NOMENCLATURE

a _y , a _z

missile accelerations

N

navigation constant

R

relative distance

s

switching signal of communication topologies

V _M

missile velocity

(X _I , Y _I , Z _I )

inertial reference co-ordinate system

(X _L , Y _L , Z _L )

line-of-sight (LOS) co-ordinate system

(X _M , Y _M , Z _M )

missile body co-ordinate system

θ _M , ψ _M

Euler angles from the LOS co-ordinate system to the body co-ordinate system

θ _L , ψ _L

LOS angles form the inertial reference co-ordinate system to the LOS co-ordinate system

${\dot{\rlambda }}_{y} ,\;{\dot{\rlambda }}_{z} $

LOS angular velocity components with respect to the LOS co-ordinate system

σ

heading error

τ

communication delay

η _i , ξ _i , δ _i

auxiliary states

${\mib {\cal A}}$

adjacency matrix

${{\rm {\rvarepsilon}}}$

set of edges

${{\rm {\cal G}}}$

communication graph

${\tilde{{\cal G}}}$

finite set of all possible graphs

$${\mib {\cal L}}$$

Laplacian matrix

$${{\rnu}}$$

set of nodes

2.0 PRELIMINARIES

In this section, some preliminaries are provided. The 3D homing engagement geometry of a missile and a target is presented firstly, based on which the typical 3D PPN guidance law is obtained. Then, some useful concepts about the graph theory are introduced to describe the communication network for the multi-missile system, and the problem studied in this paper is stated.

2.1 Formulation of 3D PPN guidance law

In this paper, the 3D homing engagement between missiles and a stationary target is considered, and the following assumptions are required before moving on⁽ Reference Zhao and Zhou ^{27 )}.

Assumption 1. The seeker and autopilot dynamics of a missile are sufficiently fast compared with the guidance loop.

Assumption 2. The missile is point mass and its total velocity is set to the constant value.

Based on the above assumptions, the 3D homing engagement geometry of a missile and a target can be shown in Fig. 1, in which (X _I , Y _I , Z _I ) is the inertial reference co-ordinate system; (X _M , Y _M , Z _M ) denotes the missile body co-ordinate system; (X _L , Y _L , Z _L ) is the line-of-sight (LOS) co-ordinate system; M and T represent the missile and the target, respectively; R denotes the relative range between the missile and the target or the so-called range-to-go; V _M is the total velocity of the interceptor missile; the notations θ _M and ψ _M stand for the Euler angles from the LOS co-ordinate system to the body co-ordinate system of the missile; θ _L and ψ _L represent the LOS angles in azimuth and elevation directions from the inertial reference co-ordinate system to the LOS co-ordinate system; and the term σ denotes the heading error, which describes the angle between the missile velocity and the LOS.

Figure 1 3D homing engagement geometry.

The 3D point-mass equations describing the homing engagement geometry can be given as⁽ Reference Song and Ha ^{28 )}

(1) $$\left\{ {\matrix{ {\dot{R}{\equals}{\minus}V_{M} \cos {\rm \rtheta }_{M} \cos {\rm \rpsi }_{M} } \hfill \cr {{\dot{\rlambda }}_{y} {\equals}{{V_{M} \sin {\rm \rtheta }_{M} } \over R}} \hfill \cr {{\dot{\rlambda }}_{z} {\equals}{\minus}{{V_{M} \cos {\rm \rtheta }_{M} \sin {\rm \rpsi }_{M} } \over R}} \hfill \cr {{\dot{\rtheta }}_{M} {\equals}{{a_{z} } \over {V_{M} }}{\plus}{{V_{M} } \over R}\cos {\rm \rtheta }_{M} \sin ^{2} {\rm \rpsi }_{M} \tan {\rm \rtheta }_{L} {\plus}{{V_{M} } \over R}\sin {\rm \rtheta }_{M} \cos {\rm \rpsi }_{M} } \hfill \cr \matrix{ {\dot{\rpsi }}_{M} {\equals}{{a_{y} } \over {V_{M} \cos {\rm \rtheta }_{M} }}{\minus}{{V_{M} } \over R}\sin {\rm \rtheta }_{M} \sin {\rm \rpsi }_{M} \cos {\rm \rpsi }_{M} \tan {\rm \rtheta }_{L} \hfill \cr \quad \quad \;{\plus}{{V_{M} } \over {R\cos {\rm \rtheta }_{M} }}\sin ^{2} {\rm \rtheta }_{M} \sin {\rm \rpsi }_{M} {\plus}{{V_{M} } \over R}\cos {\rm \rtheta }_{M} \sin {\rm \rpsi }_{M} \hfill \cr} \hfill \cr } } \right.$$

where ${\dot{\rlambda }}_{y} $ and ${\dot{\rlambda }}_{z} $ are LOS angular velocity components in the LOS co-ordinate system, and a _y and a _z denote accelerations of the missile in the missile body co-ordinate system. Then, the heading error σ can be defined from Fig. 1 as

(2) $${\rm \rsigma }{\equals}\arccos \left( {\cos {\rm \rtheta }_{M} \cos {\rm \rpsi }_{M} } \right),\;{\rm \rsigma }\in\left[ {0,\;{\rm \pi }} \right).$$

The typical PPN law in 3D engagement can be given by⁽ Reference Song and Ha ^{28 )}

(3) $$\left\{ {\matrix{ {a_{y} {\equals}{\minus}NV_{M} {\dot{\rlambda }}_{y} \sin {\rm \rtheta }_{M} \sin {\rm \rpsi }_{M} {\plus}NV_{M} {\dot{\rlambda }}_{z} \cos {\rm \rtheta }_{M} } \hfill \cr {a_{z} {\equals}{\minus}NV_{M} {\dot{\rlambda }}_{y} \cos {\rm \rpsi }_{M} } \hfill \cr } } \right.$$

in which N represents the navigation constant of the missile, and as pointed out in Ref. 20 that selecting N≥2 guarantees the convergence of the heading error.

