3.1 Derivation of shaping guidance law

The cost function is definited as

(7) \begin{align}\min J = \frac{1}{2}\int_t^{{t_{fi}}} W(\tau ){u^2}(\tau )d\tau \end{align}

where $W(t) \gt 0,t \in \left( {{t_{0i}},{t_{fi}}} \right)$ . ${t_{0i}}$ and ${t_{fi}}$ are the initial time and the final impact time of the $i$ -th missile.

Considering that the kinematics of the $i$ -th missile are established by

(8) \begin{align}{\dot y_i} = {V_{mi}}\sin {\gamma _i}\end{align}

Linearisation at point ${\gamma _{0i}}$ of segment $k$ are as follows

(9) \begin{align}{\dot y_i} = {V_{mi}}\sin\! ({\gamma _{0i}}) + {V_{mi}}\Delta {\gamma _i}\cos\!({\gamma _{0i}}) = {v_{yi}}\end{align}

where ${\gamma _i} = {\gamma _{0i}} + \Delta {\gamma _i}$ . $\Delta {\gamma _i}$ is a small value and ${\gamma _{0i}}$ is a constant in the $k$ segment. ${v_{yi}}$ is the vertical component of ${V_{mi}}$ . Derivations are derived as

(10) \begin{align}{\ddot y_i} = {V_{mi}}\dot{\Delta \gamma _i}\cos\!({\gamma _{0i}}) = {\dot v_{yi}}\end{align}

(11) \begin{align}{\dot \gamma _i} = \dot{\Delta \gamma _i}\end{align}

Then, based on Equation (3), one can be obtained as

(12) \begin{align}{\ddot y_i} = {a_i}\cos\!({\gamma _{0i}}) = {\dot v_{yi}}\end{align}

The linearised kinematic Equations (9) and (10) can be rewritten in the matrix form as

(13) \begin{align}\dot x = Ax + Bu\end{align}

where

(14) \begin{align}\begin{array}{*{20}{l}}{x \buildrel \Delta \over = {{\left[ {\begin{array}{*{20}{l}}{{y_i}}\quad {{v_{yi}}}\end{array}} \right]}^T},u \buildrel \Delta \over = {a_i}}\\[8pt]{A \buildrel \Delta \over = \left[ {\begin{array}{*{20}{l}}0\quad 1\\[5pt]0\quad 0\end{array}} \right],B \buildrel \Delta \over = \left[ {\begin{array}{*{20}{l}}0\\[5pt]{cos({\gamma _{0i}})}\end{array}} \right]}\end{array}\end{align}

Based on the linear control theory, the solution of Equation (13) can be expressed as

(15) \begin{align}x\!\left( {{t_{fi}}} \right) = \Phi \!\left( {{t_{fi}} - t} \right)x\!\left( t \right) + \int_t^{{t_{fi}}} \Phi \!\left( {{t_{fi}} - \tau } \right)Bu\!\left( \tau \right)d\tau \end{align}

(16) \begin{align}\Phi \!\left( {{t_{fi}} - t} \right) = {e^{A({t_{fi}} - t)}}\end{align}

where $\Phi \!\left( t \right)$ is the state transition matrix of the linear system as Equation (13). Expand the right side of Equation (16) as follow

(17) \begin{align}{e^{A\!\left( {{t_{fi}} - t} \right)}} = I + A\!\left( {{t_{fi}} - t} \right) + \frac{{{A^2}}}{{2!}}{\left( {{t_{fi}} - t} \right)^2} + \cdots \end{align}

Calculating the right side of Equation (15) and omitting the high order terms, we can obtain

(18) \begin{align}x\!\left( {{t_{fi}}} \right) = \left[ \begin{array}{c@{\quad}c}1 & {{t_{fi}} - t}\\[6pt]0 & 1\end{array} \right]x\!\left( t \right) + \int_t^{{t_{fi}}} \left[ \begin{array}{c@{\quad}c}1 & {{t_{fi}} - t}\\[6pt]0 & 1\end{array} \right]\left[ {\begin{array}{*{20}{l}}0\\[6pt]{cos({\gamma _{0i}})}\end{array}} \right]u\!\left( \tau \right)d\tau \end{align}

