^{2} Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin 150001, China
With the continuous development of precisionguided munitions, the guidance law design is constrained not only by the requirement on miss distances, but also by the requirement on the angle of the guidance terminal phase in many scenarios, so as to allow the maximum effectiveness of the warhead, and to achieve the best damage effect. For this reason, it is necessary to do further study on the guidance law with terminal impact angle constraints, to meet the requirement of this special guidance mission.
In 1973, Kim and Grider proposed an optimal guidance law with impact angle control for the reentry vehicle in the vertical plane based on a linear model^{[1]}. Since then, various robust guidance laws with impact angle constraints have been proposed for different scenarios. Chai et al.^{[2]} reviewed the domestic and international researches on guidance laws with terminal angle constraints, and analyzed both the advantages and disadvantages of the optimal guidance law, variable structure guidance law, improved proportional navigation guidance law and integrated guidance law. It points out the importance of guidance law with terminal angle constraints in improving the combat effectiveness of guided munitions; while there are still practical problems to be effectively addressed despite of the great progress in the research on robust guidance law. Ratnoo and Ghose^{[3]} employed the traditional proportional navigation guidance law to discuss the conditions to choose all the navigation coefficients, and it concluded that during groundtoground attack, it is available to attack with desired impact angle constraints and small miss distances but it applies to only stationary targets. Shima^{[4]} analyzed the relationship between the speed ratio and the lead angle based on the three possible modes of the collision between missiles and targets, employed sliding mode control to design guidance law, which is then compared with the proportional navigation guidance law, and simulated several maneuvering profiles of the targets. Harl and Balakrishnan^{[5]} employed the theory of secondorder sliding mode to design the guidance law satisfying the desired lineofsight (LOS) angle curve and scheduled attack time, and applied this guidance law to multimissile salvo attack. However, this method does not consider the dynamic delay of missile autopilot, and is only applicable to stationary or slow targets. Shashi et al.^{[6]} employed the traditional terminal sliding mode to design a guidance law with impact angle constraints, which is then applied to attack the stationary targets, as well as interception of constant speed and maneuvering targets. However, it does not consider the issue of singularity of the traditional sliding mode. Sachit and Debasish^{[7]} employed the theory of variable structure to study the guidance law with impact angle constraints by switching among different sliding mode controls, and analyzed the capability of missile to capture maneuvering targets. Sun et al.^{[8]} combined sliding mode and backstepping theory to study the guidance law with impact angle constraint when there is dynamic delay of missile autopilot, but backstepping calls for a quite huge computation amount.
Due to the preferable adaptivity and robustness of sliding mode variable structure control to parameter perturbation and external disturbance, it has been widely applied in missile guidance and control, and it is obvious that it is applied by all the literatures mentioned above. However, it has been the objective of the study of control system to remove the chattering behavior of sliding mode, and to improve the speed to reach the sliding mode manifold. Gao^{[9]} presented several concepts to improve the speed to reach the sliding mode manifold, including constant speed reaching law, exponential reaching law and power reaching law. However, the speed of constant speed reaching law is too slow; the convergence rate of exponential reaching law is quite fast, while there is greater system chattering when the system state is near sliding mode; power reaching law is conducive to reduce chattering, while the speed of the reaching stage is too slow when the system state is far from sliding mode, which calls for too much time^{[10]}. Yu et al.^{[11]} combined the traditional power reaching law and exponential reaching law to get a fast power reaching law, which can weaken the chattering of sliding mode, and improve the speed when the system state is far from the sliding mode manifold.
The traditional sliding mode control is designed with linear sliding mode manifold, and when the system state has reached to the sliding mode manifold, it will asymptotically converge to the equilibrium of the system, while is not finitetime convergent^{[79]}. The terminal sliding mode control realizes the finitetime convergence of the system state, with a convergence performance better than that of the traditional sliding mode control^{[12, 13]}. However, there is still the issue of singularity of the terminal sliding mode control, for which Feng et al.^{[14]} presented a nonsingular terminal sliding mode control that overcomes the issue of singularity, to improve control performance.
