2.1 Preliminaries

In multi-agent systems, the communication topology among the agents is represented by the graph $G\left( {N,\zeta, A} \right)$ , where $N = \left\{ {{N_1},{N_2},{N_3}, \ldots } \right\}$ denotes the set of all nodes, symbolising the collection of missiles. Here, $\zeta $ represents the edges, and $A = \left[ {{a_{ij}}} \right] \in {\mathbb{R}^{n \times n}}$ is the adjacency matrix. If the $i$ -th agent can receive information from the $j$ -th agent, then ${a_{ij}} = 1$ ; otherwise, ${a_{ij}} = 0$ , with the specific condition ${a_{ii}} = 0$ .

Consider nonlinear system

(1) \begin{align} {\dot x_b}\left( t \right) = f\left( {{x_b}\left( t \right)} \right)\!,\ f\!\left( 0 \right) = 0,{x_b}\left( t \right) \in {\mathbb{R}^n} \end{align}

where ${x_b} = 0$ is an equilibrium point for the nonlinear function $f\,:\,{\mathbb{R}^n} \to {\mathbb{R}^n}$ .

Lemma 1. Suppose there exists a positive definite function ${V_b}\left( {{x_b}\left( t \right)} \right)$ , which satisfies the inequality

(2) \begin{align} {\dot V_b}\left( {{x_b}\left( t \right)} \right) \le - {\tau _1}{V_b}\left( {{x_b}\left( t \right)} \right) - {\tau _2}{V_b}{({x_b}\left( t \right))^\theta }, \end{align}

where ${\tau _1} \gt 0,{\tau _2} \gt 0,0 \lt \theta \lt 1$ .

Then the system converges in finite time, and the upper bound of the setting-time

(3) \begin{align} \le \frac{1}{{{\tau _1}\left( {1 - \theta } \right)}}{\mathrm{ln}}\frac{{{\tau _1}V_b^{1 - \theta }\left( {{x_b}\left( {{t_0}} \right)} \right) + {\tau _2}}}{{{\tau _2}}}, \end{align}

where ${V_b}\left( {{x_b}\left( {{t_0}} \right)} \right)$ is the initial value of the Lyapunov function ${V_b}\left( {{x_b}\left( t \right)} \right)$ [Reference Bhat and Bernstein32, Reference Orlov33].

Lemma 2. Consider a first-order multi-agent system with $n$ individuals, where $\dot x = u$ . When the communication topology network graph $G$ is undirected and connected, the consensus protocol

(4) \begin{align} {u_i} = {\mathrm{sign}}\left( {\mathop \sum \limits_{j = 1}^n {c_{ij}}\left( {{x_j} - {x_i}} \right)} \right){\left| {\mathop \sum \limits_{j = 1}^n {c_{ij}}\left( {{x_j} - {x_i}} \right)} \right|^{{\alpha _1}}}, \end{align}

solves the finite-time consistency problem of the multi-agent system, where $0 \lt {\alpha _1} \lt 1$ , and $sign\left( - \right)\,:\,{\mathbb{R}^n} \to {\mathbb{R}^n}$ is a sign function [Reference Wang and Xiao34].

Lemma 3. If a constant $\mu $ satisfies the inequality

(5) \begin{align} f\left( {{x_i},t} \right) - f{\left( {{x_0},t} \right)_2} \le \mu {x_i} - {x_0}_2^\alpha, \end{align}

where $i \in N$ , the nonlinear function is $\alpha - H\ddot older$ continuity, where $0 \lt \alpha \lt 1$ [Reference Hölder35].

2.2 Problem formulation

Figure 1 illustrates the scenario of multiple missiles coordinating to strike a manoeuvering target. The inertial coordinate system used in this paper is the North-Up-East coordinate system, with the Y-axis perpendicular to the horizontal plane and upward, expressed as ${O^I}{x^I}{y^I}{z^I}$ . ${O^L}{x^L}{y^L}{z^L}$ represents the LOS coordinate system. ${q_\varepsilon }$ denotes the elevation angle of the LOS, ${q_\beta }$ represents the azimuth angle.

Figure 1.

Schematic diagram of three-dimensional coordinated strike.