Using shorthand notations as

(19) \begin{align}\begin{array}{*{20}{l}}{{f_1} \buildrel \Delta \over = {x_1}(t) + \left( {{t_{fi}} - t} \right){x_2}(t) - {x_1}\!\left( {{t_{fi}}} \right),}\, {{h_1}(\tau ) \buildrel \Delta \over = - cos({\gamma _{0i}})\left( {{t_{fi}} - t} \right)}\\[6pt]{{f_2} \buildrel \Delta \over = {x_2}(t) - {x_2}\!\left( {{t_{fi}}} \right),}\qquad\qquad\qquad\ {{h_2}(\tau ) \buildrel \Delta \over = - \cos\!({\gamma _{0i}})}\end{array}\end{align}

Equation (15) can be rewritten as

(20) \begin{align}\left\{ {\begin{array}{*{20}{l}}{{f_1} = \int_t^{{t_{fi}}} {h_1}(\tau )u(\tau )d\tau }\\[6pt]{{f_2} = \int_t^{{t_{fi}}} {h_2}(\tau )u(\tau )d\tau }\end{array}} \right.\end{align}

By introducing a new variable denoted as $\lambda $ , then by Equation (20) we can obtain

(21) \begin{align}f \buildrel \Delta \over = {f_1} - \lambda {f_2} = \int_t^{{t_{fi}}} \left[ {{h_1}(\tau ) - \lambda {h_2}(\tau )} \right]u(\tau )d\tau \end{align}

Furthermore, Equation (21) can be transformed as

(22) \begin{align}f = \int_t^{{t_{fi}}} \left( {{h_1}(\tau ) - \lambda {h_2}(\tau )} \right){W^{1/2}}(\tau ){W^{ - 1/2}}(\tau )u(\tau ){\rm{d}}\tau \end{align}

Based on the Schwarz inequality, the following inequality can be obtained as

(23) \begin{align}\frac{{{f^2}}}{{2\int_t^{{t_{fi}}} {{\left[ {{h_1}(\tau ) - \lambda {h_2}(\tau )} \right]}^2}{W^{ - 1}}(\tau ){\rm{d}}\tau }} \le \frac{1}{2}\int_t^{{t_{fi}}} W(\tau ){u^2}(\tau ){\rm{d}}\tau \end{align}

It is obvious that the right side of Equation (23) equals the cost function defined in Equation (12). In other words, it provides a lower bound for the minimum cost function. According to the Schwarz inequality, the condition of inequality Equation (23) becoming the equation is that the guidance command must be as

(24) \begin{align}u(t) = K\!\left[ {{h_1}(t) - \lambda {h_2}(t)} \right]{W^{ - 1}}(t)\end{align}

where $K$ is a constant to be determined. Substituting Equation (24) into Equation (19), the constant $K$ can be obtained as

(25) \begin{align}K = \frac{{{f_1}}}{{\int_t^{{t_{fi}}} h_1^2(\tau ){W^{ - 1}}(\tau )d\tau - \lambda \int_t^{{t_{fi}}} {h_1}(\tau ){h_2}(\tau ){W^{ - 1}}(\tau )d\tau }}\end{align}

To facilitate the derivation, some notations are given as follows

(26) \begin{align}\left\{ {\begin{array}{*{20}{l}}{{g_{1i}} \buildrel \Delta \over = \int_t^{{t_{fi}}} h_1^2(\tau ){W^{ - 1}}(\tau )d\tau }\\[6pt]{{g_{2i}} \buildrel \Delta \over = \int_t^{{t_{fi}}} {h_1}(\tau ){h_2}(\tau ){W^{ - 1}}(\tau )d\tau }\\[6pt]{{g_{3i}} \buildrel \Delta \over = \int_t^{{t_{fi}}} h_2^2(\tau ){W^{ - 1}}(\tau )d\tau }\end{array}} \right.\end{align}

Thus Equation (25) can be rewritten as

(27) \begin{align}K = \frac{{{f_1}}}{{{g_{1i}} - \lambda {g_{2i}}}}\end{align}

Substituting Equation (27) into Equation (24), the guidance command can be obtained as

(28) \begin{align}u(t) = \frac{{{f_1}\left[ {{h_1}(t) - \lambda {h_2}(t)} \right]{W^{ - 1}}(t)}}{{{g_{1i}} - \lambda {g_{2i}}}}\end{align}