Submitting (3) into (1) and differentiating (2) yield that

(4) $$\eqalignno{ {\dot{\rsigma }} & {\equals}{1 \over {\sqrt {1{\minus}\cos ^{2} {\rm \rsigma }} }}\left( {\sin {\rm \rtheta }_{M} \cos {\rm \rpsi }_{M} {\dot{\rtheta }}_{M} {\plus}\cos {\rm \rtheta }_{M} \sin {\rm \rpsi }_{M} {\dot{\rpsi }}_{M} } \right) \cr & {\equals}{{\left( {1{\minus}N} \right)V_{M} } \over {R\sqrt {1{\minus}\cos ^{2} {\rm \rsigma }} }}\sin ^{2} {\rm \rsigma } \cr & {\equals}{\minus}{{\left( {N{\minus}1} \right)V_{M} } \over R}\sin {\rm \rsigma } $$

Then, according to the facts of (1), (2) and (4), one obtains that

(5) $${{\partial R} \over {\partial {\rm \rsigma }}}{\equals}{{\partial R} \over {\partial t}}\left( {{{\partial {\rm \rsigma }} \over {\partial t}}} \right)^{{{\minus}1}} {\equals}{{R\cot {\rm \rsigma }} \over {N{\minus}1}}$$

Solving the equation presented in (5) in terms of the heading error σ yields that

(6) $$R\left( t \right){\equals}{{R\left( 0 \right)} \over {\left| {\sin {\rm \rsigma }\left( 0 \right)} \right|^{{{1 \over {N{\minus}1}}}} }}\left| {\sin {\rm \rsigma }\left( t \right)} \right|^{{{1 \over {N{\minus}1}}}} $$

which indicates that if missiles guided by the typical PPN law are associated with the same initial range-to-go R(0) and initial heading error σ(0), then these missiles have the same flight trajectories. This information is valuable enough to lay a firm foundation for the co-operative guidance law design.

2.2 Problem statement

This study considers the co-operative guidance problem of n + 1 missiles attacking simultaneously a stationary target. In typical scenarios of salvo attack, missiles are usually assumed to be homogeneous, which means that they perform similar aerodynamic properties. Hence, it is necessary and reasonable to regard the same velocity for multiple missiles in a salvo attack mission.

Co-operative salvo attack of multiple missiles depends on the neighbour-to-neighbour communication. Let ${{\rm {\cal G}}}$ =( $${{\rnu}}$$ , ${{\rm {\rvarepsilon}}}$ ) be a directed graph, where $${{\rnu}}$$ ={v ₀, v ₁, … , v _n } is the set of nodes, and ${{\rm {\rvarepsilon}}}$ ⊆ $${{\rnu}}$$ × $${{\rnu}}$$ is the set of edges with an ordered pair of nodes. An edge e _i,j in the directed graph is denoted by the ordered pair of nodes (ν _j , ν _i ), and e _i,j ∈ ${{\rm {\rvarepsilon}}}$ in the directed graph means that agent i can obtain information from agent j, but not vice versa. Furthermore, self-loops are not allowed, that is e _i,i ∉ ${{\rm {\rvarepsilon}}}$ . A directed path from node ν _i to ν _j is a sequence of ordered edges as (ν _i , ν _k1), (ν _k1, ν _k2), ... , (ν _kl , ν _j ), with distinct nodes ν _km , m=1, 2, … , l. A directed graph is called connected if and only if there exists a directed path between any pair of distinct nodes. Moreover, if there is exists a node called the root, and this root has a directed path to every other node of the graph, then the graph is said to contain a directed spanning tree. A nonnegative adjacency matrix ${\mib {\cal A}}{\equals}\left[ {a_{{i,\,j}} } \right]\in{\mib {\bf R}}^{{\left( {n{\plus}1} \right){\times}\left( {n{\plus}1} \right)}} $ specifies the inter-connection topology of missiles, which can be defined as

(7) $$a_{{i,j}} {\equals}\left\{ {\matrix{ 1 \hfill & {{\rm if}\;\left( {v_{j} ,v_{i} } \right)\in{\mib {\rvarepsilon}}} \hfill \cr 0 \hfill & {{\rm otherwise}} \hfill \cr } } \right.$$

Then, the Laplacian matrix $${ {\mib {\cal L}}}{\equals}\left[ {l_{{i,j}} } \right]\in{\mib {\Bbb R}}^{{\left( {n{\plus}1} \right){\times}\left( {n{\plus}1} \right)}} $$ of the graph ${\mib {\cal G}}$ is given as

(8) $$l_{{i,j}} {\equals}\left\{ {\matrix{ {{\minus}a_{{i,j}} } \hfill & {i\,\ne\,j} \hfill \cr {\mathop{\sum}\limits_{k{\equals}0,k\,\ne\,i}^n {a_{{i,k}} } } \hfill & {i{\equals}j} \hfill \cr } } \right.$$

The following lemma is satisfied for a connected graph that contains a directed spanning tree.

Lemma 1

(Ref. 29). Zero is a simple eigenvalue of the Laplacian matrix ${\mib {\cal L}}$ , and all the other eigenvalues are positive if and only if the graph ${\mib {\cal G}}$ is connected.

Suppose that there exists a finite set $${ \tilde{{\cal G}}}{\equals}\left\{ {{\mib {\cal G}}_{p} \,\colon\,p\in{\cal P}} \right\}$$ of all possible graphs, in which ${\cal P}$ is an index set for all graphs. Then, ${\mib {\cal G}}_{{s\left( t \right)}} $ can be utilised to describe the current communication graph at any time t>0, where s(t) : [0, +∞) → ${\cal P}$ represents the switching signal.

The control objective of this paper is to design a co-operative guidance law based on information from topology ${\mib {\cal G}}_{{s\left( t \right)}} $ such that all missiles can reach at a stationary target simultaneously in spite of their variety initial poses and positions.

3.0 MAIN RESULTS

Motivated by the property of the typical PPN law, a two-stage co-operative guidance strategy can be considered as a feasible solution for the salvo attack problem of multiple missiles. In the first guidance phase, a co-operative guidance law based on the communication topology is adopted to drive missiles to achieve favoured initial states (i.e. the same range-to-go and the same heading errors) for the second guidance phase. Then, missiles disconnect from each other and the typical PPN guidance law is utilised to drive missiles to accomplish the salvo attack mission.

Before moving on, the following auxiliary states are introduced first

(9) $${\rm \eta }_{i} {\equals}{{R_{i} } \over {V_{M} }},\;\;{\rm \xi }_{i} {\equals}{\minus}\hskip-2pt\cos {\rm \rsigma }_{i} ,\;\;i{\equals}0,\,\ldots\,,n$$

in which R _i and σ _i , i=0,…, n denote the range-to-go and heading error of the ith missile. Based on (1) and (2), one can obtain from (9) that

(10) $$\left\{ {\matrix{ {{\dot{\reta }}_{i} {\equals}{\rm \rxi }_{i} } \hfill \cr {{\dot{\rxi }}_{i} {\equals}u_{i} } \hfill \cr } } \right.$$

in which

(11) $$\eqaligno{ u_{i} & {\equals}{{\sin {\rm \rsigma }_{i} } \over {\sqrt {1{\minus}\cos ^{2} {\rm \rsigma }_{i} } }}\left( {{{a_{{z,i}} } \over {V_{M} }}\sin {\rm \rtheta }_{{M,i}} \cos {\rm \rpsi }_{{M,i}} {\plus}{{a_{{y,i}} } \over {V_{M} }}\sin {\rm \rpsi }_{{M,i}} } \right. \cr \left. { & {\plus}{{V_{M} } \over {R_{i} }}\left( {\sin ^{2} {\rm \rtheta }_{{M,i}} {\plus}\cos ^{2} {\rm \rtheta }_{{M,i}} \sin ^{2} {\rm \rpsi }_{{M,i}} } \right)} \right) $$

