\ {\frac{{d{N_{BDA}}(t)}}{{dt}} = {\alpha _{AB}}{N_{AB}}(t) - \frac{1}{{{\lambda _{BDA}}}}{N_{BDA}}(t) + {\alpha _{AB}}{N_{BDA}}(t)({N_{BDA}}(t) + C).} \hfill \\ {\frac{{d{N_T}(t)}}{{dt}} = - {\alpha _s}{N_S}(t){N_T}(t).} \hfill \\ \end{array}\right. \end{array}$$

Parameters Notation: C is a constant and satisfied C = N + B, N is the Scale of the system, B is the scale of the obstacles, N_s(t) is the scale of group in the searching state at time t, N_Aν (t) is the scale of group in the avoiding state at time t, N_A (t) is the scale of group in the attacking state at time t, N_BDA (t) is the scale of group in the BDA state at time t, N_T (t) is the scale of targets, λ_Aν, λl_A, λ_BDA are the average time of Avoiding, Attacking, and BDA, resp., α_S is the rate of searching a real target, and α_AS, α_AA, α_AB are the rates of encounter obstacles on searching,attacking, and BDA.

In order to illustrate the feasibility and effectiveness of the differential equation model, which was solved by Runge-Kutta method, we chose N_T = 80, λ_Aν=5min, λ_BDA=5min, and α_AS, α_AB, α_AA ∊ [0, 1] are random variables. In order to analysis the impact of the design and operating parameters, we have established expected relative efficacy EL=E(x)NT$\begin{array}{} \displaystyle {E_L} = \frac{{E(x)}}{{{N_T}}} \end{array}$ and efficiency EN=E(x)N$\begin{array}{} \displaystyle {E_N} = \frac{{E(x)}}{N} \end{array}$.

Fig. 1 describes the number of task state with respect to time under different group scale. The group LAVs system reached a state of equilibrium after a period of the game. The scale of the group will affect the operational efficacy and efficiency. With the scale of the number larger, targets decay faster, and it has higher efficacy. When the system reaches a certain scale, the efficiency is reduced.

Fig. 1

Fig. 2 describes the number of task state with respect to time at different Probability of detecting the target value. Obviously with the success probability to detect valuable targets bigger, the number of damaging enemy targets is larger, the higher its operational effectiveness. Success probability of detecting the target value is a very important factor.

Fig. 2

Fig. 3 displays the number of task states with respect to time at different attack time. We can see that shorten time to attack enemy targets in LAV, the operational tasks whose time is shortened accordingly, but the combat effectiveness impact is not great.

Fig. 3