According to the Coriolis theorem, the following equations can be derived as follows:

(6) \begin{align} {\bf{V}}_{los}^i = \frac{{\partial {{\bf{R}}_i}}}{{\partial t}} + {{\bf{\Omega }}_i} \times {{\bf{R}}_i} \end{align}

(7) \begin{align} {\bf{a}}_{los}^i = \frac{{\partial {\bf{V}}_{los}^i}}{{\partial t}} + {{\bf{\Omega }}_i} \times {\bf{V}}_{los}^i \\[6pt] \nonumber \end{align}

where ${{\bf{R}}_i} = {[{r_i},0,0]^T}$ , ${\bf{\Omega }} = {[{\dot q_{\beta, i}}{\mathrm{sin}}{q_{\varepsilon,i}},{\dot q_{\beta, i}}{\mathrm{cos}}{q_{\varepsilon, i}},{\dot q_{\varepsilon, i}}]^T}$ represents the angular velocity vector of the LOS coordinate system with respect to the inertial coordinate system. ${\bf{a}}_{los}^i = {{\bf{a}}_T} - {\bf{a}}_M^i + {\bf{d}}_{los}^i$ denotes the acceleration vectors in the LOS coordinate system. ${{\bf{a}}_T} = {[{a_{Tx}},{a_{Ty}},{a_{Tz}}]^T}$ , ${\bf{a}}_M^i = {[a_{Mx}^i,a_{My}^i,a_{Mz}^i]^T}$ represent the accelerations of the $i$ -th missile and the target in the LOS, respectively. ${\bf{d}}_{los}^i = {[d_x^i,d_y^i,d_z^i]^T}$ stands for the perturbation caused by unmodeled dynamics. Thus, Equations (6) and (7) can be derived as:

(8) \begin{align} V_{los}^i = \left[ {\begin{array}{c}{{{\dot r}_i}}\\{{r_i}{{\dot q}_{\varepsilon, i}}}\\{ - {r_i}{{\dot q}_{\beta, i}}{\mathrm{cos}}{q_{\varepsilon, i}}}\\{}\end{array}} \right] \end{align}

(9) \begin{align} a_{los}^i & = \left[ {\begin{array}{c}{{{\ddot r}_i}}\\[3pt] {{{\dot r}_i}{{\dot q}_{\varepsilon, i}} + {r_i}{{\ddot q}_{\varepsilon, i}}}\\[3pt] { - {r_i}{{\ddot q}_{\beta, i}}{\mathrm{cos}}{q_{\varepsilon, i}} - {{\dot r}_i}{{\dot q}_{\beta, i}}{\mathrm{cos}}{q_{\varepsilon, i}} + {r_i}{{\dot q}_{\beta, i}}{{\dot q}_{\varepsilon, i}}{\mathrm{sin}}{q_{\varepsilon, i}}}\\[3pt] {}\end{array}} \right] + \left[ {\begin{array}{c}{{{\dot q}_{\beta, i}}{\mathrm{sin}}{q_{\varepsilon, i}}}\\[3pt] {{{\dot q}_{\beta, i}}{\mathrm{cos}}{q_{\varepsilon, i}}}\\[3pt] {{{\dot q}_{\varepsilon, i}}}\\[3pt] {}\end{array}} \right] \times \left[ {\begin{array}{c}{{{\dot r}_i}}\\[3pt] {{r_i}{{\dot q}_{\varepsilon, i}}}\\[3pt] { - {r_i}{{\dot q}_{\beta, i}}{\mathrm{cos}}{q_{\varepsilon, i}}}\\[3pt] {}\end{array}} \right] \nonumber \\[3pt] & = \left[ {\begin{array}{*{20}{l}}{{{\ddot r}_i} - {r_i}\dot q_{\varepsilon, i}^2 - {r_i}\dot q_{\beta, i}^2{\mathrm{cos}}{q_{\varepsilon, i}}}\\[3pt] {2{{\dot r}_i}{{\dot q}_{\varepsilon, i}} + {r_i}{{\ddot q}_{\varepsilon, i}} + {r_i}{{\dot q}^2}_{\beta, i}{\mathrm{sin}}{q_{\varepsilon, i}}{\mathrm{cos}}{q_{\varepsilon, i}}}\\[3pt] { - 2{{\dot r}_i}{{\dot q}_{\beta, i}}{\mathrm{cos}}{q_{\varepsilon, i}} + 2{r_i}{{\dot q}_{\beta, i}}{{\dot q}_{\varepsilon, i}}{\mathrm{sin}}{q_{\varepsilon, i}} - {r_i}{{\ddot q}_{\beta, i}}{\mathrm{cos}}{q_{\varepsilon, i}}}\\[3pt] {}\end{array}} \right] \end{align}

The derivative of the remaining flight time can be derived as follows.