The dynamic delay of missile autopilot is a main factor influencing the precision of guidance, and it is of practical engineering significance to consider the missile
Considering the relative motion of the missile and the target in the intercepting plane
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Fig. 1. Relative motion geometry of missile and target 
In Fig. 1, The missile is denoted as
$ \begin{align} \label{eq1} \dot {r}=V_t \cos (q\phi _t )V_m (q\phi _m ) \end{align} $  (1) 
$ \begin{align} \label{eq2} r\dot {q}=V_t \sin (q\phi _t )+V_m \sin (q\phi _m ). \end{align} $  (2) 
Taking the derivative of (2), we can get
$ \begin{align} \label{eq3} \ddot {q}=\frac{2\dot {r}}{r}\dot {q}\frac{1}{r}a_m +\frac{1}{r}a_t \end{align} $  (3) 
where
The dynamics of the missile autopilot is described by the following firstorder term:
$ \begin{align} \label{eq4} \dot {a}_m =\frac{1}{\tau }a_m +\frac{1}{\tau }u \end{align} $  (4) 
where
The impact angle is the included angle between the missile
$ \begin{align} \label{eq5} \mathop {\lim }\limits_{t\to t_f } r(t)\dot {q}(t)=0 \end{align} $  (5) 
$ \begin{align} \label{eq6} \phi _m (t_f )\phi _t (t_f )=\phi _0 \end{align} $  (6) 
$ \begin{align} \label{eq7} \left {q(t_f )\phi _m (t_f )} \right<\frac{\pi }{2}. \end{align} $  (7) 
Equation (7) means that the target is in the vision scope when the missile hits it, and we can get from (2) and (5) that:
$ \begin{align} \label{eq8} V_t \sin [q(t_f )\phi _t (t_f )]=V_m \sin [q(t_f )\phi _m (t_f )]. \end{align} $  (8) 
For a missile with a specific attack mission, the expected impact angle
Remark 1. If the target is stationary, then
Assumption 1. As restrained by acceleration capability, the maximum lateral acceleration that can be actually provided by the missile and the target is limited, therefore there exists a constant
$ \begin{align} \label{eq9} \left {a_m } \right\le A_m, \left {a_t } \right\le A_1, \;\;\left {\dot {a}_t } \right\le A_2. \end{align} $  (9) 
During terminal guidance, as restrained by the power of its angle tracing system, receiver acceleration and other factors, the seeker has a minimum operating range
Assumption 2. The timevarying parameter
$ \begin{align} \label{eq10} r(t)\ge r_0. \end{align} $  (10) 
For convenience of guidance law design, some definitions and lemmas are presented as follows.
Definition 1^{[19]}. Considering a nonlinear system
$ \begin{align} \label{eq11} \dot {x}=f(x, t), f(0, t)=0, x\in {\bf R}^n \end{align} $  (11) 
where,
$ \left\{ {{\begin{array}{*{20}c} {\mathop {\lim }\limits_{t\to T(x_0 )} \varphi (t;t_0, x_0 )=0} \hfill \\ {\rm {if}} \ \ t>T(x_0 ), {\rm {then}} \ \ \varphi (t;t_0, x_0 )=0 \hfill \\ \end{array} }} \right. $ 
which means when
