3.1 Guidance law design and its stability analysis

In order to cancel the line-of-sight angle error and allow the line-of-sight angle rate to converge quickly to zero, an appropriate non-singular terminal sliding surface is designed. The specific form is as follows:

(19) \begin{equation}s{\rm{ \;=\; }}\left[ \begin{array}{l} {s_1}\\ {s_2}\end{array} \right] = \left[ \begin{array}{l} {{\dot x}_1} + {\alpha _1}{x_1} + {\alpha _2}g({x_1})\\ {{\dot x}_2} + {\alpha _1}{x_2} + {\alpha _2}g({x_2})\end{array} \right] = \dot x + {\alpha _1}x + {\alpha _2}g(x)\end{equation}

where $ {\alpha _1} > 0,{\alpha _2} > 0 $ and $ g(x) \in R $ is defined as follows:

(20) \begin{equation}g(x){\rm{ \;=\; }}\left[ \begin{array}{l} g({x_1})\\ g({x_2})\end{array} \right]\end{equation}

(21) \begin{equation}g({x_{\rm{i}}}){\rm{ \;=\; }}\left\{ \begin{array}{l} {\rm{sig}}{({x_i})^\lambda },\\[4pt] {\gamma _1}{x_i} + {\gamma _2}sign({x_i}){x_i}^2\end{array} \right.\begin{array}{*{20}{c}} {}\\[4pt] ,\end{array}\begin{array}{*{20}{c}} {}\\[4pt] {}\end{array}\begin{array}{*{20}{c}} {\left| {{x_i}} \right| > \eta }\\[4pt] {\left| {{x_i}} \right| \le \eta }\end{array}\begin{array}{*{20}{c}} {}&{i = 1,2}\end{array}\end{equation}

(22) \begin{equation}{\gamma _1}{\rm{ \;=\; }}(2 - \gamma ){\eta ^{\lambda - 1}}\end{equation}

(23) \begin{equation}{\gamma _2}{\rm{ \;=\; }}(\gamma - 1){\eta ^{\lambda - 2}}\end{equation}

where $ 0 < \lambda < 1 $ , $ 0 < \gamma < 1 $ and $ \eta $ is a small positive constant. $sig{({x_i})^\lambda } = {\left| {{x_i}} \right|^\lambda }{\mathop{\rm sgn}} ({x_i})$ .

Remark 1. The sliding surface proposed in this paper can ensure that the sliding manifolds converge to a small area instead of zero within a finite time, avoiding the system chattering phenomenon caused by crossing the sliding surface, and at the same time, ensure that the system state converges to a small area in a finite time. The following Theorem 1 reveals a detailed explanation.

The derivative of s is expressed as follows.

(24) \begin{equation}\dot s{\rm{ \;=\; }}\left[ \begin{array}{l} {{\dot s}_1}\\ {{\dot s}_2}\end{array} \right] \;=\; \ddot x + {\alpha _1}\dot x + {\alpha _2}\dot g(x){{ = f + Bu}} + d{{ + l}}\end{equation}

where

(25) \begin{equation} {{l = }}{\alpha _1}\dot x + {\alpha _2}\dot g(x)\end{equation}

(26) \begin{equation}\dot g(x){\rm{ \;=\; }}\left[ \begin{array}{l} \dot g({x_1})\\ \dot g({x_2})\end{array} \right]\end{equation}

(27) \begin{equation}\dot g({x_{\rm{i}}}){\rm{ \;=\; }}\left\{ \begin{array}{l} \lambda {\left| {{x_i}} \right|^{\lambda {\rm{ - }}1}}{{\dot x}_i},\\[4pt] {\gamma _1}{{\dot x}_i} + 2{\gamma _2}{x_i}{{\dot x}_i}sign({x_i})\end{array} \right.\begin{array}{*{20}{c}} {}\\[4pt] ,\end{array}\begin{array}{*{20}{c}} {}\\[4pt] {}\end{array}\begin{array}{*{20}{c}} {\left| {{x_i}} \right| > \eta }\\[4pt] {\left| {{x_i}} \right| \le \eta }\end{array}\begin{array}{*{20}{c}} {}&{i = 1,2}\end{array}\end{equation}

In order to make the system state trajectory converge rapidly from the initial state to the designed sliding mode surfaces, a fast power reaching law is selected as follows:

(28) \begin{equation}\dot s{\rm{ \;=\; }}\left[ \begin{array}{l} {{\dot s}_1}\\[4pt] {{\dot s}_2}\end{array} \right] = - {{{k}}_2}s - {k_1}{\rm{sig}}{(s)^\gamma }{\rm{ \;=\; }}\left[ \begin{array}{l} - {{{k}}_2}{s_1} - {k_1}{\rm{sig}}{({s_1})^\gamma }\\[4pt] - {{{k}}_2}{s_2} - {k_1}{\rm{sig}}{({s_2})^\gamma }\end{array} \right]\end{equation}

where $ {k_1} > 0,{k_2} > 0 $ .

Then design the guidance law as follows:

(29) \begin{equation}{{u = }} - {{{B}}^{ - 1}}({{f}} + \hat d + {{ad + l + }}{{{k}}_2}s{{ + }}{k_1}{\rm{sig}}{(s)^\gamma })\end{equation}

where $ \hat d $ shown as Equation (30) is the estimation term of the system disturbance d caused by the target manoeuvering. ad is an adaptive term in order to eliminate the influence of system disturbance estimation error on the guidance system and improve the convergence speed of the guidance system. The expression of the adaptive term ad is designed as Equation (31). In this equation, the first term is used to accelerate the convergence rate of the guidance system and the second term is used to reduce the impact of the system disturbance estimation error.

(30) \begin{equation}\hat d{{ = }}\left[ \begin{array}{l} {{{{\hat d}}}_1}\\[4pt] {{\hat d}_2}\end{array} \right]\end{equation}

(31) \begin{equation}{{ad = }}\left[ \begin{array}{l} {{a}}{{{d}}_1}\\[9pt] a{d_2}\end{array} \right]{{ = }}\left[ \begin{array}{l} \dfrac{{{s_1}}}{{\left| {{s_1}} \right| + \mu }}{{\hat \chi }_1} + {{\hat \delta }_1}\\[9pt] \dfrac{{{s_2}}}{{\left| {{s_2}} \right| + \mu }}{{\hat \chi }_2} + {{\hat \delta }_2}\end{array} \right]\end{equation}

where

(32) \begin{equation}\dot {\hat{\delta _i}} = {\eta _1}{{{s}}_i},i = 1,2\end{equation}

(33) \begin{equation}{\dot {\hat{\chi_i}}}=\dfrac{{{s_i}^2}}{{\left| {{s_i}} \right| + \mu }},i = 1,2\end{equation}

(34) \begin{equation}{\tilde \chi _{{i}}}{\rm{ \;=\; }}{\hat \chi _i}{\rm{ - }}{\chi _i},i = 1,2\end{equation}

Next, the stability and finite time convergence of the guidance system will be analysed and proved. Before that, the following lemma is introduced.