Then, the co-operative guidance law design problem can be formulated as a consensus problem of the multi-missile system (10). Moreover, based on (11), the following guidance law can be used to realise favoured conditions of all missiles in the first guidance phase

(12) $$\left\{ {\matrix{ {a_{{y,i}} {\equals}{\minus}{{V_{M}^{2} } \over {R_{i} }}\sin {\rm \rpsi }_{{M,i}} {\plus}{{u_{i} V_{M} \sqrt {1{\minus}\cos ^{2} {\rm \rsigma }_{i} } } \over {2\sin {\rm \rsigma }_{i} \sin {\rm \rpsi }_{{M,i}} }}} \hfill \cr {a_{{z,i}} {\equals}{\minus}{{V_{M}^{2} } \over {R_{i} }}sin\;{\rm \rtheta }_{{M,i}} \cos {\rm \rpsi }_{{M,i}} {\plus}{{u_{i} V_{M} \sqrt {1{\minus}\cos ^{2} {\rm \rsigma }_{i} } } \over {2\sin {\rm \rsigma }_{i} \sin {\rm \rtheta }_{{M,i}} \cos {\rm \rpsi }_{{M,i}} }}} \hfill \cr } } \right.$$

From (12), one can see that σ _i =0 or ξ _i =−1 is a singular point, which leads to the failure of co-operative guidance. To cope with this problem, the leader-follower strategy with a zero input leader is utilised in this paper. Simply let missile 0 be the selected leader and other missiles are performed as followers. Let δ _i =[η _i , ξ _i ]^T, i=0,..., n, then the dynamics of the multi-missile system can be obtained by

(13) $$\left\{ {\matrix{ {{\dot{\rdelta }}_{0} {\rm {\equals}}{\mib A}{\rdelta }_{0} } \hfill \cr {{\dot{\rdelta }}_{i} {\rm {\equals}}{\mib A}{\rdelta }_{i} {\plus}{\mib B}u_{i} ,i{\equals}1,...,n} \hfill \cr } } \right.$$

in which ${A}{\equals}\left[ {\matrix{ 0 & 1 \cr 0 & 0 \cr } } \right],{\mib B}{\equals}\left[ {\matrix{ 0 \cr 1 \cr } } \right].$

Hence, the control objective in the first guidance phase can be concluded as that designing distributed control protocols u _i , i=1,…, n such that

(14) $$\mathop{{\lim }}\limits_{{t\to{\plus}\infty}} \left\Vert {{\bf \rdelta }_{i} \left( t \right){\minus}{\bf \rdelta }_{0} \left( t \right)} \right\Vert{\equals}{\bf{0}} _{2} ,i{\equals}1,...,n$$

is achieved in spite of any initial conditions.

(15) $$\left\{ {\matrix{ {\mathop{{\lim }}\limits_{{t\to{\plus}\infty}} \left| {R_{i} \left( t \right){\minus}R_{0} \left( t \right)} \right|{\equals}0} \hfill \cr {\mathop{{\lim }}\limits_{{t\to{\plus}\infty}} \left| {\rsigma _{i} \left( t \right){\minus}\rsigma _{0} \left( t \right)} \right|{\equals}0} \hfill \cr } } \right.$$

Therefore, the control objective achieved in the first guidance phase results in the same range-to-go and the same heading error finally. Moreover, it can be seen from (13) that the leader missile’s input u ₀ is set to be 0, which means that the heading error σ₀(t) keeps constant in the first guidance phase. Hence, selecting a missile with favoured initial heading error σ₀(0) as the leader not only helps to avoid the singularity problem mentioned above but also to obtain favoured conditions for the work of the PNN law in the second guidance phase.

In typical leader-follower co-ordination, it is usually assumed that the leader has no neighbours while the communication network among followers is undirected. Hence, the Laplacian matrix ${\mib {\cal L}}_{{s\left( t \right)}} $ associated with the graph ${\mib {\cal G}}_{{s\left( t \right)}} $ has the following form:

(16) $${\mib {\cal L}}_{{s\left( t \right)}} {\equals}\left[ {\matrix{ 0 & {{\bf{0}} _{n}^{{\rm T}} } \cr {{\bf \rvarphi }_{{s\left( t \right)}} } & {{\mib H}_{{s\left( t \right)}} } \cr } } \right]$$

in which ϕ _s(t) ∈ℝ ⁿ and H _s(t) ∈ℝ ^n×n is a symmetric positive definite matrix.

To obtain the main results, consider an infinite sequence of nonempty, bounded and contiguous time intervals $\left[ {t_{k} ,\;t_{{k{\plus}1}} } \right),\;k{\equals}0,1,...$ , with t ₀=0 and t _k+1−t _k ≤T for some constant T>0. Suppose that in each interval $\left[ {t_{k} ,\;t_{{k{\plus}1}} } \right)$ , there exists a sequence of non-overlapping subintervals $\left[ {t_{k}^{0} ,\;t_{k}^{1} } \right),\,\ldots\,,\left[ {t_{k}^{m} ,\;t_{k}^{{m{\plus}1}} } \right),\,\ldots\,,\left[ {t_{k}^{{m_{k} {\minus}1}} ,\;t_{k}^{{m_{k} }} } \right)$ with $t_{k}^{0} {\equals}t_{k} $ and $t_{k}^{{m_{k} }} {\equals}t_{{k{\plus}1}} $ satisfying $t_{k}^{{m{\plus}1}} {\minus}t_{k}^{m} \geq T_{s} \,\gt\,0,\;0\leq m\leq m_{k} {\minus}1$ . It is assumed that in each subinterval $\left[ {t_{k}^{m} ,\;t_{k}^{{m{\plus}1}} } \right)$ , the communication graph ${\mib {\cal G}}_{{s\left( t \right)}} $ is fixed, denoted by ${\mib {\cal G}}_{{k_{m} }} $ . Then, a mild assumption of communication topologies is introduced.

In the following, for each $p\in{\mib {\cal P}}$ , eigenvalues of H _p can be denoted by ${\rm \lambda }_{p}^{i} $ , where i=1,..., n is the index of each follower missile, based on the labelling rule in Ref. 30. Let $\ell \left( p \right){\equals}\left\{ {i\,\colon\,\lambda _{p}^{i} \,\ne\,0,i{\equals}1,...,n} \right\}$ , then the following lemma can be used to obtain the relation between jointly connected graphs and eigenvalues of Laplacian matrices.