Lemma 1^{[11]}. Considering a nonlinear system of (11), if there is a continuous and positive definite function
$ \begin{align} \label{eq12} \dot {V}(x)+\mu V(x)+\lambda V^\alpha (x)\le 0\mbox{ } \end{align} $  (12) 
where
$ T\le \frac{1}{\mu (1\alpha )}\ln \frac{\mu V^{1\alpha }(x_0 )+\lambda }{\lambda }. $ 
Select state variables
$ \begin{align} \label{eq13} x_1 =q(t)q_d, x_2 =\dot {x}_1. \end{align} $  (13) 
Taking the derivative of (13), we can get the state equation of guidance system with impact angle constraints:
$ \begin{align} \label{eq14} \left[{\begin{array}{l} \dot {x}_1 \\ \dot {x}_2 \\ \end{array}} \right]=\left[{{\begin{array}{*{20}c} 0 \hfill & {1} \\ 0 \hfill & {\dfrac{2\dot {r}}{r}} \\ \end{array} }} \right]\left[{\begin{array}{l} x_1 \\ x_2 \\ \end{array}} \right]+\left[{\begin{array}{*{20}c} {0} \\ {\dfrac{1}{r}} \\ \end{array}} \right]a_m +\left[{\begin{array}{l} 0 \\ \dfrac{1}{r} \\ \end{array}} \right]a_t. \end{align} $  (14) 
Let
$ \begin{align} \label{eq15} \left {g(t)} \right=\vert \frac{1}{r}a_t \vert \le \frac{A_1 }{r_0 }=\delta. \end{align} $  (15) 
For the system of (14), select the nonsingular terminal sliding mode manifold
$ \begin{align} \label{eq16} s=x_1 +\beta {\rm sig}^\gamma (x_2 ) \end{align} $  (16) 
where
When the target is not maneuvering,
$ \begin{align} \label{eq17} \dot {s}=k_1 sk_2 {\rm sig}^\rho (s) \end{align} $  (17) 
where
Taking the derivative of (16), we can get
$ \begin{align} \label{eq18} \dot {s}=&\dot {x}_1 +\beta \gamma \left {x_2 } \right^{\gamma 1}\dot {x}_2 =\notag\\ & x_2 +\beta \gamma \left {x_2 } \right^{\gamma 1}\left(\frac{2\dot {r}}{r}x_2 \frac{1}{r}a_m +\frac{1}{r}a_t \right) . \end{align} $  (18) 
We can get from (17) and (18) that
$ \begin{align} \label{eq19} a_m =&r(\beta ^{1}\gamma {1}{\rm sig}^{2\gamma }(x_2 )\frac{2\dot {r}}{r}x_2 +\notag\\ &\beta ^{1}\gamma ^{1}\left {x_2 } \right^{1\gamma }(k_1 s+k_2 {\rm sig}^\rho (s))). \end{align} $  (19) 
As the factor
$ \begin{align} \label{eq20} \bar {a}_m =r(\beta ^{1}\gamma ^{1}{\rm sig}^{2\gamma }(x_2 )\frac{2\dot {r}}{r}x_2 +k_1 s+k_2 {\rm sig}^\rho (s)). \end{align} $  (20) 
Theorem 1. For the system of (14), when
Proof. When the guidance law of (20) is substituted into (18), we can get
$ \dot {s}=\beta \gamma \left {x_2 } \right^{\gamma 1}(k_1 sk_2 {\rm sig}^\rho (s)). $ 
Construct Lyapunov function
$ \begin{align} \label{eq21} V=s^2. \end{align} $  (21) 
Taking the derivative of (21), we can get
$ \begin{array}{l} \dot {V}=2s\dot {s} =\\ \qquad 2\beta \gamma \left {x_2 } \right^{\gamma 1}s(k_1 sk_2 {\rm sig}^\rho (s)) =\\ \qquad 2\beta \gamma \left {x_2 } \right^{\gamma 1}(k_1 s^2k_2 \vert s\vert ^{\rho +1}) =\\ \qquad \mu V\lambda V^{\textstyle{{\rho +1} \over 2}} \le 0 \\ \end{array} $ 
where
When the target is maneuvering,