From Equation (23), the minimum value of cost function can be expressed as

(29) \begin{align}J = \frac{{{{\left( {{f_1} - \lambda {f_2}} \right)}^2}}}{{2\!\left( {{g_{1i}} - 2\lambda {g_{2i}} + {\lambda ^2}{g_{3i}}} \right)}}\end{align}

Taking the derivative of $J$ with respect to $\lambda $ and imposing $dJ/d\lambda = 0$ , the optimal solution of $\lambda $ can be expressed as

(30) \begin{align}{\lambda ^*} = \frac{{{f_1}{g_{2i}} - {f_2}{g_{1i}}}}{{{f_1}{g_{3i}} - {f_2}{g_{2i}}}}\end{align}

Substituting Equation (30) into Equation (28), the optimal guidance command can be obtained as

(31) \begin{align}{u^*}(\tau ) = \frac{{\left[ {{f_1}{h_1}(\tau ){g_{3i}} - {g_{2i}}\!\left( {{f_2}{h_1}(\tau ) + {f_1}{h_2}(\tau )} \right) + {f_2}{h_2}(\tau ){g_{1i}}} \right]{W^{ - 1}}(\tau )}}{{{g_{1i}}{g_{3i}} - g_{2i}^2}}\end{align}

Substituting Equations (14) and (19) into Equation (31), the optimal trajectory shaping guidance law is expressed as

(32) \begin{align}{a_{si}} = {u^*}(t) = {k_{1i}}{y_i} + {k_{2i}}{v_{yi}} + {k_{3i}}\end{align}

where the guidance gains ${k_{1i}}$ , ${k_{2i}}$ and ${k_{3i}}$ are given as follows

(33) \begin{align}\left\{ {\begin{array}{*{20}{l}}{{k_{1i}} = \dfrac{{ - {t_{goi}}{g_{3i}} + {g_{2i}}}}{{{g_{1i}}{g_{3i}} - g_{2i}^2}}{W^{ - 1}}\left( t \right)}\\[15pt]{{k_{2i}} = \dfrac{{ - t_{goi}^2{g_{3i}} + 2{g_{2i}}{t_{goi}} - {g_{1i}}}}{{{g_{1i}}{g_{3i}} - g_{2i}^2}}{W^{ - 1}}\left( t \right)}\\[15pt]{{k_{3i}} = \dfrac{{\left( {{t_{goi}}{g_{3i}} - {g_{2i}}} \right){y_i}\!\left( {{t_{fi}}} \right) + \left( {{g_{1i}} - {t_{goi}}{g_{2i}}} \right){v_{yi}}\!\left( {{t_{fi}}} \right)}}{{{g_{1i}}{g_{3i}} - g_{2i}^2}}{W^{ - 1}}\!\left( t \right)}\end{array}} \right.\end{align}

where ${t_{goi}} = \left( {{t_{fi}} - t} \right)$ is the time-to-go of the $i$ -th missile, and ${g_{1i}}$ , ${g_{2i}}$ , ${g_{3i}}$ are defined as

(34) \begin{align}\left\{ {\begin{array}{*{20}{l}}{{g_{1i}} = \int_t^{{t_{fi}}} {{\left( {{t_{fi}} - \tau } \right)}^2}{W^{ - 1}}(\tau )d\tau }\\[10pt]{{g_{2i}} = \int_t^{{t_{fi}}} \left( {{t_{fi}} - \tau } \right)\cos\!({\gamma _{0i}}){W^{ - 1}}(\tau )d\tau }\\[10pt]{{g_{3i}} = \int_t^{{t_{fi}}} {{\cos }^2}({\gamma _{0i}}){W^{ - 1}}(\tau )d\tau }\end{array}} \right.\end{align}

From Equation (34), the analytical solution of ${g_{1i}}$ , ${g_{2i}}$ , ${g_{3i}}$ depend on the prescribed the final impact time ${t_{fi}}$ and the weighting function ${W^{ - 1}}(t)$ which is a time-varying function. In order to achieve time-cooperative guidance for multiple missiles and remove constraint of the final impact time ${t_{fi}}$ , we make the assumption as follows