(10) \begin{align} {\dot t_{go,i}} = - 1 + \frac{{{r_i}{{\ddot r}_i}}}{{{{\dot r}_i}^2}} \end{align}

The state variables of the system are then defined as ${x_1} = {r_i},{x_2} = {\dot r_i},{x_3} = {e_{{q_{\varepsilon, i}}}},{x_4} = {e_{{q_{\beta, i}}}},{x_5} = {\dot e_{{q_{\varepsilon, i}}}}, {x_6} = {\dot e_{{q_{\beta, i}}}}$ , where ${e_{{q_{\varepsilon, i}}}} = {q_{\varepsilon, i}} - {\bar q_{\varepsilon, i}}$ and ${e_{{q_{\beta, i}}}} = {q_{\beta, i}} - {\bar q_{\beta, i}}$ are the errors of the elevation and azimuth angle of LOS, respectively. ${\bar q_{\varepsilon, i}}$ and ${\bar q_{\beta, i}}$ are the desired elevation and azimuth angle, respectively, assumed to be constant. Therefore, the three-dimensional guidance system can be described as:

(11) \begin{align} \left\{ {\begin{array}{l}{{{\dot x}_1} = {x_2}}\\[3pt] {{{\dot x}_2} = {a_{Tx}} - a_{Mx}^i + {x_1}x_5^2 + {x_1}x_6^2{\mathrm{co}}{{\mathrm{s}}^2}{q_{\varepsilon, i}} + d_x^i}\\[3pt] {{{\dot x}_3} = {x_5}}\\[3pt] {{{\dot x}_4} = {x_6}}\\[3pt] {{{\dot x}_5} = \frac{1}{{{x_1}}}\left[ {{a_{Ty}} - a_{My}^i - 2{x_2}{x_5} - {x_1}x_6^2{\mathrm{sin}}{q_{\varepsilon, i}}{\mathrm{cos}}{q_{\varepsilon, i}} + d_y^i} \right]}\\[3pt] {{{\dot x}_6} = 2{x_5}{x_6}{\mathrm{tan}}{q_{\varepsilon, i}} + \left( {a_{Mz}^i - {a_{Tz}} - d_z^i} \right)/\left( {{x_1}{\mathrm{cos}}{q_{\varepsilon, i}}} \right) - 2{x_2}{x_6}/{x_1}}\\[3pt] {{{\dot x}_7} = - 1 + {x_1}/x_2^2\left[ {{a_{Tx}} - a_{Mx}^i + {x_1}x_5^2 + {x_1}x_6^2{\mathrm{co}}{{\mathrm{s}}^2}{q_{\varepsilon, i}} + d_x^i} \right]}\end{array}} \right. \end{align}

where ${q_{\varepsilon, i}} = {x_3} + {\bar q_{\varepsilon, i}}$ .

Considering the thruster failure of a missile in practice, the mathematical model of the failure can be expressed as follows [Reference Akmal, Yusoff and Arshad37]:

(12) \begin{align} {P_i} = \begin{cases}{P_c^i} & {}{0 \lt t \le t_{fal}^i}\\[3pt] {k_{fal}^iP_c^i + {\mathrm{\Delta }}_p^i} & {}{t_{fal}^i \lt t} \end{cases} \end{align}

where $P_c^i$ is the thrust command, $t_{fal}^i$ is the fault occurrence time, $k_{fal}^i$ is the thrust gain coefficient, and ${\mathrm{\Delta }}_p^i$ is the thrust deviation coefficient.

3.1 Guidance law in the LOS direction

The guidance law in the LOS direction is designed to achieve simultaneous attack. First, the missile arrival time ${t_{f,i}} = t + {t_{go,i}}$ is defined.