$ \begin{align} \label{eq22} \dot {s}=k_3 sk_4 {\rm sgn}(s) \end{align} $  (22) 
where
Taking the derivative of (16), we can get
$ \begin{align} \label{eq23} \begin{array}{l} \dot {s}=\dot {x}_1 +\beta \gamma \left {x_2 } \right^{\gamma 1}\dot {x}_2 =\\ \qquad x_2 +\beta \gamma \left {x_2 } \right^{\gamma 1}\left(\dfrac{2\dot {r}}{r}x_2 \dfrac{1}{r}a_m +\dfrac{1}{r}a_t \right). \\ \end{array} \end{align} $  (23) 
Similarly, in order to avoid singularity, we design the guidance law for the system of (14) as
$ \begin{align} \label{eq24} \tilde {a}_m =r(\beta ^{1}\gamma ^{1}{\rm sig}^{2\gamma }(x_2 )\frac{2\dot {r}}{r}x_2 +k_3 s+\eta {\rm sgn}(s)). \end{align} $  (24) 
Theorem 2. For the guidance law of (24), if the term of variable structure selected is
Proof. When the guidance law of (24) is substituted into (23), we can get
$ \dot {s}=\beta \gamma \left {x_2 } \right^{\gamma 1}(k_3 s\eta {\rm sgn}(s)+g(t)). $ 
Construct Lyapunov function
$ \begin{align} \label{eq25} V_1 =s^2 \end{align} $  (25) 
Taking the derivative of (25), we can get
$ \begin{array}{l} \dot {V}=2s\dot {s}= \\ \qquad 2\beta \gamma \left {x_2 } \right^{\gamma 1}s(k_3 s\eta {\rm sgn}(s)+g(t))\le \\ \qquad 2\beta \gamma \left {x_2 } \right^{\gamma 1}(k_3 s^2\eta \vert s\vert +\left {g(t)s} \right) \le\\ \qquad 2\beta \gamma \left {x_2 } \right^{\gamma 1}(k_3 s^2(\eta \delta )\left s \right) =\\ \qquad \mu _1 V\lambda _1 V^{\textstyle{1 \over 2}} \le 0 \\ \end{array} $ 
where
Let
$ \begin{align} &\label{eq26} \left[{\begin{array}{l} \dot {x}_1 \\ \dot {x}_2 \\ \dot {x}_3 \\ \end{array}} \right]=\left[{{\begin{array}{*{20}c} 0 \hfill&{1} &{0} \\ 0 \hfill&{\dfrac{2\dot {r}}{r}} &{\dfrac{1}{r}} \\[2mm] 0 \hfill & {0} & {\dfrac{1}{\tau }} \\ \end{array} }} \right]\left[{\begin{array}{l} x_1 \\ x_2 \\ x_3 \\ \end{array}} \right]+\notag\\ &\qquad \qquad\left[{\begin{array}{l} 0 \\ 0 \\ \dfrac{1}{\tau } \\ \end{array}} \right]u+\left[{\begin{array}{l} 0 \\ \dfrac{1}{r} \\ 0 \\ \end{array}} \right]a_t. \end{align} $  (26) 
We select the sliding mode manifold
$ \begin{align} \label{eq27} s_1 =k_0 x_1 +x_2. \end{align} $  (27) 
In order to reach the sliding mode manifold in finite time, as well as to reduce the chattering phenomenon of the control signal, we design the nonsingular terminal sliding mode manifold with linear sliding mode
$ \begin{align} \label{eq28} s_2 =s_1 +\beta _1 {\rm sig}^{\gamma _1 }(\dot {s}_1 ) \end{align} $  (28) 
where
Taking the derivative of the system of (28), we can get
$ \begin{align} \label{eq29} \begin{array}{l} \dot {s}_2 =\dot {s}_1 +\beta _1 \gamma _1 \left {\dot {s}_1 } \right^{\gamma _1 1}\ddot {s}_1 =\\ \qquad \dot {s}_1 +\beta _1 \gamma _1 \vert \dot {s}_1 \vert ^{\gamma _1 1}\left(k_0 \left(\dfrac{2\dot {r}}{r}x_2 \dfrac{1}{r}x_3 +\dfrac{1}{r}a_t \right)+\ddot {x}_2 \right) \\ \end{array} \end{align} $  (29) 