(35) \begin{align}{W^{ - 1}}\left( \tau \right) = 1\end{align}

(36) \begin{align}{\gamma _{0i}} \equiv 0\end{align}

After simplification, the shaping guidance command in Equation (32) can be transformed as

(37) \begin{align}{a_{si}} = - 6\frac{{{y_i}}}{{t_{goi}^2}} - 4\frac{{{v_{yi}}}}{{{t_{goi}}}} + 6\frac{{{y_i}({t_{fi}})}}{{t_{goi}^2}} - 2\frac{{{v_{yi}}({t_{fi}})}}{{{t_{goi}}}}\end{align}

Note that these results are identical to the optimal control guidance law as studied in Ref. [Reference Lee, Jeon and Lee2]. Combining Equation (4) with Equation (9), the following equation can be obtained as

(38) \begin{align}{\dot v_{yi}} = {V_{mi}}{\dot \gamma _i}\cos {\gamma _{0i}}\end{align}

The integral of Equation (38) from $0$ to ${t_{fi}}$ can be expressed as

(39) \begin{align}\int_0^{{t_{fi}}} {\dot v_{yi}}dt = {V_{mi}}\cos {\gamma _{0i}}\int_0^{{t_{fi}}} {\dot \gamma _i}dt\end{align}

Then, the final speed constraint ${v_{yi}}({t_{fi}})$ can be transformed into the heading angle constraint ${\gamma _i}({t_{fi}})$ by

(40) \begin{align}{v_{yi}}({t_{fi}}) = {v_{y0i}} + {V_{mi}}\left( {{\gamma _i}({t_{fi}}) - {\gamma _{0i}}} \right)\cos {\gamma _{0i}}\end{align}

Remark 1. $\Delta {\gamma _i}$ is considered as a small value in some ${\gamma _{0i}}(k)$ segment (at point ${\gamma _{0i}}$ of segment $k$ ). Once the absolute value of $\Delta {\gamma _i}$ is bigger than 5 degrees ( $\left| {\Delta {\gamma _i}} \right|{ \gt 5^ \circ }$ ), the linearised kinematics is switched in next ${\gamma _{0i}}(k + 1)$ segment. Furthermore, their relationship is satisfied by

(41) \begin{align}{\gamma _{0i}}(k + 1) = {\gamma _{0i}}(k) + \Delta {\gamma _i}\end{align}

where $k = 1,2, \cdots ,n$ , $\Delta {\gamma _i} = \pm {5^ \circ }$ .

3.2 Derivation of cooperative shaping guidance law

In this subsection, the concept of algebraic graph theory is introduced by Ref. [Reference Liu, Yan and Liu27]. Suppose that communication network between missiles can be expressed as $G = \left\{ {V,E} \right\}$ , where $V = \left\{ {1,2, \cdots ,n} \right\}$ denotes the set of vertices, $E \subseteq V \times V$ denotes the edge set of graph, the subscript $i$ denotes the $i$ -th missile, ${e_{ij}}$ is the edge of graph $G$ , and ${e_{ij}} \subseteq E$ indicates that the agents $i$ and $j$ can receive message from each other in an undirected graph. If there is a connection between any two missiles in the graph, then the graph $G$ is connected. In the directed graph, ${e_{ij}} \subseteq E$ means the missile $i$ can receive message from missile $j$ . In addition, ${A_G} = ({a_{ij}}) \in {R^{n \times n}}$ is the adjacency matrix of graph $G$ . If ${e_{ij}} \subseteq E$ , then ${a_{ij}} \gt 0$ ; ${a_{ij}} = 0$ , otherwise. It is worth noting that ${a_{ij}} = {a_{ji}}$ when the communication topology is an undirected graph. The Laplace matrix of the graph $G$ is defined as $L = ({L_{ij}}) \in {R^{n \times n}}$ , which can be expressed as

(42) \begin{align}{L_{ij}} = \left\{ {\begin{array}{*{20}{l}}{ - {a_{ij}}\;\;\;i \ne j}\\[6pt]{\sum\limits_{j = 1,j \ne i}^n {a_{ij}}\;\;\;i = j}\end{array}} \right.\end{align}

The final impact time of the $i$ th missile can be defined as

(43) \begin{align}{t_{fi}} = {t_{goi}} + t\end{align}

Let ${X_i} = {t_{fi}}$ , a multiple missiles system is described as

(44) \begin{align}{\dot X_i} = u_i^{nom}\end{align}

where $u_i^{nom}$ is the consensus protocol which should be designed with the information of the missile itself and its neighbours.