To further simplify, a virtual control instruction is designed

(13) \begin{align} {\tilde u_{x,i}} = \frac{{x_1^2x_5^2}}{{x_2^2}} - \frac{{{x_1}a_{Mx}^i}}{{x_2^2}} + \frac{{x_1^2x_6^2{\mathrm{co}}{{\mathrm{s}}^2}\left( {{x_3} + {{\bar q}_{\varepsilon, i}}} \right)}}{{x_2^2}}. \end{align}

In that way,

(14) \begin{align} {\dot t_{f,i}} = {\tilde u_{x,i}} + \tilde d_x^i, \end{align}

where $\tilde d_x^i = {x_1}\left( {{a_{Tx}} + d_x^i} \right)/x_2^2$ represents the disturbance caused by manoeuver of the target and unmodeled dynamics.

Drawing upon the first-order system described in Equation (14) and leveraging the standard finite-time consensus protocol from Lemma 2, the following robust guidance law is formulated.

(15) \begin{align} {u_i}\left( t \right) = \beta \mathop \sum \limits_{j \in {{\mathcal{N}}_i}} {a_{ij}}{\mathrm{sign}}{\left( {{x_j}\left( t \right) - {x_i}\left( t \right)} \right)^\gamma } - {d_i}{\mathrm{sign}}{\left( {{x_i}\left( t \right) - {x_0}\left( t \right)} \right)^\alpha } \end{align}

where $0 \lt \gamma \le 1,0 \lt \alpha \lt 1,{d_i} \in \left\{ {0,1} \right\}$ . In addition, ${\mathrm{sign}}{\left( x \right)^\gamma } = |x{|^\gamma }{\mathrm{sign}}\left( x \right)$ .

As illustrated in Fig. 2, when the system has no actuator faults, ${d_i} = 0$ , it represents a fully distributed system. However, in the case of thruster failure, a leader is designated to implement the cooperative guidance law using a leader-follower approach, which takes into account the reduced manoeuverability resulting from the thruster failure.

Figure 2.

Robust nonlinear cooperative guidance law.

Proof. When ${d_i} = 0$ , as indicated by Lemma 2, this distributed system converges in finite time.

When ${d_i} \ne 0,\gamma = 1$ ,

(16) \begin{align} {u_i}\left( t \right) = \beta \mathop \sum \limits_{j \in {{\mathcal{N}}_i}} {a_{ij}}{\mathrm{sign}}\left( {{x_j}\left( t \right) - {x_i}\left( t \right)} \right) - {d_i}{\mathrm{sign}}{\left( {{x_i}\left( t \right) - {x_0}\left( t \right)} \right)^\alpha }, \end{align}

where ${x_0}$ is the leader.

Assume the system is ${\dot x_i} = f\left( {{x_i}\left( t \right),t} \right) + {u_i}\left( t \right)$ , and ${\dot x_0} = f\left( {{x_0}\left( t \right),t} \right)$ . Define

(17) \begin{align} {e_i}\left( t \right) = {x_i}\left( t \right) - {x_0}\left( t \right)\!,{\mathrm{\;\;\;\;}}i \in N, \end{align}

then

(18) \begin{align} {\dot e_i}\left( t \right) = f\left( {{x_i}(t),t} \right) - f\left( {{x_0}(t),t} \right) + {u_i},{\mathrm{\;\;\;\;}}i \in N. \end{align}

According the Lemma 3,

(19) \begin{align} \Vert f\left( {{x_i},t} \right) - f{\left( {{x_0},t} \right)\Vert_2} \le \mu \Vert {x_i} - {x_0}\Vert_2^\alpha . \end{align}