Lemma 1. [Reference Si and Song39] For the nonlinear system $ {{\dot x = }}f(x,t),x \in {R^n} $ , it can assume that there exists a positive definite and continuous function V(x).The derivative of V(x) satisfies inequality (35)

(35) \begin{equation}{{\dot V}}(x) \le - \mu {{V}}(x) - \lambda {{{V}}^\alpha }(x)\end{equation}

where $ \mu > 0,\lambda > 0 $ and $ 0 < \alpha < 1 $ , if the initial time is $ {{{t}}_0} $ and $ {{x}}({{{t}}_0}) = {{{x}}_0} $ , the time from the initial state to the equilibrium point of the system satisfies the following inequality:

(36) \begin{equation}{\rm{T}} \le \frac{1}{{\mu (1 - \alpha )}}\ln \frac{{\mu {{{V}}^{1 - \alpha }}({x_0}) + \lambda }}{\lambda }\end{equation}

This inequality indicates that the system states converge in a finite-time.

Theorem 1. For the guidance system (14), the sliding mode surface is given as Equation (19), the reaching law is given as Equation (28) and the guidance law is given as Equation (29). We can achieve the following conclusions.

1. (a) In a finite-time, the sliding manifold s is converged to the small region $ \left| {\rm{s}} \right| \le {\varepsilon _1} $ , where $ {\varepsilon _1} $ is a very small positive constant.

2. (b) In a finite-time, the line-of-sight angle errors $ {q_\varepsilon } - {q_{\varepsilon d}} $ and $ {q_\beta } - {q_{\beta d}} $ are converged to the small regions $ \left| {{q_\varepsilon } - {q_{\varepsilon d}}} \right| < \eta $ and $ \left| {{q_\beta } - {q_{\beta d}}} \right| < \eta $ , where $\eta $ is a very small positive constant.

3. (c) In a finite-time, the line-of-sight angle rate $ {\dot q_\varepsilon } $ and $ {\dot q_\beta } $ are converged to the small regions $ \left| {{{\dot q}_\varepsilon }} \right| < \eta \eta $ and $ \left| {{{\dot q}_\beta }} \right| < \eta \eta $ , where

(37) \begin{equation}\eta \eta {\rm{ \;=\; }}{\alpha _1}\eta {\rm{ + }}{\alpha _2}{\eta ^\lambda }{\rm{ + }}{\varepsilon _1}\end{equation}

Proof. A Lyapunov function $ {{{V}}_1} $ is selected as Equation (38).

(38) \begin{equation}{{{V}}_1}{\rm{ \;=\; }}\frac{1}{2}{s^{{T}}}s\end{equation}

The residual estimation error of the guidance system disturbance is expressed as follows:

(39) \begin{equation}\left| {d - \hat d - {{ad}}} \right| \le \varepsilon\end{equation}

The derivative of $ {{{V}}_1} $ is expressed as follows.

(40) \begin{equation}\begin{array}{l} {{{{\dot V}}}_1}{\rm{ \;=\; }}{s^{{T}}}\dot s\\[4pt] {\rm{ = }}{s^{{T}}}({{f + Bu}} + d{{ + l}})\\[4pt] {\rm{ = }}{s^{{T}}}(d - \hat d - {{ad}} - {{{k}}_2}s - {k_1}{\rm{sig}}{(s)^\gamma })\\[4pt] \le - {{{k}}_2}{\left| {{s}} \right|^2} - {k_1}{\left| {{s}} \right|^{\gamma {\rm{ + }}1}}{\rm{ + }}\varepsilon \left| {{s}} \right|\end{array}\end{equation}

Rewrite the formula (40) into the following two forms:

(41) \begin{equation}{{{\dot V}}_1} \le - {{{k}}_2}{\left| {{s}} \right|^2} - ({k_1}{\left| {{s}} \right|^\gamma } - \varepsilon )\left| {{s}} \right|\end{equation}

(42) \begin{equation}{{{\dot V}}_1} \le - ({{{k}}_2}\left| {{s}} \right| - \varepsilon )\left| {{s}} \right| - {k_1}{\left| {{s}} \right|^{\gamma {\rm{ + }}1}}\end{equation}

When $ {k_1}{\left| {{s}} \right|^\gamma } - \varepsilon \ge 0 $ , $ \left| {\rm{s}} \right| \ge {(\varepsilon /{k_1})^{1/\gamma }} $ ,

(43) \begin{equation} {{{\dot V}}_1} \le - {{{k}}_2}{\left| {{s}} \right|^2} = - 2{{{k}}_2}{{{V}}_1}\end{equation}

When $ {{\rm{k}}_2}\left| {\rm{s}} \right| - \varepsilon \ge 0 $ , $ \left| {\rm{s}} \right| \ge (\varepsilon /{k_2}) $ ,

(44) \begin{equation} {{{\dot V}}_1} \le - {k_1}{\left| {{s}} \right|^{\gamma {\rm{ + }}1}} = - {2^{\frac{{\gamma {\rm{ + }}1}}{2}}}{{{k}}_1}{{{V}}_1}^{\frac{{\gamma {\rm{ + }}1}}{2}}\end{equation}

It can be known from Lemma 1 and the inequality (43) and (44) that the sliding manifold s is converged to the small region $ \left| {{s}} \right| \le \min \left\{ {{{(\varepsilon /{k_1})}^{1/\gamma }},} \right.\left. {(\varepsilon /{k_2})} \right\}{\rm{ \;=\; }}{\varepsilon _1} $ within a finite-time. where $ {\varepsilon _1} $ is a very small positive constant. Therefore, conclusion (a) is proved.

Case 1. $ If \left| {{{{x}}_i}} \right| \ge \eta ,i = 1,2 $ , A Lyapunov function $ {{{V}}_2} $ is selected as Equation (45).

(45) \begin{equation} {{{V}}_2}{{ = }}\frac{1}{2}{{{x}}_i}{{{x}}_i} ,{{ i}} = 1,2\end{equation}

When $ \left| {{s}} \right| \le \min \left\{ {{{(\varepsilon /{k_1})}^{1/\gamma }},} \right.\left. {(\varepsilon /{k_2})} \right\}{\rm{ \;=\; }}{\varepsilon _1} $ , according to Equation (19), the inequality (46) can be obtained.

(46) \begin{equation}{{{\dot x}}_i} \le - {\alpha _1}{{{x}}_i} - {\alpha _2}{\rm{sig}}{({x_i})^\lambda }{\rm{ + }}{\varepsilon _1}\end{equation}

The derivation of $ {{V}_2} $ is expressed as the inequality (47).