$ \begin{align} \label{eq30} \ddot {x}_2 =\frac{2(\ddot {r}x_2 +\dot {r}\dot {x}_2 )r+2\dot {r}^2x_2 }{r^2}+\frac{\dot {r}x_3 \dot {x}_3 r}{r^2}+\frac{r\dot {a}_t \dot {r}a_t }{r^2}. \end{align} $  (30) 
When (30) is substituted into (29)
$ \begin{align} \label{eq31} & \dot {s}_2 =\dot {s}_1 +\beta _1 \gamma _1 \vert \dot {s}_1 \vert ^{\gamma _1 1}\Big(\frac{2r(k_0 \dot {r}+\ddot {r})+6\dot {r}^2}{r^2}x_2 +\notag\\ &\qquad \frac{3\dot {r}k_0 r}{r^2}x_3 +\frac{1}{r\tau }x_3 +\frac{(k_0 a_t +\dot {a}_t )r3\dot {r}a_t }{r^2}\frac{1}{r\tau }u\Big) . \end{align} $  (31) 
Let
$ \begin{array}{l} \left {g_1 (t)} \right=\left {\dfrac{(k_0 a_t +\dot {a}_t )r3\dot {r}a_t }{r^2}} \right \le\\ \qquad \dfrac{k_0 \left {a_t } \right+\left {\dot {a}_t } \right}{r_0 }+\dfrac{3\left {\dot {r}a_t } \right}{r_0^2 } =\\ \qquad \dfrac{3A_1 \left {V_t \cos (q\phi _t )V_m \cos (q\phi _m )} \right}{r_0^2 }+\dfrac{k_0 A_1 +A_2 }{r_0 } \le\\ \qquad \dfrac{k_0 A_1 +A_2 }{r_0 }+\dfrac{3A_1 (\left {V_t^{\max } } \right+\left {V_m^{\max } } \right)}{r_0^2 } \le \varepsilon \\ \end{array} $ 
where
When the target is not maneuvering,
$ \begin{align} \label{eq32} \dot {s}_2 =k_5 s_2 k_6 {\rm sig}^{\rho _1 }(s_2 ) \end{align} $  (32) 
where,
We can get from (31) and (32) that
$ \begin{align} \label{eq33} u=&r\tau \beta _1^{1} \gamma _1^{1} \left {\dot {s}_1 } \right^{2\gamma _1 }{\rm sgn}(\dot {s}_1 )\notag\\ &2k_0 \tau \dot {r}x_2 +x_3 2\tau \ddot {r}x_2 k_0 \tau x_3 +\notag\\ &\frac{3\tau \dot {r}}{r}x_3 +r\tau \beta _1^{1} \gamma _1^{1} \left {\dot {s}_1 } \right^{1\gamma _1 }(k_5 s_2 +k_6 {\rm sig}^{\rho _1 }(s_2 )) . \end{align} $  (33) 
For the factor
$ \begin{align} \label{eq34} u_1 =&r\tau \beta _1^{1} \gamma _1^{1} \left {\dot {s}_1 } \right^{2\gamma _1 }{\rm sgn}(\dot {s}_1 )2k_0 \tau \dot {r}x_2 +x_3 \notag\\ & 2\tau \ddot {r}x_2 k_0 \tau x_3 +\frac{3\tau \dot {r}}{r}x_3 +r\tau (k_5 s_2 +k_6 {\rm sig}^{\rho _1 }(s_2 )). \end{align} $  (34) 
Theorem 3. For the system of (26), when
Proof. When the guidance law of (34) is substituted into (31), where
$ \begin{align} \label{eq35} \dot {s}_2 =\beta _1 \gamma _1 \vert \dot {s}_1 \vert ^{\gamma _1 1}(k_5 s_2 k_6 {\rm sig}^{\rho _1 }(s_2 )). \end{align} $  (35) 
Construct Lyapunov function
$ \begin{align} \label{eq36} V_2 =s_2^2. \end{align} $  (36) 
Taking the derivative of (36), and substituting (35), we can get
$ \begin{array}{l} \dot {V}_2 =2s_2 \dot {s}_2 =\\ \qquad 2\beta _1 \gamma _1 \vert \dot {s}_1 \vert ^{\gamma _1 1}s_2 (k_5 s_2 k_6 {\rm sig}^{\rho _1 }(s_2 )) =\\ \qquad 2\beta _1 \gamma _1 \vert \dot {s}_1 \vert ^{\gamma _1 1}(k_5 s_2^2 k_6 \vert s_2 \vert ^{\rho _1 +1}) =\\ \qquad \mu _2 V_2 \lambda _2 V_2^{\textstyle{{1+\rho _1 } \over 2}} \le 0 \\ \end{array} $ 
