2.1 Problem formulation

The geometric relationship between the relative motion of a missile and a target on a vertical plane is established in Fig. 1. In the figure, $M$ and $T$ denote the positions of the missile and the target, respectively. ${v_M}$ and ${v_T}$ indicate the velocity of the missile and the target, respectively. ${a_M}\,$ and $\,{a_T}\,$ represent the normal acceleration of the missile and the target, respectively. ${\theta _M}\,$ and $\,{\theta _T}\,$ are the flight path angle of the missile and the target, respectively. $q$ is the line-of-sight (LOS) angle between the missile and the target. $r$ is the relative distance between the missile and the target. The relative motion equation between the missile and the target can be obtained from Fig. 1

(1) \begin{equation}\dot r = {v_T}\cos \left( {q - {\theta _T}} \right) - {v_M}\cos \left( {q - {\theta _M}} \right)\!,\end{equation}

(2) \begin{equation} r\dot q = - {v_T}\sin \left( {q - {\theta _T}} \right) + {v_M}\sin \left( {q - {\theta _M}} \right)\!, \end{equation}

(3) \begin{equation}{\dot \theta _M} = {a_M}/{v_M},\end{equation}

(4) \begin{equation}{\dot \theta _T} = {a_T}/{v_T},\end{equation}

where $\,\dot r\,$ is the derivative of the relative distance with respect to time, $\dot q\,$ is the derivative of the LOS angle $q$ , it represents the LOS angular rate, and ${\dot \theta _M}\,$ and $\,{\dot \theta _T}\,$ are the derivatives of the missile’s and target’s flight path angles with respect to time.

When both sides of Equation (2) are differentiated with respect to time, the following equation can be derived:

(5) \begin{align}\ddot q = - 2\frac{{\dot r}}{r}\dot q + \frac{{{{\dot v}_M}\sin (q - {\theta _M}) - {a_M}\cos \left( {q - {\theta _M}} \right)}}{r} + \frac{{{a_T}\cos \left( {q - {\theta _T}} \right) - {{\dot v}_T}\sin (q - {\theta _T})}}{r},\nonumber\\\end{align}

where $\,{\dot v_M}\,$ and $\,{\dot v_T}\,$ are the tangential accelerations of the missile and the target; and $\,\ddot q\,$ is the derivative of the LOS angular rate with respect to time.

For the convenience of derivation, we firstly assume that the missile and the target are flying in a constant tangential speed. In fact, the guidance law is suitable for a varying speed missile because the tangential velocity is treated as the system uncertainty and is estimated and compensated by the observer together with the target manoeuvering. Then, Equation (5) can be organised as

(6) \begin{equation}\ddot q = - 2\frac{{\dot r}}{r}\dot q - \frac{{{a_{Mq}}}}{r} + \frac{{{a_{Tq}}}}{r},\end{equation}

where $\,{a_{Mq}} = {a_M}\cos \left( {q - {\theta _M}} \right)\,$ and $\,{a_{Tq}} = {a_T}\cos \left( {q - {\theta _T}} \right)\,$ are the components of the missile and the target accelerations that are perpendicular to the LOS, respectively.

In accordance with the principle of parallel approach, the purpose of a guidance law is to make the LOS angular rate $\,\dot q\,$ approach to zero by adjusting $\,{a_{Mq}}$ , thereby ensuring that the missile will attack the target accurately.

Assumption 1. Given the missile’s and the seeker’s physical limits, the missile’s and the target’s acceleration components and their first and second derivatives ( ${a_{Mq}}$ , ${a_{Tq}}$ , ${\dot a_{Tq}}$ , and ${\ddot a_{Tq}}$ ) are bounded by the following conditions:

(7) \begin{equation}\left| {{a_{Mq}}} \right| \le {A_m},\,\left| {{a_{Tq}}} \right| \le {A_t},\,\left| {{{\dot a}_{Tq}}} \right| \le {A_1},\,\left| {{{\ddot a}_{Tq}}} \right| \le {A_2};\end{equation}

where $\,{A_m}$ , ${A_t}$ , ${A_1}$ , and ${A_2}$ are unknown bounds which represent the manoeuverability; and the seeker’s minimum working distance is ${r_0}$ . The variable $r$ satisfies

(8) \begin{equation} r \ge {r_0}, \end{equation}

Considering the dynamic model of the autopilot, the autopilot model is approximated to the first-order inertia as follows:

(9) \begin{equation}{\dot a_{Mq}} = - \frac{1}{\tau }{a_{Mq}} + \frac{1}{\tau }u, \end{equation}

where $\,\tau \,$ is the autopilot dynamic lag constant of the missile, and $u$ is the guidance command acceleration to be provided to the missile.