In order to verify the finite-time stability of the system, the following Lemma is given.

Lemma 3. If the undirected graph is connected. The consensus protocol $u_i^{nom}$ shown in Equation (45) can ensure that the state of multi-missile converges in finite time.

(45) \begin{align}u_i^{nom} = sgn\!\left( {\sum\limits_{i,j = 1}^n {a_{ij}}({X_j} - {X_i})} \right){\left| {\sum\limits_{i,j = 1}^n {a_{ij}}({X_j} - {X_i})} \right|^{{\beta _i}}}\end{align}

where ${a_{ij}}$ are the elements of the weight coefficient matrix. And ${\beta _i}$ is a constant, $0 \lt {\beta _i} \lt 1$ .

Theorem 1. Subject to the system in Equation (44), in the existence of $k,\varepsilon \gt 0$ , under the action of the guidance command in Equation (46), multiple missiles can achieve consistent convergence of the cooperative variable ${t_{fi}}$ in a finite time ( $T \le {T_1} + {T_2}$ ), thereby ensuring that multiple missiles simultaneously arrive at the target.

(46) \begin{align}{a_i} = (u_i^{nom} - k{s_i} - \varepsilon sgn({s_i}) - {k_c})/{k_a}\end{align}

where $u_i^{nom} = sgn\!\left( {\sum\limits_{i,j = 1}^n {a_{ij}}({t_{fj}} - {t_{fi}})} \right){\left| {\sum\limits_{i,j = 1}^n {a_{ij}}({t_{fj}} - {t_{fi}})} \right|^{{\beta _i}}}$ , ${k_c} = 1 - \cos\!({\eta _i})\left( {1 + {{\sin }^2}({\eta _i})/10} \right) + {\sin ^2}({\eta _i}) \cos\!({\eta _i})/5$ , ${k_a} = - {r_i}\sin\! ({\eta _i})\cos\!({\eta _i})/5V_{mi}^2$ , ${\dot s_i} = - k{s_i} - \varepsilon sgn({s_i})$ .

Proof.

1. (1) System transformation

   Before proving the theorem, some preparations are made as follows.

   The derivative of ${t_{fi}}$ can be presented as

   (47) \begin{align}{\dot t_{fi}} = {\dot t_{goi}} + 1\end{align}
   From Ref. [Reference Liu, Yan and Zhang16], in order to remove the small-angle assumption of the ${\eta _i}$ , an improved estimation method of ${t_{goi}}$ is chosen as follows
   (48) \begin{align}{t_{goi}} = {r_i}\!\left( {1 + {{\sin }^2}({\eta _i})/10} \right)/{V_{mi}}\end{align}
   The derivation of ${t_{goi}}$ is derived as
   (49) \begin{align}{\dot t_{goi}} = {\dot r_i}\left( {1 + {{\sin }^2}({\eta _i})/10} \right)/{V_{mi}} + {r_i}\sin\! ({\eta _i})\cos\!({\eta _i}){\dot \eta _i}/5{V_{mi}}\end{align}
   Substituting Equations (1–4) into Equation (49), then we can obtain
   (50) \begin{align}{{\dot t}_{goi}} & = { - \cos\!({\eta _i})\left( {1 + {{\sin }^2}({\eta _i})/10} \right) + {{\sin }^2}({\eta _i})\cos\!({\eta _i})/5}\nonumber\\[6pt] & \quad - {a_i}{r_i}\sin\! ({\eta _i})\cos\!({\eta _i})/5V_{mi}^2 \end{align}
   Substituting Equation (50) into Equation (47) and using shorthand notations, Equation (47) can be rewritten as
   (51) \begin{align}{\dot t_{fi}} = {k_c} + {k_a}{a_i}\end{align}
   where
   (52) \begin{align}\left\{ {\begin{array}{*{20}{l}}{{k_c} = 1 - \cos\!({\eta _i})\left( {1 + {{\sin }^2}({\eta _i})/10} \right) + {{\sin }^2}({\eta _i})\cos\!({\eta _i})/5}\\[6pt]{{k_a} = - {r_i}\sin\! ({\eta _i})\cos\!({\eta _i})/5V_{mi}^2}\end{array}} \right.\end{align}