(47) \begin{equation}\begin{array}{l} {{{{\dot V}}}_2}{\rm{ \;=\; }}{{{x}}_i}{{{{\dot x}}}_i}\\[4pt] {\rm{ }} \le - {{{x}}_i}({\alpha _1}{{{x}}_i} + {\alpha _2}{\rm{sig}}{({x_i})^\lambda } - {\varepsilon _1})\\[4pt] {\rm{ }} \le - {\alpha _1}{\left| {{{{x}}_{{i}}}} \right|^2} - {\alpha _2}{\left| {{{{x}}_{{i}}}} \right|^{\lambda + 1}} + {\varepsilon _1}\left| {{{{x}}_{{i}}}} \right|\end{array}\end{equation}

Rewrite the inequality (47) in the following two forms:

(48) \begin{equation}{{{\dot V}}_2} \le - {\alpha _1}{\left| {{{{x}}_{{i}}}} \right|^2} - ({\alpha _2}{\left| {{{{x}}_{{i}}}} \right|^\lambda } - {\varepsilon _1})\left| {{{{x}}_{{i}}}} \right|\end{equation}

(49) \begin{equation}{{{\dot V}}_2} \le - ({\alpha _1}\left| {{{{x}}_{{i}}}} \right| - {\varepsilon _1})\left| {{{{x}}_{{i}}}} \right| - {\alpha _2}{\left| {{{{x}}_{{i}}}} \right|^{\lambda {{ + }}1}}\end{equation}

When $ {\alpha _2}{\left| {{{{x}}_{{i}}}} \right|^\lambda } - {\varepsilon _1} \ge 0 $ , $ \left| {{{{x}}_{{i}}}} \right| \ge {({\varepsilon _1}/{\alpha _2})^{1/\lambda }} $ ,

(50) \begin{equation}{{{\dot{\textrm{V}}}}_2} \le - {\alpha _1}{\left| {{{{x}}_{{i}}}} \right|^2}{\rm{ \;=\; }} - 2{\alpha _1}{{\rm{V}}_2}\end{equation}

When $ {\alpha _1}\left| {{{{x}}_{{i}}}} \right| - {\varepsilon _1} \ge 0 $ , $ \left| {{{{x}}_{{i}}}} \right| \ge ({\varepsilon _1}/{\alpha _1}) $ ,

(51) \begin{equation}{{{\dot V}}_2} \le - {\alpha _2}{\left| {{{{x}}_{{i}}}} \right|^{\lambda {\rm{ + }}1}}{\rm{ \;=\; }} - {2^{\frac{{\lambda {\rm{ + }}1}}{2}}}{\alpha _2}{{V}}_2^{\frac{{\lambda {\rm{ + }}1}}{2}}\end{equation}

Supposed that $ \min \left\{ {{{({\varepsilon _1}/{\alpha _2})}^{1/\lambda }},} \right.\left. {({\varepsilon _1}/{\alpha _1})} \right\} \le \eta $ , it can be known from Lemma 1 and the inequality (50) and (51) that $ {{{x}}_i} $ is converged to the very small region $ \left| {{{{x}}_i}} \right| < \eta ,i = 1,2 $ within a finite-time. So the line-of-sight angle errors $ {q_\varepsilon } - {q_{\varepsilon d}} $ and $ {q_\beta } - {q_{\beta d}} $ are converged to the small regions $ \left| {{q_\varepsilon } - {q_{\varepsilon d}}} \right| < \eta $ and $ \left| {{q_\beta } - {q_{\beta d}}} \right| < \eta $ within a finite-time. Conclusion (b) is proved.

Case 2. If $ \left| {{{\rm{x}}_i}} \right| < \eta ,i = 1,2 $ , when $ \left| {{s}} \right| \le \min \left\{ {{{(\varepsilon /{k_1})}^{1/\gamma }},} \right.\left. {(\varepsilon /{k_2})} \right\}{\rm{ \;=\; }}{\varepsilon _1} $ , according to Equation (19), the inequality (52) and (53) can be easily obtained.

(52) \begin{equation}{{{\dot x}}_i} \le - {\alpha _1}{{{x}}_i} - {\alpha _2}({\gamma _1}{x_i} + {\gamma _2}sign({x_i}){x_i}^2){\rm{ + }}{\varepsilon _1}\end{equation}

(53) \begin{equation}\begin{array}{l} \left| {{{{{\dot x}}}_i}} \right| \le \left| {{\alpha _1}{{{x}}_i}{\rm{ + }}{\alpha _2}({\gamma _1}{x_i} + {\gamma _2}sign({x_i}){x_i}^2)} \right|{\rm{ + }}{\varepsilon _1}\\[4pt] {\rm{ }} \le \left| {{\alpha _1}{{{x}}_i}} \right|{\rm{ + }}\left| {{\alpha _2}({\gamma _1}{x_i} + {\gamma _2}sign({x_i}){x_i}^2)} \right|{\rm{ + }}{\varepsilon _1}\\[4pt] {\rm{ }} \le {\alpha _1}\eta {\rm{ + }}{\alpha _2}{\eta ^\lambda }{\rm{ + }}{\varepsilon _1}{\rm{ \;=\; }}\eta \eta\end{array}\end{equation}

Then, $ {{{\dot x}}_i} $ is converged to the very small region $ \left| {{{{\dot{\rm x}}}_i}} \right| < \eta \eta ,i = 1,2 $ within a finite-time. So the line-of-sight angle rate $ {\dot q_\varepsilon } $ and $ {\dot q_\beta } $ are converged to the small regions $ \left| {{{\dot q}_\varepsilon }} \right| < \eta \eta $ and $ \left| {{{\dot q}_\beta }} \right| < \eta \eta $ within a finite-time. Conclusion (c) is proved.

3.2 RBF neural network disturbance observer design

In order to intercept large manoeuvering targets accurately, it is necessary to estimate the disturbance of the guidance system caused by target manoeuvering. In this paper, a new global RBF neural network is proposed to estimate the disturbance. The disturbance $ d{\rm{ \;=\; }}{\left[ {\begin{array}{*{20}{c}}{{{{d}}_1}}&{{{{d}}_2}} \end{array}} \right]^{{T}}} $ can be estimated using the RBF neural network as Equation (54).