For problems with an impact angle constraint, the impact angle is first defined as the angle between the velocity vector of the missile and the target at the end of the collision. Therefore, considering the constraint on the impact angle, the flight path angle should be strictly limited when the missile is hitting the target. Defining the end of guidance as ${t_f}$ , the expected impact angle of the missile is ${\theta _d}$ , and the desired LOS angle is ${q_d}$ at the end of guidance. Therefore, the design of the guidance law that considers the impact angle constraint refers to the missile hitting the target accurately at the desired impact angle; that is^{(Reference Zhang, Tang and Guo33)},

(10) \begin{equation}\mathop {\lim }\limits_{t \to {t_f}} r\left( t \right)\dot q\left( t \right) \to 0, \end{equation}

(11) \begin{equation}{\theta _M}\left( {{t_f}} \right) - {\theta _T}\left( {{t_f}} \right) = {\theta _d}, \end{equation}

(12) \begin{equation}\left| {{\theta _M}\left( {{t_f}} \right) - {q_d}} \right| < \frac{\pi }{2}.\end{equation}

Equation (12) indicates that the target is within the field of view of the missile’s seeker when the missile hits the target. From Equations (2) and (10):

(13) \begin{equation}{v_T}\sin \left( {{\theta _T}\left( {{t_f}} \right) - {q_d}} \right) \approx {v_M}\sin \left( {{\theta _M}\left( {{t_f}} \right) - {q_d}} \right)\!. \end{equation}

From Equation (13), a unique flight path angle of the missile ${\theta _M}\left( {{t_f}} \right)$ can be ascertained if the flight path angle of target ${\theta _T}\left( {{t_f}} \right)$ is known for the desired missile impact angle ${\theta _d}$ . Then, the desired LOS angle can be obtained from Equations (10) and (12). Hence, the control of the impact angle can be transformed into the control of the final LOS angle.

2.2 Fundamental theories

Before designing the guidance law, the definition of finite-time stability in sliding mode control is first introduced.

Definition 1 ^{(Reference Bhat and Bernstein34)}. Consider the following system:

(14) \begin{equation}\dot x\!\left( t \right) = f\!\left( {x,t} \right),x \in {{\bf{R}}^n}, \end{equation}

where $f\left( x \right):D \to {{\bf{R}}^n}$ is defined as a value of $D$ in $n$ dimensional space ${{\bf{R}}^n}$ that satisfies the local Lipschitz continuous condition. For the system Equation (14), $f\!\left( x \right):U \to {{\bf{R}}^n}$ is a continuous function for $x$ on the semi-open domain $U$ containing the origin. Finite-time convergence is represented by $\forall {x_0} \in {U_0} \subset {{\bf{R}}^n}$ , there exists a continuous function $T\!\left( x \right):{U_0}\backslash \left\{ 0 \right\} \to \left( {0, + \infty } \right)$ , thereby making the solution of the system Equation (14) $x\!\left( {{x_0},t} \right)$ satisfy: when $t \in \left[ {0,T\left( {{x_0}} \right)} \right]$ , there exists $x\left( {{x_0},t} \right) \in {U_0}\backslash \left\{ 0 \right\}$ and $\mathop {\lim }\limits_{t \to T\left( {{x_0}} \right)} x\left( {{x_0},t} \right) = 0$ , and when $t > T\left( {{x_0}} \right)$ , there exists $x\left( {{x_0},t} \right) = 0$ . Suppose the equilibrium point of the system Equation (14) is $x = 0$ , when and only when the system is strongly stable and converges in a limited time, the equilibrium point $x = 0$ of the system is then regarded as stable in a finite time. If $U = {U_0} \in {{\bf{R}}^n}$ , then the equilibrium point is stable globally in a finite time.