2. (2) Convergence of the sliding surface

   In order to achieve time-cooperative guidance for multiple missiles, considering the final impact time constraint, a sliding surface is defined as

   (53) \begin{align}{s_i} = {t_{fi}} - {t_{fi}}(0) - \int_0^t u_i^{nom}dt\end{align}
   Take the derivative of sliding surface ${s_i}$ as
   (54) \begin{align}{\dot s_i} = {\dot t_{fi}} - u_i^{nom}\end{align}
   Apply the reaching law by choosing
   (55) \begin{align}{\dot s_i} = - k{s_i} - \varepsilon sgn({s_i})\end{align}
   Take the Lyapunov function ${V_{1i}}$ as
   (56) \begin{align}{V_{1i}} = \frac{1}{2}s_i^2\end{align}
   Taking the derivative of ${V_{1i}}$ with respect to time $t$ , and substituting Equation (55) into it, we can get
   (57) \begin{align}{\dot V_{1i}} = {s_i}{\dot s_i} = {s_i}({-} k{s_i} - \varepsilon sgn({s_i})) \le - ks_i^2 - \varepsilon \left| {{s_i}} \right| \le 0\end{align}
   Based on Equation (56), the following equation is obtained
   (58) \begin{align}\left| {{s_i}} \right| = \sqrt {2{V_{1i}}} \end{align}
   Substituting Equation (58) into Equation (57), we have
   (59) \begin{align}{\dot V_{1i}} \le - ks_i^2 - \varepsilon \!\left| {{s_i}} \right| = - 2k{V_{1i}} - \sqrt 2 \varepsilon V_{1i}^{\frac{1}{2}}\end{align}
   According to Lemma 1 , the sliding surface ${s_i}$ can converge to zero in a finite time ${T_1}$ , where
   (60) \begin{align}{T_1} \le (1/k)\ln \!\left( {\left( {2kV_{1i}^{\frac{1}{2}}\left( {{x_0}} \right) + \sqrt 2 \varepsilon } \right)/\sqrt 2 \varepsilon } \right)\end{align}