where
When the target is maneuvering, we select an exponential reaching law for the sliding mode manifold of (28):
$ \begin{align} \label{eq37} \dot {s}_2 =k_7 s_2 k_8 {\rm sgn}(s_2 ) \end{align} $  (37) 
where
For the system of (26), we can design the guidance law as
$ \begin{align} \label{eq38} \begin{array}{c} u_2 =r\tau \beta _1^{1} \gamma _1^{1} \left {\dot {s}_1 } \right^{2\gamma _1 }{\rm sgn}(\dot {s}_1 )2k_0 \tau \dot {r}x_2 +x_3 \\ \qquad \quad 2\tau \ddot {r}x_2 k_0 \tau x_3 +\dfrac{3\tau \dot {r}}{r}x_3 +r\tau (k_7 s_2 +\zeta {\rm sgn}(s_2 )) \\ \end{array} \end{align} $  (38) 
where
Theorem 4. For the guidance law of (38), when
Proof. When the guidance law of (38) is substituted into (31), we can get
$ \begin{align} \label{eq39} \dot {s}_2 =\beta _1 \gamma _1 \vert \dot {s}_1 \vert ^{\gamma _1 1}(k_7 s_2 \zeta {\rm sgn}(s_2 )+g_1 (t)). \end{align} $  (39) 
Construct Lyapunov function
$ \begin{align} \label{eq40} V_3 =s_2^2. \end{align} $  (40) 
Taking the derivative of (40), and substituting (39), we can get
$ \begin{align} \label{eq41} \begin{array}{l} \dot {V}_3 =2s_2 \dot {s}_2 =\\ \qquad \beta _1 \gamma _1 \left {\dot {s}_1 } \right^{\gamma _1 1}(k_7 s_2 \zeta {\rm sgn}(s_2 )+g_1 (t)) =\\ \qquad2\beta _1 \gamma _1 \left {\dot {s}_1 } \right^{\gamma _1 1}(k_7 s_2^2 \zeta \left {s_2 } \right+g_1 (t)s_2 ) \le\\ \qquad2\beta _1 \gamma _1 \left {\dot {s}_1 } \right^{\gamma _1 1}(k_7 s_2^2 \zeta \left {s_2 } \right+\left {g_1 (t)s_2 } \right) \le\\ \qquad 2\beta _1 \gamma _1 \left {\dot {s}_1 } \right^{\gamma _1 1}(k_7 s_2^2 (\zeta \varepsilon )\left {s_2 } \right) =\\ \qquad\mu _3 V_1 \lambda _3 V_1^{\textstyle{1 \over 2}} \le 0 \\ \end{array} \end{align} $  (41) 
where
Remark 2. In practical applications, the rate of relative range
$ \begin{align} \label{eq42} \begin{array}{l} u_1 =r\tau \beta _1^{1} \gamma _1^{1} \left {\dot {s}_1 } \right^{2\gamma _1 }{\rm sgn}(\dot {s}_1 )2k_0 \tau \dot {r}x_2 +x_3 \\ \quad k_0 \tau x_3 +\dfrac{3\tau \dot {r}}{r}x_3 +r\tau (k_5 s_2 +k_6 {\rm sig}^{\rho _1 }(s_2 )) .\\ \end{array} \end{align} $  (42) 
And the proposed guidance law (38) can be rewritten as (43).
$ \begin{align} \label{eq43} \begin{array}{l} u_2 =r\tau \beta _1^{1} \gamma _1^{1} \left {\dot {s}_1 } \right^{2\gamma _1 }{\rm sgn}(\dot {s}_1 )2k_0 \tau \dot {r}x_2 +x_3 \\ \quad k_0 \tau x_3 +\dfrac{3\tau \dot {r}}{r}x_3 +r\tau (k_7 s_2 +\zeta {\rm sgn}(s_2 )). \\ \end{array} \end{align} $  (43) 
Remark 3. When the target is not maneuvering, we select rapid power reaching law, which can not only improve the speed to reach the sliding mode manifold, but also inhibit its chattering behavior.