Lemma 1 ^{(Reference Bhat and Bernstein35)}. Suppose the existence of a continuous differentiable positive definite function ${\kern 1pt} V\!\left( x \right){\kern 1pt} $ defined in a neighborhood of the origin. For the real numbers $c \gt 0$ and $0 \lt \beta \lt 1$ , there is $\dot V\left( x \right) + c{V^\beta }\left( x \right) \le 0$ and $x \in U\backslash \left\{ 0 \right\}$ . Then, the system could converge to origin in finite time, and the upper bound of the convergence time is satisfied as follows:

(15) \begin{equation}T\left( {{x_0}} \right) \le \frac{{{V^{1 - \beta }}\left( {{x_0}} \right)}}{{c\left( {1 - \beta } \right)}}. \end{equation}

3.1 Target manoeuvering estimator

Consider the following system:

(16) \begin{equation}\dot x = f + bu + d. \end{equation}

where $f$ is the known state variable of the system; $u$ and $b{\kern 1pt} $ are the control input and coefficient, respectively; and $d$ represents target manoeuvering, which is regarded as system interference.

For System Equation (15), the general disturbance observer is defined as

(17) \begin{equation}\left\{ \begin{array}{l}{{\dot z}_0} = {z_1} + {k_1}{\left| {x - {z_0}} \right|^{1 - 1/p}}{\mathop{\rm sgn}} \left( {x - {z_0}} \right) + f + bu\\[5pt] {{\dot z}_1} = {k_2}{\left| {x - {z_0}} \right|^{1 - 2/p}}{\mathop{\rm sgn}} \left( {x - {z_0}} \right)\end{array} \right., \end{equation}

where ${\kern 1pt} {k_1},{k_2} > 0$ and $p > 2$ are parameters to be designed; and ${z_0}$ and ${z_1}$ are estimations of $x$ and $d$ , respectively.

Theorem 1 ^{(Reference He and Lin36)}. Variables ${\eta _1} = x - {z_0}$ and ${\eta _2} = d - {z_1}$ are defined as estimation errors. If the derivative of system interference satisfies $\left| {\dot d} \right| \le {\dot d^{\max }}$ with ${\kern 1pt} {\dot d^{\max }}$ being a positive constant, then the vector can converge to the following domain in finite time:

(18) \begin{equation}\left\| \eta \right\| \le {\left( {\frac{{{{\dot d}^{\max }}\left\| \textbf{{B}} \right\|}}{{{\lambda _{\min }}\left( \textbf{{Q}} \right)}}} \right)^{\left( {p - 1} \right)/\left( {p - 2} \right)}}, \end{equation}

where $\boldsymbol{\eta} = {\left[ {{\eta _1},{\eta _2}} \right]^T}$ , ${\lambda _{\min }}\left( \cdot \right)$ refers to the minimum eigenvalue of matrix $\left( \cdot \right)$ and the vector 2-norm $\left\| {\cdot} \right\|$ , and

(19) \begin{equation}\textbf{{Q}} = \left[ \begin{array}{c@{\quad}c} {{k_1}{k_2} + k_1^3\frac{{p - 1}}{p}} & { - k_1^2\frac{{p - 1}}{p}} \\[8pt] { - k_1^2\frac{{p - 1}}{p}} & {k\frac{{p - 1}}{p}} \end{array} \right],\,\textbf{{B}} = \left[ \begin{array}{c} { - {k_1}} \\ 2 \end{array} \right], \end{equation}

Proof of Theorem 1. According to Equations (16) and (17), the estimation error variable can be acquired as

(20) \begin{equation}\left\{ \begin{array}{l}{{\dot \eta }_1} = {\eta _2} - {k_1}{\left| {{\eta _1}} \right|^{1 - 1/p}}{\mathop{\rm sgn}} \left( {{\eta _1}} \right)\\[5pt] {{\dot \eta }_2} = {{\dot a}_{Tq}} - {k_2}{\left| {{\eta _1}} \right|^{1 - 2/p}}{\mathop{\rm sgn}} \left( {{\eta _1}} \right)\end{array} \right., \end{equation}