(17) $$\bigcup\limits_{t\in\left[ {t_{k} ,\;t_{{k{\plus}1}} } \right)} {\ell \left( {s\left( t \right)} \right){\equals}\left\{ {1,...,n} \right\}} $$

Based on the communication topology ${\mib {\cal G}}_{{s\left( t \right)}} $ , the consensus protocol is designed as

(18) $$\eqalignno{ u_{i} \left( t \right) & {\equals}{\minus}{\mib B}^{{\rm T}} {\mib P}\mathop{\sum}\limits_{j{\equals}1,j\,\ne\,i}^n {a_{{i,j}} \left( t \right)\left( {{\bf \rdelta }_{i} \left( {t{\minus}{\rm \tau }\left( t \right)} \right){\minus}{\bf \rdelta }_{j} \left( {t{\minus}{\rm \tau }\left( t \right)} \right)} \right)} \cr & {\minus}{\mib B}^{{\rm T}} {\mib P}a_{{i,0}} \left( t \right)\left( {{\bf \rdelta }_{i} \left( {t{\minus}{\rm \tau }\left( t \right)} \right){\minus}{\bf \rdelta }_{0} \left( {t{\minus}{\rm \tau }\left( t \right)} \right)} \right) $$

where P ∈ℝ^2×2 is a positive definite matrix, and τ(t) is a uniform time-varying communication delay satisfying the following assumption.

To analyse the leader–follower consensus of the multi-missile system (13), let ${\tilde{\delta}_{i}} {\equals}{\bf \rdelta }_{i} {\minus}{\bf \rdelta }_{0} ,i{\equals}1,...,n$ be the state error of the ith missile. Then, the dynamics of $\tilde{\delta}_{i}$ can be obtained based on (18) as

(19) $${\dot{\tilde{\rdelta }}}_{i} \left( t \right){\rm {\equals}}{A\tilde{\rdelta }}_{i} \left( t \right){\minus}{\mib BB}^{{\rm T}} {\mib P}\mathop{\sum}\limits_{j{\equals}1,j\,\ne\,i}^n {l_{{i,j}} \left( t \right){\tilde{\rdelta }}_{j} \left( {t{\minus}{\rm \tau }\left( t \right)} \right)} $$

Let $${\tilde{\rdelta }}{\equals}\left[ {{\tilde{\rdelta }}_{1}^{{\rm T}} ,...,{\tilde{\rdelta }}_{n}^{{\rm T}} } \right]^{{\rm T}} $$ , then the network dynamics can be rewritten in the compact form as

(20) $${\dot{\tilde{\rdelta }}}\left( t \right){\rm {\equals}}\left( {{I}_{n} \,\otimes\,{A}} \right){\tilde{\rdelta }}\left( t \right){\minus}\left( {{ H}_{{s\left( t \right)}} \,\otimes\,{\mib BB}^{{\rm T}} {\mib P}} \right){\tilde{\rdelta }}\left( {t{\minus}{\rm \tau }\left( t \right)} \right)$$

The following theorem presents sufficient conditions to guarantee the consensus of the multi-missile system (13) based on the control law in (18).

(21) $${\mib PA}{\plus}{\mib A}^{{\rm T}} {\mib P}{\plus}{\mib P}{\plus}2{\rm \tau }_{m} {\mib A}^{{\rm T}} {\mib A}\leq 0$$

(22) $${\minus}2{\rm \lambda }_{{\min }} {\plus}{\rm \tau }_{m} {\rm \lambda }_{{\max }}^{2} {\mib PBB}^{{\rm T}} {\mib P}\,\lt\,0$$

(23) $${\minus}\left( {1{\minus}{\rm \tau }_{{dm}} } \right){\mib P}{\plus}2{\rm \tau }_{m} {\rm \lambda }_{{\max }}^{2} {\mib PBB}^{{\rm T}} {\mib BB}^{{\rm T}} {\mib P}\,\lt\,0$$

are satisfied, where ${\rm \lambda }_{{\min }} $ and ${\rm \lambda }_{{\max }} $ are the smallest non-zero and the biggest eigenvalues associated with all possible communication topologies.

To start the consensus analysis, consider the following Lyapunov–Krasovskii function:

(24) $$V\left( t \right){\equals}V_{1} \left( t \right){\plus}V_{2} \left( t \right)$$

with

(25) $$\left\{ {\matrix{ {V_{1} \left( t \right){\equals}{\tilde{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib I}_{n} \,\otimes\,{\mib P}} \right){\tilde{\rdelta }}\left( t \right)} \hfill \cr {V_{2} \left( t \right){\equals}{\int}_{t{\minus}\tau \left( t \right)}^t {{\tilde{\rdelta }}^{{\rm T}} \left( {\rm \romega } \right)\left( {{\mib I}_{n} \,\otimes\,{\mib P}} \right){\tilde{\rdelta }}\left( {\rm \romega } \right){\rm d\romega }} {\plus}{\int}_{{\minus}\tau _{m} }^0 {{\int}_{t{\plus}\rrho }^t {{\dot{\tilde{\rdelta }}}^{{\rm T}} \left( {\rm \romega } \right){\dot{\tilde{\rdelta }}}\left( {\rm \romega } \right){\rm d\romega d\rrho }} } } \hfill \cr } } \right.$$

From (25), it can be seen that V(t) is continuously differentiable at any time except for the switching time. Then, at any non-switching time t, the time derivative of V ₁(t) and V ₂(t) along the trajectory of system (20) can be given by