3. (3) Convergence of finite-time cooperative errors

   When ${s_i} = 0$ , from Equation (54), the following equation can be obtained as

   (61) \begin{align}{\dot t_{fi}} = u_i^{nom}\end{align}
   Take the Lyapunov function ${V_{2i}}$ as
   (62) \begin{align}{V_{2i}} = \frac{1}{4}\sum\limits_{i,j = 1}^n {a_{ij}}{\left( {{t_{fj}} - {t_{fi}}} \right)^2} = \frac{1}{2}T_f^TL{T_f}\end{align}
   where ${T_f} = {\left[ {{t_{f1}},{t_{f2}}, \cdots ,{t_{fn}}} \right]^T}$ is the vector consisting of the final impact time of each missile and $L$ is the Laplace matrix. Based on the symmetry $L$ due to the assumption of undirected graph, the partial derivative of ${V_{2i}}$ with respect to ${t_{fi}}$ can be presented as
   (63) \begin{align}\frac{{\partial {V_{2i}}}}{{\partial {t_{fi}}}} = - \sum\limits_{j = 1}^n {a_{ij}}\left( {{t_{fj}} - {t_{fi}}} \right)\end{align}
   Taking the derivative of ${V_{2i}}$ with respect to $t$ , combining it with Equations (63), (61) and Lemma 5 , the following equation can be obtained as
   (64) \begin{align}{\frac{{d{V_{2i}}}}{{dt}}} & = \sum\limits_{i = 1}^n \frac{{\partial {V_{2i}}}}{{\partial {t_{fi}}}}{{\dot t}_{fi}}\nonumber\\[6pt]& = - \sum\limits_{i = 1}^n {a_{ij}}\!\left( {{t_{fj}} - {t_{fi}}} \right) \cdot {{\left( {sgn\!\left( {\sum\limits_{j = 1}^n {a_{ij}}\!\left( {{t_{fj}} - {t_{fi}}} \right)} \right)\sum\limits_{j = 1}^n {a_{ij}}\!\left( {{t_{fj}} - {t_{fi}}} \right)} \right)}^{{\beta _i}}}\\[6pt]& = - \sum\limits_{i = 1}^n {{\left( {{{\left( {\sum\limits_{j = 1}^n {a_{ij}}\!\left( {{t_{fj}} - {t_{fi}}} \right)} \right)}^2}} \right)}^{\frac{{1 + {\beta _i}}}{2}}}\nonumber\\[6pt]& \le - {{\left( {\sum\limits_{i = 1}^n {{\left( {\sum\limits_{j = 1}^n {a_{ij}}\!\left( {{t_{fj}} - {t_{fi}}} \right)} \right)}^2}} \right)}^{\frac{{1 + {\beta _i}}}{2}}}\nonumber\end{align}
   when ${V_{2i}} \ne 0$ , we obtain
   (65) \begin{align}\frac{{\sum\limits_{i = 1}^n {{\left( {\sum\limits_{j = 1}^n {a_{ij}}\left( {{t_{fj}} - {t_{fi}}} \right)} \right)}^2}}}{{{V_{2i}}}} = \frac{{T_f^T{L^T}L{T_f}}}{{\frac{1}{2}T_f^TL{T_f}}}\end{align}
   According to Lemma 4 , Equation (65) can be transformed as
   (66) \begin{align}\frac{{T_f^T{L^T}L{T_f}}}{{\frac{1}{2}T_f^TL{T_f}}} \geq 2{\lambda _2}(L)\end{align}
   Substituting Equation (66) into Equation (64), the following inequality is obtained
   (67) \begin{align}{\frac{{d{V_{2i}}}}{{dt}}} & \le - {{\left( {\frac{{\sum\limits_{i = 1}^n {{\left( {\sum\limits_{j = 1}^n {a_{ij}}\left( {{t_{fj}} - {t_{fi}}} \right)} \right)}^2}}}{{{V_{2i}}}}{V_{2i}}} \right)}^{\frac{{1 + {\beta _i}}}{2}}}\nonumber\\[3pt]& \le - {{\left( {2{\lambda _2}(L)} \right)}^{\frac{{1 + {\beta _i}}}{2}}}V_{2i}^{\frac{{1 + {\beta _i}}}{2}}\end{align}
   According to Lemma 2 , the final impact time of the missiles can converge to zero in a finite time ${T_2}$ , where
   (68) \begin{align}{T_2} \le \left( {\frac{1}{{{\lambda _2}{{(L)}^{\frac{{1 + {\beta _i}}}{2}}}(1 - {\beta _i})}}} \right)V_{2i}^{\frac{{1 - {\beta _i}}}{2}}({x_0})\end{align}

Table 1.

The initial conditions of missiles

Figure 2.

Communication topology for three missiles.

4.1 Multiple missiles cooperative attack

In this subsection, we compared with PNG algorithm to further demonstrate the superiority of the finite-time TSCGL in Equation (46). The results are presented in Fig. 3, and the attack results are given in Table 2.

Figure 3.

Simulation results of case 1: (a) Trajectory of missiles, (b) Heading angle, (c) Guidance command, (d) Relative distance, (e) Time-to-go.

Table 2.

Attack results of case 1

From Fig. 3 and Table 2, both the finite-time TSCGL and PNG can attack the target. The miss-distances of the finite-time TSCGL are less than $0.36m$ . The miss-distances of PNG are at least $1.05m$ larger than finite-time TSCGL. Compared with PNG, finite-time TSCGL has higher attack accuracy. The trajectories of finite-time TSCGL are more curved than that of PNG. The final heading angle errors between the finite-time TSCGL and PNG are bigger than at least $16.59deg$ , which implies that finite-time TSCGL has better damage performance. At the beginning of the guidance process, a larger guidance command is required to adjust the posture of the missiles to be consistent. The time-to-go of the finite-time TSCGL can converge within a short time. Compared with PNG, the finite-time TSCGL has faster convergence speed. The final impact time of finite-time TSCGL are $72s$ with their errors less than $0.1s$ . The final impact time of PNG are $66.68s$ , $70.58s$ , $68.58s$ and their differences are approximately $4s$ . In conclusion, the finite-time TSCGL can achieve cooperative attack with higher attack accuracy, better damage performance and faster convergence speed.