Remark 4. When the target is maneuvering, we select exponential reaching law, and adjust the term of variable structure, to compensate the disturbance to the target in the system. The convergence rate of exponential reaching law is quite fast, while there will be greater system chattering when the system state is near sliding mode. Highorder sliding mode can alleviate the chattering resulted from sign function; however, in order to eliminate chattering, we further replace the sign function in the designed guidance law with the saturation function
$ sat(s)=\left\{ {\begin{array}{l} 1, \qquad s>\Delta \\ \dfrac{s}{\Delta}, \quad\vert s\vert \le \Delta \\ 1, \quad s\le \Delta \\ \end{array}} \right. $ 
where
Taking a missile air intercept as an example, in the reference inertial coordinate system, the velocities of the missile and target are constants, i.e., 600 m/s and 200 m/s, respectively. The missile
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Fig. 2. Responses under the guidance law (20) for nonmaneuvering target 
4.1 Not considering the dynamic delay of missile autopilot
We verify the effectiveness of the proposed guidance law (20) and guidance law (24), without considering the dynamic delay of missile autopilot. During simulation, the parameters of the guidance law are selected as:
Here, under the proposed guidance law (20), we consider that the missile intercepts a nonmaneuvering target. The responses of LOS angular rate, LOS angle, sliding mode surface and missile normal acceleration are shown in Figs. 2(a)2(d), respectively. From Figs. 2(a) and 2(b), it can be seen that the LOS angular rate converges to zero fast in finite time and the LOS angle converges to the desired LOS angle in finite time. So, the proposed guidance law can guarantee the missile intercepts the target with the desired LOS angle successfully. From Fig. 2 (c), we can observe that the sliding mode surface without chattering reaches to zero in finite time. As shown in Fig. 2 (d), the proposed guidance law can guarantee the missile normal acceleration converges to zero fast, and there is no chattering phenomenon. Under the proposed guidance law, the interception time, missdistance and LOS angle error are 15.018 s, 0.328 m and 0.235Ãƒâ€šÃ‚Â°, respectively, which are also given to illustrate the effectiveness of the proposed guidance law (20).
For the case without considering the dynamic delay of missile autopilot, the simulation results are shown in Figs. 2(a)2(d). In practical problems, the dynamic delay of missile autopilot is not ignorable. Hence, we select
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Fig. 3. Responses under the guidance law (20) for considering the dynamic delay of missile autopilot 
To illustrate the performance of the designed guidance law (24), we consider three different target acceleration profiles as given below.
Case 1. Cosine maneuvering
Case 2. Step maneuvering
Case 3. Constant maneuvering
The missdistances, LOS angle errors and interception times are given in Table 1. The curves of LOS angular rate, LOS angle, sliding mode surface and missile normal acceleration are shown in Figs. 4(a)4(d), respectively.
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Fig. 4. Responses under the guidance law (24) for three cases 
From Figs. 4(a)4(c), in the three target acceleration profiles, we can see that the LOS angular rates and sliding mode surfaces converge to zero fast. But, for the three cases, the curves of the LOS angular rates have peak, which lead to small peak in the curves of the missile normal accelerations shown in Fig. 4 (d). From Fig. 4 (d), we can observe that there are acceleration saturations before 10 s in all the three cases, while the accelerations are decreasing correspondingly as the LOS angular rates tend to zero. From Fig. 4 (b), the proposed guidance law can guarantee the LOS angles converge to the desired LOS angle. From Table 1, simulation data verify the high precision guidance performance of the proposed guidance law (24).