(54) \begin{equation}{{{d}}_i}{\text{ = }}\sum\limits_{j = 1}^{{m}} {w_{{{ij}}}^*} {{h}}_{{{ij}}}^* + {\varepsilon _{i0}} = {W_i}{^{*T}}{{{h}}_i}^* + {\varepsilon _{i0}}{\text{ = }}{{{d}}_i}^{\text{*}} + {\varepsilon _{i0}},i = 1,2\end{equation}

Where $ {{{W}}_{{i}}}^{{*}}{\rm{ \;=\; }}{\left[ {w_{i1}^{{*}}\begin{array}{*{20}{c}} {w_{i2}^{\rm{*}}}&{...}&{w_{{{im}}}^{\rm{*}}} \end{array}} \right]^T},i = 1,2 $ are optimal output weight vectors of the output layer of the RBF neural network. $ {\varepsilon _{i0}},i = 1,2 $ are the optimal estimation error. $ {{{d}}_i}^{{*}},i = 1,2 $ are the optimal approximation of the disturbance $ {{\rm{d}}_i},i = 1,2 $ . $ {{{h}}_{{i}}}^{\rm{*}}{\rm{ \;=\; }}{\left[ {h_{i1}^*\begin{array}{*{20}{c}} {h_{i2}^*}&{...}&{h_{{{im}}}^*}\end{array}} \right]^T},i = 1,2 $ are optimal vectors whose elements are given by Equation (55) which represents the Gaussian activation function of RBF neural network hidden layer neurons.

(55) \begin{equation}{{h}}_{{{ij}}}^*{\rm{ \;=\; }}\exp \left(\frac{{ - {{\left\| {x - c_{{{ij}}}^*} \right\|}^2}}}{{2\sigma _{ij}^{*2}}}\right),i = 1,2\end{equation}

Where $c_{{ij}}^{*}$ is the optimal centre and $\sigma _{ij}^{*}$ is the optimal width for the ${{{j}}_{{th}}}$ Gaussian function.

The RBF neural network disturbance estimator is chosen as Equation (56).

(56) \begin{equation}{{{\hat d}}_i}{\rm{ \;=\; }}\sum\limits_{j = 1}^{{m}} {{{\hat w}_{{{ij}}}}} {{{\hat h}}_{ij}} = {\hat W_i}^T{{{\hat h}}_i},i = 1,2,{{j}} = 1,2,3 \cdot \cdot \cdot m\end{equation}

Where $\hat{W}_{\rm{i}}{\rm{ \;=\; }}{\left[ {{{\hat w}_{i1}}\begin{array}{*{20}{c}} {{{\hat w}_{i2}}}&{...}&{{{\hat w}_{{{im}}}}} \end{array}} \right]^T},i = 1,2$ that adaptively adjust online are output weight vector of the RBF neural network linear output layer. ${{\hat{\rm h}}_{\rm{i}}}{\rm{ \;=\; }}{\left[ {{{\hat h}_{{\rm{i}}1}}\begin{array}{*{20}{c}} {{{\hat h}_{i2}}}&{...}&{{{\hat h}_{{\rm{im}}}}}\end{array}} \right]^T}$ $,i = 1,2$ that also adaptively adjusts online are the activation function vectors of the hidden layer neurons of the RBF neural network. The elements of ${{{\hat h}}_{{i}}}$ are given by Equation (57).

(57) \begin{equation}{{{\hat h}}_{{{ij}}}}{\rm{ \;=\; }}\exp \left(\frac{{ - {{\left\| {x - {{\hat c}_{ij}}} \right\|}^2}}}{{2\hat \sigma _{ij}^2}}\right),i = 1,2,{{j}} = 1,2,3 \cdot \cdot \cdot m\end{equation}

Let the optimal approximation of the disturbance ${{{d}}_i}^{\rm{*}},i = 1,2$ subtract the adaptive approximation of the disturbance ${\hat d_i},i = 1,2$ . Then we use linearisation techniques to expand it into a partially linear form of Taylor expansion series. Equation (58) is obtained.

(58) \begin{equation}{{{d}}_i}^* - {\hat d_i}{\rm{ \;=\; }}{W_i}{^{*T}}{{{h}}_i}^* - {\hat W_i}^T{{{\hat h}}_i}{\rm{ \;=\; }} - {\tilde W_i}^T{{{\hat h}}_i} - {\hat W_i}\frac{{\partial {{{h}}_i}}}{{\partial {c_i}}}{\tilde c_i} - {\hat W_i}\frac{{\partial {{\rm{h}}_i}}}{{\partial {\sigma _i}}}{\tilde \sigma _i} + {\varepsilon _{i1}},i = 1,2\end{equation}

Where ${{{\tilde W}}_{{i}}}{{ = }}{{{\hat W}}_i} - {{{W}}_i}^{\rm{*}},i = 1,2$ are the deviation of weight vectors between the adaptive weight vectors ${{{\hat W}}_i}$ and the optimal weight vectors ${{{W}}_i}^{\rm{*}}$ for the linear output layer of RBF neural network. ${{{\tilde c}}_{{i}}}{\rm{ \;=\; }}{\hat c_i} - {{{c}}_i}^{\rm{*}},i = 1,2$ are the deviation centre vectors of the Gaussian function between the adaptive centre vectors ${\hat c_i}$ and the optimal centre vectors ${{{c}}_i}^{\rm{*}}$ . ${\tilde \sigma _{{i}}}{\rm{ \;=\; }}{\hat \sigma _i} - {\sigma _i}^{\rm{*}},i = 1,2$ are the deviation of the width vectors between the adaptive width vectors ${\hat \sigma _i}$ and the optimal width vectors ${\sigma _i}^{\rm{*}}$ . ${\varepsilon _{i1}},i = 1,2$ are the error for the Taylor series. $\frac{{\partial {{{h}}_i}}}{{\partial {c_i}}},i = 1,2$ are given by Equation (59) and $\frac{{\partial {{{h}}_i}}}{{\partial {\sigma _i}}},i = 1,2$ are given by Equation (60), respectively.

(59) \begin{equation}\frac{{\partial {{{h}}_{{i}}}}}{{\partial {c_i}}}{\rm{ \;=\; }}{{{h'_{ci}}}} = \left[ {\begin{array}{*{20}{c@{\quad}c@{\quad}c@{\quad}c}} {\frac{{\partial {{{h}}_{i1}}}}{{\partial {c_{i1}}}}}&0&0&0\\ 0&{\frac{{\partial {{\rm{h}}_{i2}}}}{{\partial {c_{i2}}}}}&0&0\\ \cdots & \cdots & \cdots & \cdots \\ \cdots & \cdots & \cdots &{\frac{{\partial {{{h}}_{im}}}}{{\partial {c_{im}}}}}\end{array}} \right] \in {R^{m \times m}},i = 1,2\end{equation}

(60) \begin{equation}\frac{{\partial {{{h}}_{{i}}}}}{{\partial {\sigma _i}}}{\rm{ \;=\; }}{{{h'_{\sigma i}}}} = \left[ {\begin{array}{*{20}{c@{\quad}c@{\quad}c@{\quad}c}} {\frac{{\partial {{\rm{h}}_{i1}}}}{{\partial {\sigma _{i1}}}}}&0&0&0\\ 0&{\frac{{\partial {{\rm{h}}_{i2}}}}{{\partial {\sigma _{i2}}}}}&0&0\\ \cdots & \cdots & \cdots & \cdots \\ \cdots & \cdots & \cdots &{\frac{{\partial {{{h}}_{im}}}}{{\partial {\sigma _{im}}}}}\end{array}} \right] \in {R^{m \times m}},i = 1,2\end{equation}

Substituting Equation (29) into Equation (24), Equation (61) can be obtained.