For differential Equation (16), construct the Lyapunov function as follows

(21) \begin{equation}{V_1} = \frac{{{k_2}p}}{{p - 1}}{\left| {{\eta _1}} \right|^{2\left( {p - 1} \right)/p}} + \frac{1}{2}\eta _2^2 + \frac{1}{2}{\left( {{k_1}{{\left| {{\eta _1}} \right|}^{\left( {p - 1} \right)/p}}{\mathop{\rm sgn}} \left( {{\eta _1}} \right) - {\eta _2}} \right)^2}, \end{equation}

which can be written as the following vector form

(22) \begin{equation}{V_1} = {\boldsymbol{\eta }^T}\textbf{{P}}\boldsymbol{\eta }, \end{equation}

where

(23) \begin{equation}\textbf{{P}} = \frac{1}{2}\left[ {\begin{array}{c@{\quad}c}{\frac{{{k_2}p}}{{p - 1}} + k_1^2} &{ - {k_1}}\\[5pt]{ - {k_1}} &2\end{array}} \right]\!, \end{equation}

As ${k_2} > 0$ and $p > 2$ are positive definite and radially unbounded, that is

(24) \begin{equation}{\lambda _{\min }}\left( \textbf{{P}} \right){\left\| \boldsymbol{\eta} \right\|^2} \le {V_1} \le {\lambda _{\max }}\left( \textbf{{P}} \right){\left\| \boldsymbol{\eta} \right\|^2}, \end{equation}

where ${\lambda _{\max }}\left( \cdot \right)$ refers to the maximum eigenvalue of matrix $\left( \cdot \right)$ .

Advert that ${V_1}$ is continuously differentiable, except on the set $\Omega = \left\{ {\left( {{\eta _1},{\eta _2}} \right) \in {{{R}}^2}\left| {{\eta _1} = 0} \right.} \right\}$ . However, if ${\eta _1} = 0$ , ${\eta _2} \ne 0$ , one can imply $\,{\dot \eta _1} = {\eta _2} \ne 0$ , this means that the trajectories of system Equation (20) just cross the area $\Omega \;$ and cannot maintain on it, except when the initial values $\,{\eta _1} = 0$ , ${\eta _2} = 0$ , have been achieved. Thus, the time derivative of $\,{V_1}\,$ can be calculated by derivation as

(25) \begin{align}{{\dot V}_1} & = \left( {\frac{{{k_2}p}}{{p - 1}} + \frac{1}{2}k_1^2} \right)\frac{{2p - 2}}{p}{\left| {{\eta _1}} \right|^{\left( {p - 2} \right)/p}}{\mathop{\rm sgn}} \left( {{\eta _1}} \right)\nonumber\\[4pt] &\quad\left( {{\eta _2} - {k_1}{{\left| {{\eta _1}} \right|}^{\left( {p - 1} \right)/p}}{\mathop{\rm sgn}} \left( {{\eta _1}} \right)} \right)\nonumber\\[4pt] &\quad + \left( {2{\eta _2} - {k_1}{{\left| {{\eta _1}} \right|}^{\left( {p - 1} \right)/p}}{\mathop{\rm sgn}} \left( {{\eta _1}} \right)} \right)\left( {{{\dot a}_{Tq}} - {k_2}{{\left| {{\eta _1}} \right|}^{\left( {p - 2} \right)/p}}{\mathop{\rm sgn}} \left( {{\eta _1}} \right)} \right)\nonumber\\[4pt] &\quad - {k_1}\frac{{p - 1}}{p}{\left| {{\eta _1}} \right|^{ - 1/p}}\left( {{\eta _2} - {k_1}{{\left| {{\eta _1}} \right|}^{\left( {p - 1} \right)/p}}{\mathop{\rm sgn}} \left( {{\eta _1}} \right)} \right),\nonumber\\[4pt] &= - {\left| {{\eta _1}} \right|^{ - 1/p}}\left( {{k_1}{k_2}{{\left| {{\eta _1}} \right|}^{2\left( {p - 1} \right)/p}} + \frac{{p - 1}}{p}k_1^3{{\left| {{\eta _1}} \right|}^{2\left( {p - 1} \right)/p}}} \right.\\[4pt] &\quad\left. { - 2\frac{{p - 1}}{p}k_1^2{{\left| {{\eta _1}} \right|}^{\left( {p - 1} \right)/p}}{\mathop{\rm sgn}} \left( {{\eta _1}} \right){\eta _2} + \frac{{p - 1}}{p}{\kern 1pt} {k_1}\eta _2^2} \right)\nonumber\\[4pt] &\quad + \left( {2{\eta _2} - {k_1}{{\left| {{\eta _1}} \right|}^{\left( {p - 1} \right)/p}}{\mathop{\rm sgn}} \left( {{\eta _1}} \right)} \right){{\dot a}_{Tq}}\nonumber\\[4pt] &\le - {\left| {{\eta _1}} \right|^{ - 1/p}}{\boldsymbol{\eta}^T}\textbf{{P}}\boldsymbol{\eta} + {{\dot d}^{\max }}\left( x \right)\textbf{{B}}\boldsymbol{\eta}\nonumber \end{align}