(26) $$\eqalignno{ \dot{V}_{1} \left( t \right) & {\equals}2{\tilde{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib I}_{n} \,\otimes\,{\mib P}} \right){\dot{\tilde{\rdelta }}}\left( t \right) \cr & {\equals}{\tilde{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib I}_{n} \,\otimes\,\left( {{\mib PA}{\plus}{\mib A}^{{\rm T}} {\mib P}} \right)} \right){\tilde{\rdelta }}\left( t \right){\minus}{\tilde{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib H}_{{s\left( t \right)}} \,\otimes\,2{\mib PBB}^{{\rm T}} {\mib P}} \right){\tilde{\rdelta }}\left( {t{\minus}{\rm \tau }\left( t \right)} \right) \cr & {\equals}{\tilde{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib I}_{n} \,\otimes\,\left( {{\mib PA}{\plus}{\mib A}^{{\rm T}} {\mib P}} \right)} \right){\tilde{\rdelta }}\left( t \right){\minus}{\tilde{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib H}_{{s\left( t \right)}} \,\otimes\,2{\mib PBB}^{{\rm T}} {\mib P}} \right){\tilde{\rdelta }}\left( t \right) \cr & {\plus}{\int}_{t{\minus}\tau \left( t \right)}^t {{\tilde{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib H}_{{s\left( t \right)}} \,\otimes\,2{\mib PBB}^{{\rm T}} {\mib P}} \right){\dot{\tilde{\rdelta }}}\left( {\rm \romega } \right){\rm d\romega }} \cr & \leq {\tilde{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib I}_{n} \,\otimes\,\left( {{\mib PA}{\plus}{\mib A}^{{\rm T}} {\mib P}} \right)} \right){\tilde{\rdelta }}\left( t \right){\minus}{\tilde{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib H}_{{s\left( t \right)}} \,\otimes\,2{\mib PBB}^{{\rm T}} {\mib P}} \right){\tilde{\rdelta }}\left( t \right) \cr & {\plus}{\int}_{t{\minus}\tau \left( t \right)}^t {{\tilde{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib H}_{{s\left( t \right)}}^{{\rm T}} {\mib H}_{{s\left( t \right)}} \,\otimes\,{\mib PBB}^{{\rm T}} {\mib PPBB}^{{\rm T}} {\mib P}} \right){\tilde{\rdelta }}\left( t \right){\rm {\plus}}{\dot{\tilde{\rdelta }}}^{{^{{\rm T}} }} \left( {\rm \romega } \right){\dot{\tilde{\rdelta }}}\left( {\rm \romega } \right){\rm d\romega }} \cr & \leq {\tilde{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib I}_{n} \,\otimes\,\left( {{\mib PA}{\plus}{\mib A}^{{\rm T}} {\mib P}} \right)} \right){\tilde{\rdelta }}\left( t \right){\minus}{\tilde{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib H}_{{s\left( t \right)}} \,\otimes\,2{\mib PBB}^{{\rm T}} {\mib P}} \right){\tilde{\rdelta }}\left( t \right) \cr & {\plus}{\rm \tau }_{m} {\tilde{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib H}_{{s\left( t \right)}}^{{\rm T}} {\mib H}_{{s\left( t \right)}} \,\otimes\,{\mib PBB}^{{\rm T}} {\mib PPBB}^{{\rm T}} {\mib P}} \right){\tilde{\rdelta }}\left( t \right){\plus}{\int}_{t{\minus}\tau _{m} }^t {{\dot{\tilde{\rdelta }}}^{{^{{\rm T}} }} \left( {\rm \romega } \right){\dot{\tilde{\rdelta }}}\left( {\rm \romega } \right){\rm d\romega }} $$

and

(27) $$\eqalignno{ \dot{V}_{2} \left( t \right) & {\equals}{\tilde{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib I}_{n} \,\otimes\,{\mib P}} \right){\tilde{\rdelta }}\left( t \right){\minus}\left( {1{\minus}{\dot{\rtau }}\left( t \right)} \right){\tilde{\rdelta }}^{{\rm T}} \left( {t{\minus}{\rm \rtau }\left( t \right)} \right)\left( {{\mib I}_{n} \,\otimes\,{\mib P}} \right){\tilde{\rdelta }}\left( {t{\minus}{\rm \rtau }\left( t \right)} \right) \cr & {\plus}{\rm \rtau }_{m} {\dot{\tilde{\rdelta }}}^{{\rm T}} \left( t \right){\dot{\tilde{\rdelta }}}\left( t \right){\minus}{\int}_{t{\minus}\rtau _{m} }^t {{\dot{\tilde{\rdelta }}}^{{\rm T}} \left( {\rm \romega } \right){\dot{\tilde{\rdelta }}}\left( {\rm \romega } \right){\rm d\romega }} \cr & \:\leq {\tilde{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib I}_{n} \,\otimes\,{\mib P}} \right){\tilde{\rdelta }}\left( t \right){\minus}\left( {1{\minus}{\rm \rtau }_{{dm}} } \right){\tilde{\rdelta }}^{{\rm T}} \left( {t{\minus}{\rm \rtau }\left( t \right)} \right)\left( {{\mib I}_{n} \,\otimes\,{\mib P}} \right){\tilde{\rdelta }}\left( {t{\minus}{\rm \rtau }\left( t \right)} \right) \cr & {\plus}2{\rm \rtau }_{m} {\tilde{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib I}_{n} \,\otimes\,{\mib A}^{{\rm T}} {\mib A}} \right){\tilde{\rdelta }}\left( t \right){\minus}{\int}_{t{\minus}\rtau _{m} }^t {{\dot{\tilde{\rdelta }}}^{{\rm T}} \left( {\rm \romega } \right){\dot{\tilde{\rdelta }}}\left( {\rm \romega } \right){\rm d\romega }} \cr & {\plus}2{\rm \rtau }_{m} {\tilde{\rdelta }}^{{\rm T}} \left( {t{\minus}{\rm \rtau }\left( t \right)} \right)\left( {{\mib H}_{{s\left( t \right)}}^{{\rm T}} {\mib H}_{{s\left( t \right)}} \,\otimes\,{\mib PBB}^{{\rm T}} {\mib BB}^{{\rm T}} {\mib P}} \right){\tilde{\rdelta }}\left( {t{\minus}{\rm \rtau }\left( t \right)} \right) $$

Therefore, based on (26) and (27), one has

(28) $$\eqalignno{ \dot{V}\left( t \right) & {\equals}\dot{V}_{1} \left( t \right){\plus}\dot{V}_{2} \left( t \right) \cr & \leq {\tilde{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib I}_{n} \,\otimes\,\left( {{\mib PA}{\plus}{\mib A}^{{\rm T}} {\mib P}{\plus}{\mib P}{\plus}2\rtau _{m} {\mib A}^{{\rm T}} {\mib A}} \right)} \right){\tilde{\rdelta }}\left( t \right) \cr & {\minus}{\tilde{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib H}_{{s\left( t \right)}} \,\otimes\,2{\mib PBB}^{{\rm T}} {\mib P}} \right){\tilde{\rdelta }}\left( t \right) \cr & {\plus}{\rm \rtau }_{m} {\tilde{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib H}_{{s\left( t \right)}}^{{\rm T}} {\mib H}_{{s\left( t \right)}} \,\otimes\,{\mib PBB}^{{\rm T}} {\mib PPBB}^{{\rm T}} {\mib P}} \right){\tilde{\rdelta }}\left( t \right) \cr & {\minus}\left( {1{\minus}{\rm \rtau }_{{dm}} } \right){\tilde{\rdelta }}^{{\rm T}} \left( {t{\minus}{\rm \rtau }\left( t \right)} \right)\left( {{\mib I}_{n} \,\otimes\,{\mib P}} \right){\tilde{\rdelta }}\left( {t{\minus}{\rm \rtau }\left( t \right)} \right) \cr & {\plus}2{\rm \rtau }_{m} {\tilde{\rdelta }}^{{\rm T}} \left( {t{\minus}{\rm \rtau }\left( t \right)} \right)\left( {{\mib H}_{{s\left( t \right)}}^{{\rm T}} {\mib H}_{{s\left( t \right)}} \,\otimes\,{\mib PBB}^{{\rm T}} {\mib BB}^{{\rm T}} {\mib P}} \right){\tilde{\rdelta }}\left( {t{\minus}{\rm \rtau }\left( t \right)} \right) $$