Consider the Lyapunov function

(20) \begin{align} V\left( t \right) = \frac{1}{2}\mathop \sum \limits_{i = 1}^N e_i^T\left( t \right){e_i}\left( t \right), \end{align}

then

(21) \begin{align*} \dot V\left( t \right) & = \mathop \sum \limits_{i = 1}^N \,e_i^T{\dot e_i} \nonumber\\[3pt]& \le \mathop \sum \limits_{i = 1}^N \,\mu \Vert {e_i}\Vert^{1 + \alpha } - \beta {e^T}\left( {L \otimes {I_n}} \right)e - \mathop \sum \limits_{i = 1}^N \,{d_i}\Vert{e_i}\Vert^{1 + \alpha }\nonumber\\[3pt]\end{align*}\begin{align}&\le - {\left( {\mathop \sum \limits_{i = 1}^N {{\left( {{d_{{\mathrm{min}}}} - \mu } \right)}^{\frac{2}{{1 + \alpha }}}}\Vert{e_i}\Vert^2} \right)^{\frac{{1 + \alpha }}{2}}}\nonumber\\[3pt]& = - cV{(t)^{\frac{{1 + \alpha }}{2}}} \end{align}

where $e = {({e_1},{e_2}, \cdots, {e_N})^T}$ , $c = {2^{\frac{{1 + \alpha }}{2}}}\left( {{d_{{\mathrm{min}}}} - \mu } \right)$ , and $ \otimes $ denotes the Kronecker product. Thus, according to Lemma 1, it can be concluded that the system converges in finite time.

Therefore, whether it is a fully distributed system or a leader-follower system, the time-to-go will converge within a finite time. This completes the proof.

3.2 Guidance laws in the normal direction of LOS

In this subsection, finite-time guidance laws for both the elevation and azimuth directions of the LOS are designed and analysed to satisfy the attack angle constraint.

Design a nonlinear sliding surface,

(22) \begin{align} s = {x_5} - \left( {a/\left( {1 + {e^{k{x_3}}}} \right) - 0.5a} \right), \end{align}

and take the reaching law $\dot s = - {k_1}s - {k_2}{\mathrm{sign}}{(s)^\rho }$ . Thus,

(23) \begin{align} a_{My}^i = - 2{x_2}{x_5} - {x_1}x_6^2{\mathrm{sin}}{q_{\varepsilon, i}}{\mathrm{cos}}{q_{\varepsilon, i}} + {k_1}{x_1}s + {k_2}{x_1}{\mathrm{sign}}{(s)^\rho } + \frac{{ak{e^{k{x_3}}}{x_1}{x_5}}}{{{{(1 + {e^{k{x_3}}})}^2}}} + \tilde q_y^i \end{align}

where $\tilde d_y^i = \left( {{a_{Ty}} + d_y^i} \right)/{x_1}$ represents the disturbance caused by manoeuver of the target and unmodeled dynamic.

In the subsequent analysis, the finite-time convergence of the proposed guidance law will be demonstrated from two perspectives. Firstly, it will be established that reaching the sliding surface occurs in finite time. Secondly, it will be proven that the system converges to the equilibrium point exponentially.

Proof. Considering the Lyapunov function ${V_1} = \frac{1}{2}{s^2}$ , its derivative can be derived as:

(24) \begin{align}{\dot V_1} & = s\dot s \nonumber\\[3pt]& = - {k_1}{s^2} - {k_2}|s{|^{\rho + 1}} \nonumber\\[3pt]& = - 2{k_1}{V_1} - {2^{\left( {\rho + 1} \right)/2}}{k_2}V_1^{\left( {\rho + 1} \right)/2}. \end{align}

Based on Lemma 1, it can be concluded that when the designed reaching law is employed, the system will converge to the sliding surface within a finite time. The upper limit of the convergence time depends on the initial state of the system.

(25) \begin{align} T \le \frac{1}{{2{k_1}\left( {\left( {1 - \rho } \right)/2} \right)}}{\mathrm{ln}}\frac{{2{k_1}V_1^{\left( {1 - \rho } \right)/2}\left( 0 \right) + {2^{\left( {\rho + 1} \right)/2}}{k_2}}}{{{2^{\left( {\rho + 1} \right)/2}}{k_2}}} \end{align}

Next, the convergence process from the sliding surface to the equilibrium state is analysed. When the system slides on the sliding surface, $s = 0$ , which means

(26) \begin{align} {x_5} = a/\left( {1 + {e^{k{x_3}}}} \right) - 0.5a \end{align}

Considering the Lyapunov function ${V_2} = \frac{1}{2}x_3^2$ , its derivative can be derived as:

(27) \begin{align} {\dot V_2} &= {x_3}{\dot x_3} \nonumber\\[3pt]&= a{x_3}\left( {\frac{1}{{1 + {e^{k{x_3}}}}} - \frac{1}{2}} \right) \end{align}

where $a,k \gt 0$ .