(61) \begin{equation}{\dot s_{{i}}}{\rm{ \;=\; }}{{{d}}_i} - {\hat d_i} - a{d_i} - {{{k}}_2}{s_i} - {k_1}{\rm{sig}}{({s_i})^\gamma },i = 1,2\end{equation}

Substituting Equations (31) and (58) into Equation (61), Equation (62) can be obtained.

(62) \begin{equation}\begin{array}{l} {{\dot s}_{{i}}}{\rm{ \;=\; }} - {{\tilde W}_i}^T{{{{\hat h}}}_i} - {{\hat W}_i}{{{{h_{\sigma i}}}} }{{\tilde c}_i} - {{\hat W}_i}{{{{h'_{\sigma i}}}}}{{\tilde \sigma }_i} + {\delta _{{i}}}\\[3pt] - \frac{{{s_i}}}{{\left| {{s_i}} \right| + \mu }}{{\hat \chi }_i} - {{\hat \delta }_i} - {{{k}}_2}{s_i} - {k_1}{\rm{sig}}{({s_i})^\gamma },i = 1,2\end{array}\end{equation}

Where ${\delta _{{i}}}{\rm{ \;=\; }}{\varepsilon _{0i}} + {\varepsilon _{1i}},i = 1,2$ are total RBF neural network estimation error for the system disturbance. It is the sum of the optimal estimation error ( ${\varepsilon _{i0}},i = 1,2$ ) and the error for the Taylor series ( ${\varepsilon _{i1}},i = 1,2$ ).

Define ${\tilde \delta _{{i}}},i = 1,2$ to be the deviation of the total RBF neural network estimation error for the system disturbance between the adaptive estimation error ${\hat \delta _i}$ and the actual estimation error ${\delta _i}$ , which is expressed by Equation (63).

(63) \begin{equation}{\tilde \delta _{{i}}}{\rm{ \;=\; }}{\hat \delta _i} - {\delta _i},i = 1,2\end{equation}

Substituting Equation (63) into Equation (62), Equation (64) can be obtained.

(64) \begin{equation}{\dot s_{{i}}}{\rm{ \;=\; }} - {\tilde W_i}^T{{{\hat h}}_i} - {\hat W_i}{{{h'_{\!\!ci}}}}{\tilde c_i} - {\hat W_i}{{{h'_{\!\!\sigma i}}}}{\tilde \sigma _i} - {\tilde \delta _{{i}}} - \frac{{{s_i}}}{{\left| {{s_i}} \right| + \mu }}{\hat \chi _i} - {{{k}}_2}{s_i} - {k_1}{\rm{sig}}{({s_i})^\gamma },i = 1,2\end{equation}

In order to prove the stability of the guidance system expressed in Equation (14) under the guidance law expressed in Equation (29), the Lyapunov function is selected in Equation (65).

(65) \begin{equation}{{{V}}_{0{{i}}}}{\rm{ \;=\; }}\frac{1}{2}{s_i}^2{\rm{ + }}\frac{1}{2}{\tilde \chi _i}^2{\rm{ + }}\frac{1}{{2{\eta _1}}}{\tilde \delta _i}^2{\rm{ + }}\frac{1}{{2{\eta _2}}}{\tilde W_i}^T{\tilde W_i}{\rm{ + }}\frac{1}{{2{\eta _3}}}{\tilde c_i}^T{\tilde c_i} + \frac{1}{{2{\eta _4}}}{\tilde \sigma _i}^T{\tilde \sigma _i},i = 1,2\end{equation}

Where ${\eta _1}$ , ${\eta _2}$ , ${\eta _3}$ and ${\eta _4}$ are designed parameters, ${\tilde \chi _{{i}}}{\rm{ \;=\; }}{\hat \chi _i}{\rm{ - }}{\chi _i},i = 1,2$ are the error between the adaptive gain ${\hat \chi _i}$ and the fixed gain ${\chi _i}$ .

Take the derivative of the Lyapunov function and get Equation (66).

(66) \begin{equation}{{{\dot V}}_{0{{i}}}}{\rm{ \;=\; }}{{{s}}_i}{{{\dot s}}_i}{\rm{ + }}{\tilde \chi _i}{\dot {\tilde \chi} _i}{\rm{ + }}\frac{1}{{{\eta _1}}}{\tilde \delta _i}{\dot {\tilde \delta} _i}{\rm{ + }}\frac{1}{{{\eta _2}}}{\tilde W_i}^T{\dot {\tilde {W_i}}}{\rm{ + }}\frac{1}{{{\eta _3}}}{\tilde c_i}^T{\dot {\tilde {c_i}}} + \frac{1}{{{\eta _4}}}{\tilde \sigma _i}^T{\dot {\tilde \sigma} _i},i = 1,2\end{equation}

Substituting in Equation (64) into Equation (66), Equation (67) can be obtained.

(67) \begin{align} {{{{\dot V}}}_{0{{i}}}} & = {{{s}}_i}\left( - {{\tilde W}_i}^T{{{{\hat h}}}_i} - {{\hat W}_i}{{{{h'_{\!\!ci}}}}}{{\tilde c}_i} - {{\hat W}_i}{{{{h'_{\!\!\sigma i}}}}}{{\tilde \sigma }_i} - {{\tilde \delta }_{{i}}} - \frac{{{s_i}}}{{\left| {{s_i}} \right| + \mu }}{{\hat \chi }_i} - {{{k}}_2}{s_i} - {k_1}{\rm{sig}}{({s_i})^\gamma }\right)\nonumber\\ & \quad + {{\tilde \chi }_i}{{\dot {\tilde \chi }}_i}{\rm{ + }}\frac{1}{{{\eta _1}}}{{\tilde \delta }_i}{{\dot {\tilde \delta} }_i}{\rm{ + }}\frac{1}{{{\eta _2}}}{{\tilde W}_i}^T{{\dot {\tilde W}}_i}{\rm{ + }}\frac{1}{{{\eta _3}}}{{\tilde c}_i}^T{{\dot {\tilde c}}_i} + \frac{1}{{{\eta _4}}}{{\tilde \sigma }_i}^T{{\dot {\tilde \sigma} }_i},i = 1,2\end{align}

Substituting ${\dot {\tilde {\delta _{{i}}}}}{\rm{ \;=\; }}{\dot {\hat {\delta _i}}}$ , ${\dot {\tilde {\chi _{{i}}}}}{\rm{ \;=\; }}{\dot {\hat {\chi _{{i}}}}}$ , ${\dot {\tilde {W_{{i}}}}}{\rm{ \;=\; }}{\dot {\hat {W_{{i}}}}}$ , ${{{\dot {\tilde {c_{{i}}}}}}}{\rm{ \;=\; }}{{{\dot {\hat {c_i}}}}}$ and ${\dot {\tilde {\sigma _{{i}}}}}{\rm{ \;=\; }}{\dot {\hat {\sigma _i}}}$ into Equation (67), Equation (68) can be obtained.