As $\,{k_1},{k_2} > 0\,$ an $\,p > 2$ , obviously, the matrix $\textbf{{Q}}$ is Hurwitz. Moreover, from the fact that $\,\left\| \boldsymbol{\eta} \right\| = \sqrt {{{\left| \eta \right|}^{2\left( {p - 1} \right)/p}} + \eta _2^2} \ge {\left| {{\eta _1}} \right|^{\left( {p - 1} \right)/p}}$ , one can imply $\,{\left| {{\eta _1}} \right|^{ - 1/p}} \ge {\left\| \boldsymbol{\eta} \right\|^{ - 1/\left( {p - 1} \right)}}$ . Then, from Equations (24) and (25) one can get

(26) \begin{equation}\begin{array}{l}{{\dot V}_1} \le - {\left| {{\eta _1}} \right|^{ - 1/p}}{\lambda _{min}}\left( \textbf{{Q}} \right){\left\| \boldsymbol{\eta} \right\|^2} + {{\dot d}^{\max }}\left( x \right)\left\| \textbf{{B}} \right\|\left\| \boldsymbol{\eta} \right\|\\[5pt] {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \le - \left( {{\lambda _{min}}\left( \textbf{{Q}} \right){{\left\| \boldsymbol{\eta} \right\|}^{\left( {p - 2} \right)/\left( {p - 1} \right)}} - {{\dot d}^{\max }}\left( x \right)\left\| \textbf{{B}} \right\|} \right)\left\| \boldsymbol{\eta} \right\|,\\[5pt] {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \le - \left( {{\lambda _{min}}\left( \textbf{{Q}} \right){{\left\| \boldsymbol{\eta} \right\|}^{\left( {p - 2} \right)/\left( {p - 1} \right)}} - {{\dot d}^{\max }}\left( x \right)\left\| \textbf{{B}} \right\|} \right)\frac{{V_1^{1/2}}}{{\sqrt {{\lambda _{\max }}\left( \textbf{{P}} \right)} }}\end{array} \end{equation}

If $\,{\lambda _{min}}\left( \textbf{{Q}} \right){\left\| \boldsymbol{\eta} \right\|^{\left( {p - 2} \right)/\left( {p - 1} \right)}} - {\dot d^{\max }}\left( x \right)\left\| \textbf{{B}} \right\| > 0$ , Equation (26) can be written as

(27) \begin{equation}{\dot V_1} \le - \frac{{\gamma V_1^{1/2}}}{{\sqrt {{\lambda _{\max }}\left( \textbf{{P}} \right)} }}, \end{equation}

where $\,\gamma = {\lambda _{min}}\left( \textbf{{Q}} \right){\left\| \boldsymbol{\eta} \right\|^{\left( {p - 2} \right)/\left( {p - 1} \right)}} - {\dot d^{\max }}\left( x \right)\left\| \textbf{{B}} \right\| > 0$ , In the light of Lemma 1, the area

(28) \begin{equation}\left\| \eta \right\| \le {\left( {\frac{{{{\dot d}^{\max }}\left( x \right)\left\| \textbf{{B}} \right\|}}{{{\lambda _{\min }}\left( \textbf{{Q}} \right)}}} \right)^{\left( {p - 1} \right)/\left( {p - 2} \right)}}, \end{equation}

can be approached in finite time, which also proves that the estimation errors will be converge in a small area around zero in finite time. The proof is completed.