Considering different cases, it can be observed that when ${x_3} \gt 0$ , $\frac{1}{{1 + {e^{k{x_3}}}}} - \frac{1}{2} \lt 0$ , hence ${\dot V_2} \lt 0$ . Similarly, when ${x_3} \lt 0$ , $\frac{1}{{1 + {e^{k{x_3}}}}} - \frac{1}{2} \gt 0$ , again ${\dot V_2} \lt 0$ . Moreover, if and only if when ${x_3} = 0$ , ${\dot V_2} = 0$ . Therefore, ${\dot V_2}$ is negative definite, indicating that the system is uniformly asymptotically stable.

On the sliding surface, it also holds that $\dot s = 0$ . Based on Equation (22), it can be further derived that

(28) \begin{align} {\dot x_5} & = - \frac{{ak{e^{k{x_3}}}}}{{{{(1 + {e^{k{x_3}}})}^2}}}{x_5} \nonumber\\&= - ak\frac{{{e^a}\left( {1 + k\left( {{x_3} - a} \right)} \right) + O\left( {x_3^2} \right)}}{{{{(1 + {e^{ka}}\left( {1 + k\left( {{x_3} - a} \right)} \right))}^2} + O\left( {x_3^2} \right)}} \end{align}

At the equilibrium point, it can be further simplify to:

(29) \begin{align} {\dot x_5} & = - ak\frac{{1 + k{x_3} + O\left( {x_3^2} \right)}}{{{{(2 + k{x_3})}^2} + O\left( {x_3^2} \right)}}{x_5} \nonumber\\& \approx - ak\frac{{1 + k{x_3} + O\left( {x_3^2} \right)}}{{4 + 4k{x_3} + O\left( {x_3^2} \right)}}{x_5}\nonumber\\& \approx - \frac{{ak}}{4}{x_5} \end{align}

Integrating both sides of the equation, it can be obtained that:

(30) \begin{align} \mathop \int \limits_{x_5^0}^{x_5^f} \frac{1}{{{x_5}}}d{x_5} = \mathop \int \limits_0^T - \frac{{ak}}{4}dt \end{align}

By solving the differential align, near the equilibrium point, the system state can be represented as:

(31) \begin{align} x_{5}^{\,f} = x_5^0{e^{ - \frac{{ak}}{4}T}} \end{align}

Therefore, the system exhibits exponential convergence from the sliding surface to the equilibrium state. This completes the proof.

Likewise, for the LOS azimuthal orientation, the sliding surface is configured as

(32) \begin{align} s = {x_6} - \left( {a/\left( {1 + {e^{k{x_4}}}} \right) - 0.5a} \right). \end{align}

Therefore, the command in the azimuthal direction is

(33) \begin{align} a_{Mz}^i = {x_1}{\mathrm{cos}}{q_{\varepsilon, i}}\left[ {\frac{{2{x_2}{x_6}}}{{x1}} - 2{x_5}{x_6}{\mathrm{tan}}{q_{\varepsilon, i}} - {k_1}s - {k_2}{\mathrm{sign}}{{(s)}^\rho } - \frac{{ak{e^{k{x_4}}}{x_6}}}{{{{(1 + {e^{k{x_4}}})}^2}}}} \right] + \tilde d_z^i \end{align}

where $\tilde d_z^i = \left( {{a_{Tz}} + d_z^i} \right){x_1}{\mathrm{cos}}{q_{\varepsilon, i}}$ is the disturbance caused by the target manoeuver and unmodeled dynamics.

Using the proposed sliding mode guidance law ensures that the system converges to the sliding surface in finite time, and the convergence from the sliding surface to the equilibrium state is exponential. The proof follows similar reasoning as presented above.

After obtaining the guidance commands in LOS frame system from Equations (13), (23) and (33), inner loop commands in the ballistic frame system can be derived through the frame transformations, including thrust $P$ , lift $L$ , drag $D$ and lateral force $Z$ . More details can be found in Ref. (Reference Wang, Liu, Wang and Tang39). Further, attitude commands can be computed, such as angle-of-attack and sideslip, and angle tracking is achieved through the inner-loop attitude controller [Reference Tao, Lin, Song and Li40].