(68) \begin{align} {{{{\dot V}}}_{0{{i}}}} &= {{{s}}_i}\left( - {{\tilde W}_i}^T{{{{\hat h}}}_i} - {{\hat W}_i}{{{{h'}}}_{ci}}{{\tilde c}_i} - {{\hat W}_i}{{{{h'}}}_{\sigma i}}{{\tilde \sigma }_i} - {{\tilde \delta }_{{i}}} - \frac{{{s_i}}}{{\left| {{s_i}} \right| + \mu }}{{\hat \chi }_i} - {{{k}}_2}{s_i} - {k_1}{\rm{sig}}{({s_i})^\gamma }\right) \nonumber\\ & \quad + {{\tilde \chi }_i}{{\dot {\hat {\chi_i}} }}{\rm{ + }}\frac{1}{{{\eta _1}}}{{\tilde \delta }_i}{{\dot {\hat {\delta_i}} }}{\rm{ + }}\frac{1}{{{\eta _2}}}{{\tilde W}_i}^T{{\dot {\hat {W_i}}}}{\rm{ + }}\frac{1}{{{\eta _3}}}{{\tilde c}_i}^T{{\dot {\hat {c_i}}}} + \frac{1}{{{\eta _4}}}{{\tilde \sigma }_i}^T{{\dot {\hat {\sigma_i}} }},i = 1,2 \end{align}

By rearranging and combining Equation (68), Equation (69) can be obtained.

(69) \begin{align} {{{{\dot V}}}_{0{{i}}}} &= {{{s}}_i}( - {{{k}}_2}{s_i} - {k_1}{\rm{sig}}{({s_i})^\gamma }) - \frac{{{s_i}^2}}{{\left| {{s_i}} \right| + \mu }}({{\hat \chi }_i}{\rm{ - }}{{\tilde \chi }_i}){\rm{ + }}{{\tilde \delta }_i}\left(\frac{1}{{{\eta _1}}}{{\dot {\hat {\delta _i}} }} - {{{s}}_i}\right){\rm{ + }}{{\tilde W}_i}^T \left(\frac{1}{{{\eta _2}}}{{\dot {\hat {W_i}}}} - {{{{\hat h}}}_i}{s_i}\right) \nonumber\\[1pt] & \quad + {{\tilde c}_i}^T \left(\frac{1}{{{\eta _3}}}{{\dot {\hat {c_i}}}} - {{{{h'_{\!\!ci}}}}}{{{s}}_i}{{\hat W}_i}\right) + {{\tilde \sigma }_i}^T\left(\frac{1}{{{\eta _4}}}{{\dot {\hat {\sigma_i}} }} - {{{{h'_{\!\!\sigma i}}}}}{{{s}}_i}{{\hat W}_i}\right),i = 1,2\end{align}

To guarantee that the guidance system expressed in Equation (14) under the guidance law expressed in Equation (29) is globally stable and the first derivative of the Lyapunov function ${{{\dot V}}_{0{{i}}}} < 0$ is satisfied, the time evolution of the adaptive estimation error ${\hat \delta _i}$ , the adaptive weight vectors ${{{\hat W}}_i}$ , the adaptive centre vectors ${\hat c_i}$ and the adaptive width vectors ${\hat \sigma _i}$ are chosen as Equation (70).

(70) \begin{equation}{\dot {\hat {\delta_{{i}}}}}{\rm{ \;=\; }}{\eta _1}{{{s}}_{{i}}}, {\dot{\hat {W_i}}}{\rm{ \;=\; }}{\eta _2}{{{\hat h}}_i}{s_i}{{ , }} {\dot {\hat {c_{{i}}}}}{\rm{ \;=\; }}{\eta _3}{{{h'_{\!\!ci}}}}{{{s}}_i}{\hat W_i}, {\dot {\hat {\sigma _i}}}{\rm{ \;=\; }}{\eta _4}{{{h'_{\!\!\sigma i}}}}{{{s}}_i}{\hat W_i}, i = 1,2\end{equation}

Substituting Equations (34) and (70) into Equation (69), Equation (71) can be obtained.

(71) \begin{equation}{{{\dot V}}_{0{{i}}}}{\rm{ \;=\; }}{{{s}}_i}(\!- {{{k}}_2}{s_i} - {k_1}{\rm{sig}}{({s_i})^\gamma }) - \frac{{{s_i}^2}}{{\left| {{s_i}} \right| + \mu }}{\chi _i},i = 1,2\end{equation}

According to Equation (71), the stability of the guidance system is guaranteed since the first derivative of the Lyapunov function ${{{\dot V}}_{0{{i}}}} < 0$ with the gain ${\chi _i},i = 1,2$ as a positive value.

The summary of the presented three-dimensional finite-time guidance law schemeis based on sliding mode adaptive RBF neural network against highly manoeuvering targets is as follows:

(72) \begin{equation}\begin{array}{l} {{u = }} - {{{B}}^{ - 1}}{{(f}} + \hat d + {{ad + l + }}{{{k}}_2}s{\rm{ + }}{k_1}{\rm{sig}}{(s)^\gamma })\\[5pt] {\rm{ = }}\left[ \begin{array}{l} {{{u}}_1}\\[5pt] {{{u}}_2} \end{array} \right]{\rm{ \;=\; }}\left[ \begin{array}{l} - \frac{1}{{{b_1}}}{\rm{(}}{{{f}}_1}{\rm{ + }}{{{{\hat d}}}_1}{{ + a}}{{{d}}_1}{\rm{ + }}{\alpha _1}{{\dot x}_1} + {\alpha _2}\dot g({x_1}){\rm{ + }}{{{k}}_2}{s_1}{\rm{ + }}{k_1}{\rm{sig}}{({s_1})^\gamma })\\[5pt] - \frac{1}{{{b_2}}}{\rm{(}}{{{f}}_2}{\rm{ + }}{{\hat d}_2}{\rm{ + }}a{d_2}{\rm{ + }}{\alpha _1}{{\dot x}_2} + {\alpha _2}\dot g({x_2}){\rm{ + }}{{{k}}_2}{s_2}{\rm{ + }}{k_1}{\rm{sig}}{({s_2})^\gamma }) \end{array} \right]\\[5pt] {\rm{ = }}\left[ \begin{array}{l} - \frac{1}{{{b_1}}}( - \frac{{2\dot R}}{R}{x_3} - {x_4}^2\sin {q_\varepsilon }\cos {q_\varepsilon }{\rm{ + }}{{\hat W}_1}^T{{{{\hat h}}}_1}{\rm{ + }}\\[5pt] \frac{{{s_1}}}{{\left| {{s_1}} \right| + \mu }}{{\hat \chi }_1} + {{\hat \delta }_1}{\rm{ + }}{\alpha _1}{{\dot x}_1} + {\alpha _2}\dot g({x_1}){\rm{ + }}{{{k}}_2}{s_1}{\rm{ + }}{k_1}{\rm{sig}}{({s_1})^\gamma })\\[5pt] - \frac{1}{{{b_2}}}( - \frac{{2\dot R}}{R}{x_4} + 2{x_3}{x_4}\tan {q_\varepsilon }{\rm{ + }}{{\hat W}_2}^T{{{{\hat h}}}_2}{\rm{ + }}\\[5pt] \frac{{{s_2}}}{{\left| {{s_2}} \right| + \mu }}{{\hat \chi }_2} + {{\hat \delta }_2}{\rm{ + }}{\alpha _1}{{\dot x}_2} + {\alpha _2}\dot g({x_2}){\rm{ + }}{{{k}}_2}{s_2}{\rm{ + }}{k_1}{\rm{sig}}{({s_2})^\gamma }) \end{array} \right]\end{array}\end{equation}