3.2 Second-order sliding mode control

In general, we use the first-order sliding mode method to deal with problems with a relative degree of 1^{(Reference Levant37)}. However, a higher-order sliding mode method should be used for problems with larger degrees. In higher-order sliding mode, the sliding surface and its successive derivatives will converge to zero; that is,

(29) \begin{equation}s,\dot s,\ddot s,...{s^\kappa } = 0, \end{equation}

where $\kappa $ is the relative degree. The second-order sliding mode method is widely adopted in guidance system design. This method can be used to deal with problems with a relative degree of 2 or to eliminate chattering with a relative degree of 1. One example of an extensive second-order sliding mode method is the super-twisting algorithm^{(Reference Lee, Kim and Tahk15)}, in which the sliding surface and its derivative make countless twists around the origin and converge to zero in finite time. Despite being a useful method that has been considerably studied, the super-twisting algorithm exhibits a drawback, i.e., the sliding surface is forced to oscillate near the equilibrium point (zero), which results in command chattering. NSO-SMGL, which is based on a backstepping technique, is deduced in this section. This approach exhibits the advantage of ensuring that the sliding surface and its derivative converge to zero in finite time while guaranteeing that the sliding surface will not cross zero until the ultimate time.

First, the desired sliding surface is defined as follows:

(30) \begin{equation}s = f = 0. \end{equation}

The first and second derivatives of the sliding surface Equations (30) are

(31) \begin{equation}\dot s = h, \end{equation}

(32) \begin{equation}\ddot s = f + bu + d, \end{equation}

where $h$ , $f$ , and $b$ are known functions of the system; and $d$ represents target manoeuvering, which is assumed as a bounded uncertainty. The guidance command ${\kern 1pt} u{\kern 1pt} $ emerges in the second derivative of the sliding surface; thus, surface Equations (30) has a relative degree of 2.

The objective is to search for a formula for sliding surface Equations (30) and its derivative Equations (31). Both equations converge to zero at an expected time ${t_r}$ . A backstepping technique is used to achieve the objective. We set ${\kern 1pt} \dot s{\kern 1pt} $ as a virtual command, and ${\kern 1pt} \dot s{\kern 1pt} $ should be designed to send ${\kern 1pt} s{\kern 1pt} $ to zero. The Lyapunov function ${\kern 1pt} {V_2}$ is selected.

(33) \begin{equation}{V_2} = \frac{1}{2}{s^2} \end{equation}

Considering the derivative of Equation (33) with respect to time yields

(34) \begin{equation}{\dot V_2} = s\dot s. \end{equation}

To ensure that sliding surface Equations (30) will converge to zero in finite time, ${\dot V_2}$ must be negative infinity. Then, ${\kern 1pt} \dot s{\kern 1pt} $ can be set as

(35) \begin{equation}\dot s = - \frac{{ns}}{{{t_r} - t}},n > 1, \end{equation}

where ${t_r}{\kern 1pt} $ is the expected convergence time. Substituting Equations (35) into Equations (34) yields

(36) \begin{equation}{\dot V_2} = - \frac{{n{s^2}}}{{{t_r} - t}}. \end{equation}

Substituting Equation (33) into Equation (36) derives

(37) \begin{equation}{\dot V_2} = - \frac{{2n{V_2}}}{{{t_r} - t}}. \end{equation}

Integrating Equation (37) from the initial value $\left( {{V_2}\left( {{t_0}} \right),{t_0}} \right)$ into the current value $\left( {{V_2}\left( t \right),t} \right)$ can result in

(38) \begin{equation}{V_2}\left( t \right) = \frac{{{V_2}\left( {{t_0}} \right)}}{{{{\left( {{t_r} - {t_0}} \right)}^{2n}}}}{\left( {{t_r} - t} \right)^{2n}}. \end{equation}