4.2 Compared with other methods of estimating time-to-go

In this subsection, another method of estimating ${t_{go}}$ in Ref. [Reference Jeon, Lee and Tahk31] is provided to compare that in Equation (48), which is expressed as

(70) \begin{align}{t_{goi}} = {r_i}\!\left( {1 + \eta _i^2/10} \right)/{V_{mi}}\end{align}

The results are illustrated in Fig. 4, and the miss-distance are given in Table 3. As can be seen in Fig. 4, multiple missiles can attack the target cooperatively. A striking difference was noted when the method of estimating ${t_{go}}$ was considered individually in Equation (48) and Equation (70). From Table 3, the miss-distances of Equation (48) are less than $0.36m$ . The miss-distances of Equation (70) are at least $1.95m$ larger than Equation (48). By comparison, Equation (48) demonstrates the superiority in attack accuracy. The time-to-go of the two methods can converge within a short time, which indicates that the proposed finite-time TSCGL has fast convergence speed. Similarly, their final impact time are approximately the same, which shows that the proposed finite-time TSCGL can achieve simultaneous attack. Consequently, the different methods of estimating time-to-go mainly affect the attack accuracy rather than the convergence speed and final impact time, further indicating that the proposed finite-time TSCGL has good expandability.

4.3 Cooperative attack with Monte Carlo simulations

In order to further verify the robustness of the finite-time TSCGL in Equation (46), 300 Monte Carlo simulations were performed with different conditions, and the bias are listed in Table 4. The initial conditions of the missiles and the target are the same as in Table 1. The simulation results are illustrated in Figs 5 and 6.

It can be seen from Fig. 5 that multiple missiles can cooperatively attack the target in all Monte Carlo bias scenarios. The distributions of miss distance in Fig. 6 are less than $1.83m$ . In Table 5, the average mean and standard deviation of the miss-distances are $0.87m$ , $0.39m$ . The maximum and minimum are $1.80m$ , $0.11m$ . All the statistics can meet the accuracy requirement. Consequently, finite-time TSCGL has strong robustness.

Table 3.

Miss-distance of case 2

Figure 4.

Simulation results of case 2: (a) Trajectory of missiles, (b) Heading angle, (c) Guidance command, (d) Relative distance, (e) Leading angle, (f) Time-to-go.

4.4 Expansion to a target with constant speed

Inspired by the concept of predicted interception point (PIP) in Ref. [Reference Liu, Yan and Zhang16], a target with a constant speed is considered to verify the generality of the proposed finite-time TSCGL. The position of virtual stationary target can be expressed as follows:

(71) \begin{align}\left\{ {\begin{array}{*{20}{l}}{{X_{TP}} = {X_T} + {V_T}\cos {\theta _T}{t_{{\rm{go}}}}}\\[6pt]{{Y_{TP}} = {Y_T} + {V_T}\sin {\theta _T}{t_{{\rm{go}}}}}\end{array}} \right.\end{align}

where ( ${X_{TP}}$ , ${Y_{TP}}$ ) is the PIP, ( ${X_T}$ , ${Y_T}$ ) is current position of the target, and ${\theta _T}$ is the heading angle of the target.

Table 4.

Bias of Monte Carlo simulations

Table 5.

Monte Carlo results of the miss-distance

Figure 5.

Trajectory of missiles.

Figure 6.

Miss-distance of case 3: (a) Missile 1, (b) Missile 2, (c) Missile 3.

The target moves with the heading angle of $0deg$ and speed of $20m/s$ . It can be seen from Fig. 7 that finite-time TSCGL can simultaneously attack the target with a constant speed under the PIP theory. The miss-distance are $1.31m$ , $1.37m$ and $1.39m$ , which implies that finite-time TSCGL has high attack accuracy. The time-to-go can converge within a short time, which shows that finite-time TSCGL has fast convergence speed. Therefore, the finite-time TSCGL demonstrates good expandability.

Figure 7.

Simulation results of case 4: (a) Trajectory of missiles, (b) Heading angle, (c) Guidance command, (d) Relative distance, (e) Time-to-go.