4.1 Effectiveness verification

To verify the effectiveness of the proposed guidance law, four missiles in the same initial position and state are considered to attack targets in two manoeuvering situations at their desired terminal line-of-sight angles. Initial conditions of the missile and the target for the simulation are shown as Table 1. The expected terminal line-of-sight angles of the four missiles attacking the target are shown in Table 2. Two different cases for target manoeuvers are as follows Case 1 and Case 2.

Case 1: ${a_{T\varepsilon }}{\rm{ \;=\; }}200\sin (0.5t){\rm{m}}/{{\rm {\rm s}}^2}$ , ${a_{T\beta }}{\rm{ \;=\; }}200\cos (0.5t){\rm{m}}/{{\rm s}^2}$ .

Case 2: ${a_{T\varepsilon }}{\rm{ \;=\; }} -\!200{\rm{m}}/{{\rm s}^2}$ , ${a_{T\beta }}{\rm{ \;=\; }}200{\rm{m}}/{{\rm s}^2}$ .

For Case 1, Fig. 2 and Table 3 show the simulation results of the proposed guidance law. The relative motion curves of the missile and the target are elaborated in Fig. 2(a). It can be known that four missiles with different expected terminal line-of-sight angles were all able to successfully intercept the target. The curves of ${q_\varepsilon }$ and ${q_\beta }$ for the four missiles are demonstrated in Fig. 2(b) and (c). It can be obtained that ${q_\varepsilon }$ and ${q_\beta }$ converge to different expected terminal line-of-sight angles precisely in a finite-time. The curves of ${\dot q_\varepsilon }$ and ${\dot q_\beta }$ for the four missiles are shown in Fig. 2(d) and (e). It is clearly revealed that the guidance system has the best convergence performance under the proposed guidance law in this paper. Figure 2(f) and (g) indicate the curves of ${{{u}}_\varepsilon }$ and ${{{u}}_\beta }$ , which are within reasonable bounds. From Table 3, it can be known that all four missiles in the same initial state and at the same initial position under the proposed guidance law in this paper are able to accurately hit a manoeuvering target with the specified terminal line-of-sight angle. The maximum miss distance for the four missiles with different expected terminal sight angles is less than 0.002m, and the errors of the terminal line-of-sight angles are all controlled within $0.006^{\circ}$ . The excellent control ability of the proposed guidance law in this paper to miss distance and expected terminal line-of-sight angle error is verified.

Table 3. Simulation results of four expected line-of-sight angles attack targets with Case 1

Figure 3. Simulation results of four expected line-of-sight angles attack targets with Case 2.

For Case 2, when ${a_{T\varepsilon }}{\rm{ \;=\; }} -\!200{\rm{m}}/{{\rm s}^2}$ , ${a_{T\beta }}{\rm{ \;=\; }}200{\rm{m}}/{{\rm s}^2}$ , Fig. 3 and Table 4 display the simulation results of the proposed guidance law. The relative motion curves of the missile and the target are revealed in Fig. 3(a). The curves of ${q_\varepsilon }$ and ${q_\beta }$ for the four missiles are manifested in Fig. 3(b) and (c). The curves of ${\dot q_\varepsilon }$ and ${\dot q_\beta }$ for the four missiles are shown in Fig. 3(d) and (e). The curves of ${{{u}}_\varepsilon }$ and ${{{u}}_\beta }$ for the four missiles are displayed in Fig. 3(f) and (g). Figure 3 shows the same situation as Fig. 2, so it will not be repeated here. From Table 4, it can be known that all four missiles in the same initial state and at the same initial position under the proposed guidance law in this paper are able to accurately hit a manoeuvering target with the specified terminal line-of-sight angle. The maximum miss distance for the four missiles with different expected terminal sight angles is less than 0.0001m, and the errors of the terminal line-of-sight angles are all controlled within $0.007^{\circ}$ . The excellent control ability of the proposed guidance law in this paper to miss distance and expected terminal line-of-sight angle error is verified.

4.2 Superiority verification

To further verify the superiority of the proposed guidance law, the conventional proportional navigation guidance law and nonlinear terminal sliding mode guidance law [Reference Kumar and Ghose29] are selected for comparison. Initial conditions of the missile and the target for the simulation are shown as Table 1. Select the expected terminal line-of-sight angles ${q_{\varepsilon d}}{\rm{ \;=\; }}30^\circ $ , ${q_{\beta d}}{\rm{ \;=\; }}10^\circ $ . Two different cases for target manoeuvers are as follows Case 1 and Case 2.

Case 1: ${a_{T\varepsilon }}{\rm{ \;=\; }}200\sin (0.5t){\rm{m}}/{{\rm s}^2}$ , ${a_{T\beta }}{\rm{ \;=\; }}200\cos (0.5t){\rm{m}}/{{\rm s}^2}$ .

Case 2: ${a_{T\varepsilon }}{\rm{ \;=\; }} -\!200{\rm{m}}/{{\rm s}^2}$ , ${a_{T\beta }}{\rm{ \;=\; }}200{\rm{m}}/{{\rm s}^2}$ .

The traditional proportional guidance law is shown as Equation (73).

(73) \begin{equation}{u_{PN{{GL}}}} = \left[ \begin{array}{l} {{{n}}_1}\left| {\dot R} \right|{{\dot q}_\varepsilon }\\[5pt] {n_2}\left| {\dot R} \right|{{\dot q}_\beta }\end{array} \right]\end{equation}

Where ${{{n}}_1}$ and ${{{n}}_2}$ are guidance parameters of the ${u_{PNGL}}$ .

Table 4. Simulation results of four expected line-of-sight angles attack targets with Case 2

Figure 4. Simulation results of three different guidance laws with Case 1.