Noting that H _s(t) is symmetric, hence there exists an orthogonal matrix W _s(t) such that ${\mib W}_{{s\left( t \right)}} {\mib H}_{{s\left( t \right)}} {\mib W}_{{s\left( t \right)}}^{{\rm T}} {\equals}{\bf \Lambda } _{{s\left( t \right)}} \,\colon\,{\equals}{\rm diag}\left\{ {\lambda _{{s\left( t \right)}}^{1} ,...,\lambda _{{s\left( t \right)}}^{n} } \right\}$ . Let ${\bf \bar{\rdelta }}\left( t \right){\equals}\left( {{\mib W}_{{s\left( t \right)}} \,\otimes\,{\mib I}_{2} } \right){\tilde{\rdelta }}\left( t \right)$ , and it follows from (28) that

(29) $$\eqalignno{ \dot{V}\left( t \right)\leq & {\bf \bar{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\mib I}_{n} \,\otimes\,\left( {{\mib PA}{\plus}{\mib A}^{{\rm T}} {\mib P}{\plus}{\mib P}{\plus}2{\rm \rtau }_{m} {\mib A}^{{\rm T}} {\mib A}} \right)} \right){\bf \bar{\rdelta }}\left( t \right) \cr {\minus} & {\bf \bar{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\bf \Lambda } _{{s\left( t \right)}} \,\otimes\,2{\mib PBB}^{{\rm T}} {\mib P}} \right){\bf \bar{\rdelta }}\left( t \right) \cr {\plus} & {\rm \rtau }_{m} {\bf \bar{\rdelta }}^{{\rm T}} \left( t \right)\left( {{\bf \Lambda } _{{s\left( t \right)}}^{2} \,\otimes\,{\mib PBB}^{{\rm T}} {\mib PPBB}^{{\rm T}} {\mib P}} \right){\bf \bar{\rdelta }}\left( t \right) \cr {\minus} & \left( {1{\minus}{\rm \rtau }_{{dm}} } \right){\bf \bar{\rdelta }}^{{\rm T}} \left( {t{\minus}{\rm \rtau }\left( t \right)} \right)\left( {{\mib I}_{n} \,\otimes\,{\mib P}} \right){\bf \bar{\rdelta }}\left( {t{\minus}{\rm \rtau }\left( t \right)} \right) \cr {\plus} & 2{\rm \rtau }_{m} {\bf \bar{\rdelta }}^{{\rm T}} \left( {t{\minus}{\rm \rtau }\left( t \right)} \right)\left( {{\bf \Lambda } _{{s\left( t \right)}}^{2} \,\otimes\,{\mib PBB}^{{\rm T}} {\mib BB}^{{\rm T}} {\mib P}} \right){\bf \bar{\rdelta }}\left( {t{\minus}{\rm \rtau }\left( t \right)} \right) \cr \leq & \mathop{\sum}\limits_{i\in\ell \left( {s\left( t \right)} \right)} {{\bf \bar{\rdelta }}_{i}^{{\rm T}} \left( t \right)\left( {{\minus}2{\rm \lambda }_{{\min }} {\mib PBB}^{{\rm T}} {\mib P}{\plus}{\rm \rtau }_{m} {\rm \lambda }_{{\max }}^{2} {\mib PBB}^{{\rm T}} {\mib PPBB}^{{\rm T}} {\mib P}} \right){\bf \bar{\rdelta }}_{i} \left( t \right)} \cr {\plus} & \mathop{\sum}\limits_{i\in\ell \left( {s\left( t \right)} \right)} {{\bf \bar{\rdelta }}_{i}^{{\rm T}} \left( {t{\minus}{\rm \rtau }\left( t \right)} \right)\left( {{\minus}\left( {1{\minus}{\rm \rtau }_{{dm}} } \right){\mib P}{\plus}2{\rm \rtau }_{m} \lambda _{{\max }}^{2} {\mib PBB}^{{\rm T}} {\mib BB}^{{\rm T}} {\mib P}} \right){\bf \bar{\rdelta }}_{i} \left( {t{\minus}{\rm \rtau }\left( t \right)} \right)} $$

where the fact of (21) is utilised to obtain the above inequality. From (29), it concludes based on (22) and (23) that there exists a constant κ>0 such that

(30) $$\dot{V}\left( t \right)\leq {\minus}\kappa \mathop{\sum}\limits_{i\in\ell \left( {s\left( t \right)} \right)} {\left( {{\bf \bar{\rdelta }}_{i}^{{\rm T}} \left( t \right){\bf \bar{\rdelta }}_{i} \left( t \right){\plus}{\bf \bar{\rdelta }}_{i}^{{\rm T}} \left( {t{\minus}{\rm \rtau }\left( t \right)} \right){\bf \bar{\rdelta }}_{i} \left( {t{\minus}{\rm \rtau }\left( t \right)} \right)} \right)} \,\lt\,0$$

is achieved. Hence, $\mathop{{\lim }}\limits_{{t\to{\plus}\infty}} V\left( t \right)$ exists.