The overall flowchart of the simulation is shown as Fig. 3.

Figure 3.

Flowchart of inner and outer loop integrated simulation.

3.3 Design of ESO

In order to estimate and compensate for disturbances generated by target manoeuvers and unmodeled dynamics, an ESO is applied for LOS direction and LOS normal direction, respectively.

Regarding the disturbance $\tilde d_x^i$ in the LOS direction, the ESO is designed as follows:

(34) \begin{align} \left( {\begin{array}{*{20}{l}}{{{\dot z}_{x1}} = {z_{x2}} - {\beta _{x1}}\left( {{t_{f,i}} - {z_{x1}}} \right) + {{\tilde u}_{x,i}}}\\[3pt] {{{\dot z}_{x2}} = {\beta _2}\left( {{t_{f,i}} - {z_{x1}}} \right)}\end{array}} \right.\end{align}

where ${\beta _{x1}} = 2{w_{xo}},{\beta _{x2}} = w_{xo}^2$ , and ${w_{xo}}$ is the bandwidth of the observer. According to the proof by Gao et al. [Reference Gao38], appropriate parameter design can achieve accurate estimation of disturbances, that is, ${z_{x1}} \to {t_{f,i}}$ and ${z_{x2}} \to \tilde d_x^i$ .

Similarly, in order to compensate for target manoeuvers and other disturbances, In the elevation direction of LOS, the ESO is designed as:

(35) \begin{align} \left( {\begin{array}{l}{{{\dot z}_{y1}} = {z_{y2}} + {\beta _{y1}}\left( {{x_3} - {z_{y1}}} \right)}\\[3pt] {{{\dot z}_{y2}} = {z_{y3}} + {\beta _{y2}}\left( {{x_3} - {z_{y1}}} \right) + \left( { - a_{My}^i - 2{x_2}{x_5} - {x_1}x_6^2{\mathrm{sin}}{q_{\varepsilon, i}}{\mathrm{cos}}{q_{\varepsilon, i}}} \right)/{x_1}}\\[3pt] {{{\dot z}_{y3}} = {\beta _{y3}}\left( {{x_3} - {z_{y1}}} \right)}\end{array}} \right..\end{align}

In the azimuth direction of LOS, the observer is designed as follows,

(36) \begin{align}\left( {\begin{array}{l}{{{\dot z}_{z1}} = {z_{z2}} + {\beta _{z1}}\left( {{x_4} - {z_{z1}}} \right)}\\[3pt] {{{\dot z}_{z2}} = {z_{z3}} + {\beta _{z2}}\left( {{x_4} - {z_{z1}}} \right) + 2{x_5}{x_6}{\mathrm{tan}}{q_{\varepsilon, i}} + a_{Mz}^i/\left( {{x_1}{\mathrm{cos}}{q_{\varepsilon, i}}} \right) - 2{x_2}{x_6}/{x_1}}\\[3pt] {{{\dot z}_{z3}} = {\beta _{z3}}\left( {{x_4} - {z_{z1}}} \right)}\end{array}} \right. \end{align}

where ${\beta _{y1}} = 3{w_{yo}},{\beta _{y2}} = 3w_{yo}^2,{\beta _{y3}} = 3w_{yo}^3$ , ${\beta _{z1}} = 3{w_{zo}},{\beta _{z2}} = 3w_{zo}^2,{\beta _{z3}} = 3w_{zo}^3$ , and ${w_{yo}}$ , ${w_{zo}}$ are the bandwidth of the observer, respectively. By setting the parameters reasonably, it can be achieved that ${z_{y3}} \to \tilde d_y^i$ and ${z_{z3}} \to \tilde d_z^i$ .

4.1 Initial conditions

In the simulation, the parameters of the scenario and the hyperparameters of the guidance law are shown in Table 1. When utilising the proposed nonlinear surface, the maximum available overload in the X-Y-Z direction under the LOS frame is set to $\left( { \pm 1.5g, \pm 10g, \pm 10g} \right)$ , respectively. However, in the case of adopting the linear sliding surface, the overload is adjusted to $\left( { \pm 1.5g, \pm 30g, \pm 30g} \right)$ , respectively. This adjustment ensures the efficacy of the method using the linear sliding surface. Here, $g$ represents the acceleration due to gravity.