For the case without considering the dynamic delay of missile autopilot, the simulation results are shown in Fig. 4 to demonstrate the effectiveness of the guidance law (24). Further, for the case 1, the parameter
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Fig. 5. Responses under the guidance law (24) for considering the dynamic delay of missile autopilot 
4.2 Considering the dynamic delay of missile autopilot
We verify the effectiveness of the designed guidance laws (42) and (43), considering the dynamic delay of missile autopilot. During simulation, the parameters of the guidance law are selected as:
For a nonmaneuvering target, the proposed guidance law (42) is applied. The responses of LOS angular rate, LOS angle, sliding mode surface and missile normal acceleration are shown in Figs. 6(a)6(d), respectively. From Figs. 6(a)6(c), it can be seen that the LOS angular rate and sliding mode surface converge to zero fast. As shown in Fig. 6 (b), the proposed guidance law can guarantee the LOS angle converges to the desired LOS angle and the LOS angle error is zero. As shown in Fig. 6 (d), while considering the dynamic delay of missile autopilot, the proposed guidance law can guarantee the missile normal acceleration converges to zero fast and there is no chattering phenomenon. Under the proposed guidance law (42), the interception time, missdistance and LOS angle error are 21.68 s, 0.02 m and 0.05Ãƒâ€šÃ‚Â°. It can be obtained that the guidance precision of the guidance law (42) is higher than that of guidance law (20).
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Fig. 6. Responses under the guidance law (42) for nonmaneuvering target 
To verify the performance of the designed guidance law (43) for the maneuvering target, we also consider the abovementioned three different target acceleration profiles. The missdistances, LOS angle errors and interception times are given in Table 2. The curves of LOS angular rate, LOS angle, sliding mode surface and missile normal acceleration are shown in Figs. 7(a)7(d), respectively.
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Fig. 7. Responses under the guidance law (43) for three cases 
From the Figs. 7(a) and 7(c), in the three target acceleration profiles, we can see that the LOS angular rates and sliding mode surfaces converge to zero fast. In addition, we can also observe that the curves of LOS angular rates considering the missile
At present, much study has been devoted to the guidance law design about the dynamic delay and impact angle constraints. However, there are not many guidance laws which are designed by applying the nonsingular terminal sliding mode control. In [20], a guidance law based on the nonlinear backstepping method with autopilot lag and impact angle constraint was proposed, which was expressed as
$ \begin{align} \label{eq44} A_c =\dfrac{\tau }{\cos \theta _m }\left\{ {\begin{array}{l} \left(\dfrac{k_1 \cos ^2\theta _m }{R(t)}+c_1 k_1 k_2 \left {\dot {R}(t)} \right\right)x_1 +\left(\dfrac{1}{\tau }\dfrac{(k_2 +2)\left {\dot {R}(t)} \right}{R(t)}  k_1 c_1 \right)x_3 \cos \theta _m +\\ \left(\dfrac{\cos ^2\theta _m }{R(t)}+c_1 (2+k_2 )\left {\dot {R}(t)} \right+ c_1 k_1 R(t)+k_1 (1+k_2 )\left {\dot {R}(t)} \right+\dfrac{(2k_2 +4)\left {\dot {R}(t)} \right^2}{R(t)}\right)x_2 +\\ c_1 \varepsilon _1 {\rm sat}(z_3 )\left(\dfrac{(k_2 +2)\left {\dot {R}(t)} \right}{R(t)}+k_1 \right)\varepsilon _1 {\rm sgn}z_4 \\ \end{array}} \right\} \end{align} $  (44) 
where
For Case 1, the initial conditions are chosen as the same as in the previous simulation. Simulation results are shown in Figs. 8(a)8(d) for the guidance laws (44) and (43).
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Fig. 8. Responses under the guidance law (43) and guidance law (44) for Case 1 
From Figs. 8(a) and 8(b), the LOS angular rate and LOS angle under the guidance law (43) faster converge to their corresponding desired values than that under the guidance law (44). In addition, the convergence accuracy under the guidance law (43) is higher than that under the guidance law (44). As shown in Fig. 8(c), it can be obtained that the convergence rate of the guidance law (43) is faster than that of the guidance law (44). From Fig. 8 (d), although the missile normal accelerations for the two guidance laws are similar, the chattering problem of the missile normal acceleration under the guidance law (44) is a little bit serious.
5 ConclusionsWhile not considering the missile
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