Considering the infinite sequences V(t _k ), k=0,1,... and using Cauchy’s convergence criteria, one has for any α>0, there exists a positive number k _α such that

(31) $$\;\;\left| {{\int}_{t_{k} }^{t_{{k{\plus}1}} } {\dot{V}\left( \romega \right){\rm d}\romega } } \right|{\equals}{\int}_{t_{k}^{0} }^{t_{k}^{1} } {{\minus}\dot{V}\left( \romega \right){\rm d}\romega } {\plus} \cdots {\plus}{\int}_{t_{k}^{{m_{k} {\minus}1}} }^{t_{k}^{{m_{k} }} } {{\minus}\dot{V}\left( \romega \right){\rm d}\romega } \,\lt\,\alpha ,\forall k\geq k_{\alpha } $$

For each integral, one obtains

(32) $$\eqalignno{ & \quad {\int}_{t_{k}^{j} }^{t_{k}^{{j{\plus}1}} } {{\minus}\dot{V}\left( {\rm \romega } \right){\rm d\romega }} \cr & \geq {\rm \kappa }{\int}_{t_{k}^{j} }^{t_{k}^{{j{\plus}1}} } {\mathop{\sum}\limits_{i\in\ell \left( {s\left( {t_{k}^{j} } \right)} \right)} {\left( {{\bf \bar{\rdelta }}_{i}^{{\rm T}} \left( {\rm \romega } \right){\bf \bar{\rdelta }}_{i} \left( {\rm \romega } \right){\plus}{\bf \bar{\rdelta }}_{i}^{{\rm T}} \left( {{\rm \romega }{\minus}{\rm \rtau }\left( {\rm \romega } \right)} \right){\bf \bar{\rdelta }}_{i} \left( {{\rm \romega }{\minus}{\rm \rtau }\left( {\rm \romega } \right)} \right)} \right)} {\rm d\romega }} \cr & \geq {\rm \kappa }{\int}_{t_{k}^{j} }^{t_{k}^{j} {\plus}T_{s} } {\mathop{\sum}\limits_{i\in\ell \left( {s\left( {t_{k}^{j} } \right)} \right)} {\left( {{\bf \bar{\rdelta }}_{i}^{{\rm T}} \left( {\rm \romega } \right){\bf \bar{\rdelta }}_{i} \left( {\rm \romega } \right){\plus}{\bf \bar{\rdelta }}_{i}^{{\rm T}} \left( {{\rm \romega }{\minus}{\rm \rtau }\left( {\rm \romega } \right)} \right){\bf \bar{\rdelta }}_{i} \left( {{\rm \romega }{\minus}{\rm \rtau }\left( {\rm \romega } \right)} \right)} \right)} {\rm d\romega }} $$

Hence, it follows from (32) that

(33) $$\kappa \mathop{\sum}\limits_{j{\equals}0}^{m_{k} {\minus}1} {{\int}_{t_{k}^{j} }^{t_{k}^{j} {\plus}T_{s} } {\mathop{\sum}\limits_{i\in\ell \left( {s\left( {t_{k}^{j} } \right)} \right)} {\left( {{\bf \bar{\rdelta }}_{i}^{{\rm T}} \left( {\rm \romega } \right){\bf \bar{\rdelta }}_{i} \left( {\rm \romega } \right){\plus}{\bf \bar{\rdelta }}_{i}^{{\rm T}} \left( {{\rm \romega }{\minus}{\rm \rtau }\left( {\rm \romega } \right)} \right){\bf \bar{\rdelta }}_{i} \left( {{\rm \romega }{\minus}{\rm \rtau }\left( {\rm \romega } \right)} \right)} \right)} {\rm d\romega }} } \,\lt\,{\rm \alpha }$$

As switches in the interval $\left[ {t_{k} ,\;t_{{k{\plus}1}} } \right)$ are finite, therefore, one has for ∀k≥k _α

(34) $$\mathop{{\lim }}\limits_{{t\to{\plus}\infty}} {\int}_t^{t{\plus}T_{s} } {\mathop{\sum}\limits_{j{\equals}0}^{m_{k} {\minus}1} {\mathop{\sum}\limits_{i\in\ell \left( {s\left( {t_{k}^{j} } \right)} \right)} {\left( {{\bf \bar{\rdelta }}_{i}^{{\rm T}} \left( {\rm \romega } \right){\bf \bar{\rdelta }}_{i} \left( {\rm \romega } \right){\plus}{\bf \bar{\rdelta }}_{i}^{{\rm T}} \left( {{\rm \romega }{\minus}{\rm \rtau }\left( {\rm \romega } \right)} \right){\bf \bar{\rdelta }}_{i} \left( {{\rm \romega }{\minus}{\rm \rtau }\left( {\rm \romega } \right)} \right)} \right)} } {\rm d\romega }} {\equals}0$$

Based on Lemma 2 and Assumption 3, it follows from (34) that there exist some positive integers β₁,...,β _n such that

(35) $$\mathop{{\lim }}\limits_{{t\to{\plus}\infty}} {\int}_t^{t{\plus}T_{s} } {\mathop{\sum}\limits_{i{\equals}1}^n {{\rm \beta }_{i} \left( {{\bf \bar{\rdelta }}_{i}^{{\rm T}} \left( {\rm \romega } \right){\bf \bar{\rdelta }}_{i} \left( {\rm \romega } \right){\plus}{\bf \bar{\rdelta }}_{i}^{{\rm T}} \left( {{\rm \romega }{\minus}{\rm \rtau }\left( {\rm \romega } \right)} \right){\bf \bar{\rdelta }}_{i} \left( {{\rm \romega }{\minus}{\rm \rtau }\left( {\rm \romega } \right)} \right)} \right){\rm d\romega }} } {\equals}0$$

The fact $\dot{V}\left( t \right)\leq 0$ implies that $${\bf \bar{\rdelta }}\left( t \right)$$ and $${\tilde{\rdelta }}\left( t \right)$$ are bounded. Invoking Barbalat’s lemma, one can obtain from (35) that $\mathop{{\lim }}\limits_{{t\to{\plus}\infty}} {\tilde{\rdelta }}\left( t \right){\equals}\mathop{{\lim }}\limits_{{t\to{\plus}\infty}} {\bf \bar{\rdelta }}\left( t \right){\equals}0$ . This completes the proof of Theorem 1.

In general, the missile guidance is a finite-time control problem. However, due to the asymptotical convergence of the control law (18), the first guidance phase is expressed in terms of infinite-time form in this study. Considering this observation, a terminal condition should be prescribed to terminate the first guidance phase. Hence, an implementable version of the proposed guidance law can be summarised as