Table 1.

Simulation initial parameters

4.2 Numerical simulation

In this subsection, simulations are conducted separately for both ideal and failure conditions.

In this scenario, three missiles are simultaneously launched from different positions to engage a moving target, satisfying terminal time and angle constraints. The simulation results are illustrated in Fig. 4. As shown in Fig. 4(b), the remaining flight time converges within $1.5$ seconds and remains constant until the end of the strike. Consequently, the acceleration in the X-axis direction, displayed in Fig. 4(e), maintains a zero value after the remaining flight time convergence.

Figure 4.

Spatiotemporal cooperative guidance under ideal conditions.

As depicted in Fig. 4(c) and (d), while ensuring the same final convergence time, the linear SMC necessitates larger guidance commands in the early stages, as shown in Fig. 4(e), resulting in a faster initial convergence rate. However, as the guidance error reduces in the later stages, the convergence rate significantly slows down. In contrast, the proposed nonlinear SMC requires guidance commands that are only one-third of those needed by the linear guidance law. The system converges at a relatively stable rate, resulting in a smoother flight trajectory, as illustrated in Fig. 4(a). Furthermore, Fig. 4(f) demonstrates that, with the same convergence time, the proposed nonlinear guidance law can handle more stringent input saturation constraints and achieve smaller terminal guidance errors.

Figure 5 compares simulation outcomes corresponding to various sliding mode surfaces. Figure 6 depicts these surfaces. The findings demonstrate that adjusting the hyperparameters of the nonlinear sliding surface appropriately enables striking a balance between input saturation constraints and convergence speed.

Figure 5.

The influence of hyperparameters on guidance results.

Figure 6.

Comparison of sliding surface.

In this scenario, assume that at simulation time $t = 5{\mathrm{s}}$ , Missile 1 experiences a thruster failure. That is, $t_{fal}^1 = 5,k_{fal}^1 = 0,{\mathrm{\Delta }}_p^1 = 0$ . According to the law of conservation of energy, although Missile 1 has lost all of its propulsion, the kinetic energy acquired earlier is sufficient to support it in striking the target. Figure 7 illustrates the simulation results using both the proposed RCGL and the fully distributed cooperative guidance law (DCGL). The inclusion of the faulty Missile 1 in cooperative decision-making notably degrades the overall guidance performance of the cooperative system, as evident from Fig. 7(d). When employing the proposed leader-follower hybrid cooperative guidance law, Missiles 2 and 3 track Missile 1 along the line of sight. However, with the fully DCGL, the coordination of time-to-go persists, ultimately leading to compromised angle maintenance.

Figure 7.

Spatiotemporal cooperative guidance under thruster failure condition.

Table 2 displays the terminal guidance outcomes under the thruster failure condition. Entries marked with $ - - {\mathrm{\;}}$ indicate state divergence. The results illustrate that the proposed robust cooperative guidance law successfully manages thruster complete failure, ensuring dependable cooperative attack capabilities.

Table 2.

Terminal errors under thruster failure

4.3 HIL experiment

The HIL experiment serves as a valuable tool to identify potential issues prior to actual flight tests. It allows for evaluating the validity of model assumptions and considering factors that may be overlooked in real-world scenarios. Furthermore, the HIL experiment facilitates real-time assessment of the performance and robustness of guidance and control algorithms. This proactive approach not only minimises the dependency on flight tests but also mitigates associated risks, providing opportunities to optimise system performance. Figure 8 illustrates the HIL system architecture, while Fig. 9 details the experimental equipment used.

Figure 8.

HIL system architecture.

Figure 9.

HIL experimental equipment.

The results of the HIL experiment are depicted in Fig. 10. They indicate that the proposed robust cooperative guidance law effectively mitigates signal transmission delays and unmodeled disturbances, demonstrating robust real-time performance. These findings validate the effectiveness of the proposed guidance law on the hardware platform, thereby supporting its practical application.

Figure 10.

Spatiotemporal coordination of HIL experiments.