After a lot of simulations, the control effect of ${u_{PNGL}}$ is optimised by adjusting the parameters. The final guidance parameters ${{{n}}_1}{\rm{ \;=\; }}4$ and ${{{n}}_2}{\rm{ \;=\; }}4$ are chosen.

The sliding surface of nonlinear terminal sliding mode guidance law (NTSMGL) is chosen as Equation (74).

(74) \begin{equation} {{s}} = \left[ \begin{array}{l} {s_1}\\[5pt] {s_2} \end{array} \right] = x + \frac{1}{\beta }{\dot x^{p/q}}\end{equation}

Where $\beta > 0$ , p and q are positive odd numbers, and they satisfy $p > {{q}}$ , $1 < p/q < 2$ .

The NTSMGL is chosen as Equation (75).

(75) \begin{equation}{u_{NTSMGL}} = - {{{B}}^{ - 1}}\left({{f}} + \frac{{\beta {{q}}}}{p}{\dot x^{2 - p/q}} + ks + H{\mathop{\rm sgn}} (s)\right)\end{equation}

NTSMGL can make the guidance system state converge in finite time and is robust to system disturbance. It can intercept manoeuvering targets.

After a lot of simulations, the control effect of ${u_{NTSMGL}}$ is optimised by adjusting the parameters. Finally Select the guidance parameter of ${u_{NTSMGL}}$ as $\beta {{ \;=\; 10}}$ , $p{{ \;=\; }}9$ , ${{q \;=\; 7}}$ , ${{k}} \;=\; 5.5$ , $H{\rm{ \;=\; }}\left[ {\begin{array}{*{20}{c}} {0.01}&0\\[5pt] 0&{0.01}\end{array}} \right]$ .

For Case 1, when ${a_{T\varepsilon }}{\rm{ \;=\; }}200\sin (0.5t){\rm{m}}/{{\rm s}^2}$ , ${a_{T\beta }}{\rm{ \;=\; }}200\cos (0.5t){\rm{m}}/{{\rm s}^2}$ . Figure 4 and Table 5 reveal the simulation results of sinusoidal wave manoeuvering target interception under three different guidance laws, which are the proposed guidance law in this paper, the PNGL and the NTSMGL. The relative motion curves of the missile and the target under three different guidance laws are shown in Fig. 4(a). All three different guidance laws allow the missile to successfully intercept sinusoidal manoeuverable targets, even if the missiles’ flight paths are different. Figure 4(b)–(e) shows the curves of ${q_\varepsilon }$ , the curves of ${q_\varepsilon }$ partial enlarged detail, the curves of ${q_\beta }$ and the curves of ${q_\beta }$ partial enlarged detail, respectively. It can be clearly seen from Fig. 4(b)–(e) that both the proposed guidance law in this paper and NTSMGL guarantee the convergence of ${q_\varepsilon }$ and ${q_\beta }$ . However, PNGL cannot ensure the convergence of ${q_\varepsilon }$ and ${q_\beta }$ . In addition, ${q_\varepsilon }$ and ${q_\beta }$ under the proposed guidance law in this paper obviously has faster convergence speed and higher accuracy than those under NTSMGL. Figure 4(f)–(i) shows the curves of ${\dot q_\varepsilon }$ , the curves of ${\dot q_\varepsilon }$ partial enlarged detail, the curves of ${\dot q_\beta }$ and the curves of ${\dot q_\beta }$ partial enlarged detail, respectively. It can be clearly seen from Fig. 4(f)–(i) that the guidance law proposed in this paper makes ${\dot q_\varepsilon }$ and ${\dot q_\beta }$ converge to near zero with high accuracy. Although NTSMGL also makes ${\dot q_\varepsilon }$ and ${\dot q_\beta }$ converge to near zero, divergence phenomenon appears at the moment when the missile and the target are about to encounter. And PNGL does not makes ${\dot q_\varepsilon }$ and ${\dot q_\beta }$ converge to near zero. Figure 4(j)–(k) shows the curves of ${{{u}}_\varepsilon }$ and ${{{u}}_\beta }$ . The elevation and azimuth acceleration commands of the missile ${{{u}}_\varepsilon }$ and ${{{u}}_\beta }$ are in a reasonable range under three different guidance laws. From Table 5 it can be known that under the guidance law proposed in this paper, the missile’s miss distance to the target is much smaller than the other two guidance laws. At the same time, the line-of-sight dip angle error and line-of-sight drift angle error are far less than the other two guidance laws. Compared with NTSMGL, the flight time of the missile under action of the guidance law proposed in this paper is shorter, which makes the interception probability of the missile with shorter flight time increase in the terminal guidance phase. Although the flight time of the missile under action of PNGL is the shortest, it cannot control the line-of-sight angle, and the miss distance is large. In conclusion, by comparing the simulation results of the three guidance laws, the guidance law proposed in this paper has better control performance.

Table 5. Simulation results of three different guidance laws with Case 1

Figure 5. Simulation results of three different guidance laws with Case 2.

For Case 2, when ${a_{T\varepsilon }}{\rm{ \;=\; }} -\!200{\rm{m}}/{{\rm s}^2}$ , ${a_{T\beta }}{\rm{ \;=\; }}200{\rm{m}}/{{\rm s}^2}$ . Figure 5 and Table 6 reveal the simulation results of constant manoeuvering target interception under three different guidance laws, which are the proposed guidance law in this paper, the conventional PNGL and the NTSMGL. The relative motion curves of the missile and the target under three different guidance laws are shown in Fig. 5(a). PNGL failed to intercept the constant manoeuvering target, while the other two guidance laws succeeded. Figure 5(b)–(e) indicates the curves of ${q_\varepsilon }$ , the curves of ${q_\varepsilon }$ partial enlarged detail, the curves of ${q_\beta }$ and the curves of ${q_\beta }$ partial enlarged detail, respectively. Figure 5(f)–(i) displays the curves of ${\dot q_\varepsilon }$ , the curves of ${\dot q_\varepsilon }$ partial enlarged detail, the curves of ${\dot q_\beta }$ and the curves of ${\dot q_\beta }$ partial enlarged detail, respectively. Figure 5(j)–(k) shows the curves of ${{{u}}_\varepsilon }$ and ${{{u}}_\beta }$ . Figure 5(b)–(k) shows the same situation as Fig. 4(b)–(k), so it will not be repeated here. It can be seen from Table 6 that, under the guidance law proposed in this paper, the miss distance of the missile to the target is much less than NTSMGL, while PNGL cannot intercept the constant manoeuver target with the acceleration of 20g. At the same time, the line-of-sight dip angle error and line-of-sight drift angle error under action of the guidance law proposed in this paper are far less than NTSMGL, while PNGL has no ability to control the line-of-sight angle. In conclusion, by comparing the simulation results of the three guidance laws, the guidance law proposed in this paper has better control performance.

Table 6. Simulation results of three different guidance laws with Case 2