(36) $$\matrix{ {\left\{ {\matrix{ {a_{{y,i}} {\equals}{\minus}NV_{M} {\dot{\rlambda }}_{{y,i}} \sin {\rm \rtheta }_{{M,i}} \sin {\rm \rpsi }_{{M,i}} {\plus}NV_{M} {\dot{\rlambda }}_{{z,i}} \cos {\rm \rtheta }_{{M.i}} } \hfill \cr {a_{{z,i}} {\equals}{\minus}NV_{M} {\dot{\rlambda }}_{{y,i}} \cos {\rm \rpsi }_{{M,i}} } \hfill \cr } } \right.} \hfill & {{\rm if}\;\left\{ {\matrix{ {\left| {{\rm \eta }_{i} {\minus}{\rm \eta }_{j} } \right|\leq {\rm {\rvarepsilon}}_{1} } \hfill \cr {\left| {{\rm \rxi }_{i} {\minus}{\rm \rxi }_{j} } \right|\leq {\rm {\rvarepsilon}}_{2} } \hfill \cr } } \right.} \hfill \cr {\left\{ {\matrix{ {a_{{y,i}} {\equals}{\minus}{{V_{M}^{2} } \over {R_{i} }}\sin {\rm \rpsi }_{{M,i}} {\plus}{{u_{i} V_{M} \sqrt {1{\minus}\cos ^{2} {\rm \rsigma }_{i} } } \over {2\sin {\rm \rsigma }_{i} \sin {\rm \rpsi }_{{M,i}} }}} \hfill \cr {a_{{z,i}} {\equals}{\minus}{{V_{M}^{2} } \over {R_{i} }}sin\;{\rm \rtheta }_{{M,i}} \cos {\rm \rpsi }_{{M,i}} {\plus}{{u_{i} V_{M} \sqrt {1{\minus}\cos ^{2} {\rm \rsigma }_{i} } } \over {2\sin {\rm \rsigma }_{i} \sin {\rm \rtheta }_{{M,i}} \cos {\rm \rpsi }_{{M,i}} }}} \hfill \cr } } \right.} \hfill & {{\rm otherwise}} \hfill \cr } $$

in which ε₁ and ε₂ are two small enough positive constants. It can be seen from (36) that the consensus-based guidance law is utilised to realise agreement of the heading error and range-to-go among all missiles in the first guidance phase. When the terminal condition is touched off, all missiles will be guided by the PPN law to achieve salvo attack.

According to the property of the typical PPN law, if ε₁ and ε₂ are small enough, the impact time errors among missiles can be neglected in nature. However, one should be noted that if ε₁ and ε₂ are too small, the initial range-to-go for the terminal guidance phase will be very short or even become zero before |η _i −η _j |≤ε₁ and |ξ _i −ξ _j |≤ε₂ are satisfied, which may also lead to the failure of salve attack. To address this problem and guarantee successful salvo attack, $$\left\Vert {\mib P} \right\Vert$$ should be selected appropriately according to the limitation of the bounded manoeuvrability of missiles to obtain a favoured initial range-to-go for the terminal guidance phase, based on which one can decrease the values of ε₁ and ε₂ to obtain more precise salvo attack.

4.0 SIMULATIONS AND RESULTS

In this section, numerical simulations are carried out to demonstrate the effectiveness of the proposed co-operative law. We consider an engagement scenario in which four missiles are expected to simultaneously attack a stationary target located at (0, 0, 0) m, and missile 0 is selected as the leader in this scenario. It is presumed that the missile’s velocity is 200 m/s. The navigation gain utilised in the second guidance phase is chosen as N=4. The limitations of missile’s accelerations are |a _y,i |≤50 m/s² and |a _z,i |≤50 m/s². The numerical simulations are performed with respect to various terminal conditions for the first guidance phase: case 1: ε₂=0.001; case 2: ε₂=0.003; case 3: ε₂=0.005. Moreover, the other terminal condition is selected as ε₁=0.05.

The initial conditions of multiple missiles are shown in Table 1. The communication topologies among these missiles are presented in Fig. 2. It can be obviously seen from Fig. 2 that the union graphs $${\mib {\cal G}}_{1} \bigcup {{\mib {\cal G}}_{2} } \bigcup {{\mib {\cal G}}_{3} } $$ and $${\mib {\cal G}}_{4} \bigcup {{\mib {\cal G}}_{5} } \bigcup {{\mib {\cal G}}_{6} } $$ are both jointly connected. Hence, the switching signal s(t) can be formulated as

(37) $$s\left( t \right){\equals}\left\{ {\matrix{ 1 \hfill & {k\leq t\,\lt\,k{\plus}{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \hfill \cr 2 \hfill & {k{\plus}{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}\leq t\,\lt\,k{\plus}{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \hfill \cr 3 \hfill & {k{\plus}{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}\leq t\,\lt\,k{\plus}1} \hfill \cr 4 \hfill & {k{\plus}1\leq t\,\lt\,\left( {k{\plus}1} \right){\plus}{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \hfill \cr 5 \hfill & {\left( {k{\plus}1} \right){\plus}{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}\leq t\,\lt\,\left( {k{\plus}1} \right){\plus}{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \hfill \cr 6 \hfill & {\left( {k{\plus}1} \right){\plus}{\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}\leq t\,\lt\,\left( {k{\plus}1} \right){\plus}1} \hfill \cr } k{\equals}0,1,2,...} \right.$$

which is shown in Fig. 3. Additionally, it can be obtained that ${\rm \lambda }_{{\max }} {\equals}2.0$ and ${\rm \lambda }_{{\min }} {\equals}1.0$ . The time delay is set to be τ(t)=0.09 + 0.01 cos(t/10), hence τ _m =0.1 and τ _dm =0.01. Solving LMIs (21)–(23) yields that

(38) $${\mib P}{\equals}\left[ {\matrix{ {0.3711} & {0.1760} \cr {0.1760} & {1.6931} \cr } } \right]$$

Table 1 Initial conditions of multiple missiles

Figure 2 Communication topologies.

Figure 3 Time history of the switching signal.

The simulation results for different cases are given in Figs 4–6, where the time histories of the relative range, the heading error and the missile achieved accelerations are presented in detail. It can be seen from these figures that based on the proposed co-operative guidance law the salvo attack can be achieved in spite of various terminal conditions of the first guidance phase. During the first guidance phase, the consensus of the multi-missile system is realised for different cases, resulting in the same relative range and heading error achieved by all missiles. Then, the typical PPN law takes charge of all missiles to accomplish the salvo attack mission, eventually. Moreover, it can be seen from the time histories of the missile accelerations that the missile accelerations achieved in the first guidance phase change as the communication topology switches and large amplitude of the missile accelerations are required at the beginning of the co-operative guidance to regulate the states of missiles. Then, owing to the typical PPN law, all accelerations converge to zero in the second guidance phase.

Figure 4 Simulation results for Case 1.

Figure 5 Simulation results for Case 2.

Figure 6 Simulation results for Case 3.

To further explore the effect of various terminal conditions on the whole guidance phase, the quantitative analysis regarding the switching time of the two guidance phase, the impact time and the consensus precision are summarised in Table 2, where the consensus precision is defined as $\mathop{\sum}\limits_{i{\equals}1}^3 {\left| {{\rm \rsigma }_{i} \left( {t_{s} } \right){\minus}{\rm \rsigma }_{0} \left( {t_{s} } \right)} \right|} $ with t _s being the switching time. From the simulation results, it is evidently shown that decreasing the value of ε₂ can increase the consensus precision, resulting in the increased impact time in turn. This is mainly due to that lower the value of ε₂ requires more time to regulate the states of missiles to satisfy the terminal condition for the first guidance phase.

Table 2 Summary of the quantitative results