Notably, when ${\kern 1pt} n{\kern 1pt} $ is greater than 0.5, ${\kern 1pt} {\left( {{t_r} - t} \right)^{2n}}{\kern 1pt} $ in Equation (38) will converge to zero because ${\kern 1pt} t{\kern 1pt} $ is close to ${\kern 1pt} {t_r}$ . Therefore, the Lyapunov function (and consequently, the sliding surface) is guaranteed to converge to zero at the expected time $\,{t_r}\,$ provided that $n > 0.5$ . Equation (35) depends on the values of the sliding surface and time $\,t$ ; it can also ensure that $\,\dot s$ converges to zero at $\,{t_r}$ , as proven in^{(Reference Kumar, Rao and Ghose19)}. Furthermore, achieving such goal for $\,\dot s\,$ places an additional constraint on parameter $\,n$ , which should be satisfied as $\,n > 1$ .

Equation (35) provides an expected trajectory for $\,\dot s\,$ to follow. Notably, condition Equation (35) should not necessarily be satisfied at the initial time because the initial value of $\,\dot s\,$ completely depends on the initial conditions of the problem. Therefore, guidance command $\,u\,$ must be used to send $\,\dot s\,$ from its initial value to the trajectory defined by Equation (35) in finite time, and then $\,\dot s\,$ must be maintained on the trajectory. Given that the system must comply with certain actions along trajectory Equation (35) to ensure that $\,s\,$ and $\,\dot s\,$ converge to zero, the trajectory should be reached in finite time.

To ensure that trajectory Equation (35) will be reached in finite time, the new sliding surface should be considered as follows:

(39) \begin{equation}{s_2} = \dot s + \frac{{ns}}{{{t_r} - t}} = 0. \end{equation}

The time derivative of surface Equation (39) results in

(40) \begin{equation}{\dot s_2} = f + d + \frac{{n\dot s\left( {{t_r} - t} \right) + ns}}{{{{\left( {{t_r} - t} \right)}^2}}} + bu. \end{equation}

As indicated in Equation (40), the new sliding surface Equation (39) has a relative degree of 1 to guidance command $\,u$ .

The expression for $\,u\,$ must be obtained to make $\,{s_2}\;$ converge to zero in finite time $\,t_r^{\ast} $ . The Lyapunov function is selected as follows:

(41) \begin{equation}{V_2} = \frac{1}{2}s_2^2. \end{equation}

The time derivative of Equation (41) yields

(42) \begin{equation}{\dot V_2} = {s_2}{\dot s_2} = {s_2}\left[ {f + \frac{{n\dot s\left( {{t_r} - t} \right) + ns}}{{{{\left( {{t_r} - t} \right)}^2}}} + d + bu} \right] \end{equation}

To ensure that the Lyapunov function will converge to zero in finite time, $u\,$ is exhibited as

(43) \begin{equation}u = - \frac{1}{b}\left( {f + \frac{{n\dot s\left( {{t_r} - t} \right) + ns}}{{{{\left( {{t_r} - t} \right)}^2}}} + \hat d + \eta\, {\mathop{\rm sgn}} \left( {\dot s + \frac{{ns}}{{{t_r} - t}}} \right)} \right)\!. \end{equation}

where $\eta $ is an appropriately positive constant, and $n > 2$ . $\hat d\,$ is the estimation of target manoeuvering. If $\,\hat d \to d\,$ is in finite time, then ${\dot V_2}\,$ is a negative definite function with guidance command Equation (43), which satisfies the following inequality :

(44) \begin{equation}{\dot V_2} < - \eta \left| {{s_2}} \right|\!, \end{equation}

where $\,\eta \,$ is the design parameter that satisfies $\,\eta > 0$ and influences the convergence speed of the system.

The consequence from the preceding derivations can be summarised in Theorem 2.

In terms of the application of guidance command Equation (43), the sign function is under guidance command. To eliminate the overload command chattering problem brought by the sign function, we use the following continuous saturation function to smooth it:

(45) \begin{equation}sat\left( {S,\phi } \right) = \left\{ \begin{array}{l}S/\phi {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left| S \right| \le \phi \\[5pt]{\mathop{\rm sgn}} \left( S \right){\kern 1pt} {\kern 1pt} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left| S \right| > \phi \end{array} \right., \end{equation}

where $\phi $ is the defibrillation factor, and $\phi = 0.01$ is used in the simulation.