3. Guidance scheme design

#### 3.1. Guidance scheme based on the sliding mode control

A sliding surface is determined by

$$s = z\_1^{(1)} \dot{q}.\tag{16}$$

In accordance with Eq. (16), forcing s to zero represents that q\_ or r prompts to 0. In terms of quasi-parallel approach guideline, the line-of-sight angular velocity will be adjusted to 0 to guarantee that the interceptor hits the target [5, 6].

Theorem 1. A sliding mode control-based guidance (SMCG) scheme described by

$$a\_{m\boldsymbol{q}} = \left[ \left( N - \frac{z\_1^{(2)}}{z\_1^{(3)}} \right) \middle| z\_2^{(1)} \Big| + 2|\Delta \boldsymbol{v}| \right] \dot{\boldsymbol{q}} + \varepsilon \text{sgn}(\dot{\boldsymbol{q}}),\tag{17}$$

where z ð Þ1 <sup>2</sup> , z ð Þ2 <sup>1</sup> , and z ð Þ3 <sup>1</sup> are deduced from Eq. (11) with Eqs. (12–14), N > 2 is an integer, j j Δv is obtained from Eq. (15), and ε is atq's upper boundary, guarantees that s ¼ z ð Þ1 <sup>1</sup> q\_ is driven to 0.

Proof. Compute Eq. (11) with Eq. (9) and define v ¼ z ð Þ1 <sup>2</sup> and r ¼ z ð Þ1 <sup>1</sup> .v and r are obtained as

$$
\overline{r} = \overline{r} + \widetilde{r}, \overline{v} = \overline{v} + \widetilde{v}, \tag{18}
$$

where derivations between estimations and real values are denoted by "~v" and "~r". In terms of the deduction in Section 2, we have

$$|\tilde{r}| \le |\Delta r|\_{\prime} \, |\tilde{v}| \le |\Delta v|. \tag{19}$$

Define a Lyapunov function:

$$V\_1 = 0.5 \text{s}^2.\tag{20}$$

Since v < 0, an approach scheme is defined as

Adaptive Robust Guidance Scheme Based on the Sliding Mode Control in an Aircraft Pursuit-Evasion Problem http://dx.doi.org/10.5772/intechopen.72177 31

$$\dot{s} = \frac{\overline{r}}{r} \{ - \left( N - 2 + \frac{z\_1^{(2)}}{z\_1^{(3)}} - \frac{r}{\overline{r}} \right) |\overline{v}| \dot{\eta} - 2 \{ |\Delta v| - \widetilde{v} \} \dot{\eta} - \left[ \varepsilon \text{sgn}(\dot{\eta}) - a\_{l\eta} \right] \}. \tag{21}$$

Then,

z ð Þ1

z ð Þ2

z ð Þ3

8 ><

>:

3.1. Guidance scheme based on the sliding mode control

guarantee that the interceptor hits the target [5, 6].

ð Þ3

Proof. Compute Eq. (11) with Eq. (9) and define v ¼ z

In terms of the deduction in Section 2, we have

Since v < 0, an approach scheme is defined as

Define a Lyapunov function:

amq <sup>¼</sup> <sup>N</sup> � <sup>z</sup>

3. Guidance scheme design

30 Adaptive Robust Control Systems

A sliding surface is determined by

where z

ð Þ1 <sup>2</sup> , z ð Þ2 <sup>1</sup> , and z <sup>1</sup> ð Þ¼ <sup>t</sup><sup>0</sup> <sup>r</sup>0, zð Þ<sup>1</sup>

Equations (12)–(14) are employed to calculate Eq. (11). Boundaries of v and rare computed as

n o

n o

ð Þ2 <sup>1</sup> � z ð Þ1 <sup>1</sup> ; z ð Þ1 <sup>1</sup> � z ð Þ3 1

ð Þ2 <sup>2</sup> � z ð Þ1 <sup>2</sup> ; z ð Þ1 <sup>2</sup> � z ð Þ3 2

s ¼ z ð Þ1

Theorem 1. A sliding mode control-based guidance (SMCG) scheme described by

ð Þ2 1 z ð Þ3 1

z ð Þ1 2 � � � � � � <sup>þ</sup> <sup>2</sup>j j <sup>Δ</sup><sup>v</sup>

<sup>1</sup> are deduced from Eq. (11) with Eqs. (12–14), N > 2 is an integer, j j Δv is

<sup>2</sup> and r ¼ z

ð Þ1

r ¼ r þ ~r, v ¼ v þ ~v, (18)

j~rj ≤ j j Δr , j~vj ≤ j j Δv : (19)

: (20)

ð Þ1

" #

!

obtained from Eq. (15), and ε is atq's upper boundary, guarantees that s ¼ z

where derivations between estimations and real values are denoted by "~v" and "~r".

V<sup>1</sup> ¼ 0:5s

2

In accordance with Eq. (16), forcing s to zero represents that q\_ or r prompts to 0. In terms of quasi-parallel approach guideline, the line-of-sight angular velocity will be adjusted to 0 to

<sup>1</sup> ð Þ¼ <sup>t</sup><sup>0</sup> <sup>r</sup><sup>0</sup> <sup>þ</sup> <sup>δ</sup>r0, zð Þ<sup>2</sup>

<sup>1</sup> ð Þ¼ <sup>t</sup><sup>0</sup> <sup>r</sup><sup>0</sup> � <sup>δ</sup>r0, zð Þ<sup>3</sup>

j j¼ Δr max z

j j¼ Δv max z

<sup>2</sup> ð Þ¼ t<sup>0</sup> v0, (12)

<sup>2</sup> ð Þ¼ t<sup>0</sup> v<sup>0</sup> þ δv0, (13)

<sup>2</sup> ð Þ¼ t<sup>0</sup> v<sup>0</sup> � δv0: (14)

<sup>1</sup> q\_: (16)

q\_ þ εsgnð Þ q\_ , (17)

ð Þ1

<sup>1</sup> .v and r are obtained as

<sup>1</sup> q\_ is driven to 0.

(15)

,

:

$$\dot{V}\_{1} = -\frac{\overline{r}^{2}}{r} \left( N - 2 + \frac{z\_{1}^{(2)}}{z\_{1}^{(3)}} - \frac{r}{\overline{r}} \right) |\overline{v}| \dot{q}^{2} - 2\frac{\overline{r}^{2}}{r} (|\Delta v| - \ddot{v}) \dot{q}^{2} - \frac{\overline{r}^{2}}{r} \left[ \varepsilon \text{sgn}(\dot{q}) - a\_{\text{lq}} \right] \dot{q}. \tag{22}$$

Equation (11) is solved with Eq. (12) or (14). Then, we get

$$0 < z\_1^{(3)} \le \overline{r}.\tag{23}$$

Equation (11) is solved with Eq. (9) or (13). Because ∣~r0∣ ≤ δr0, we have

$$z\_1^{(2)} \ge r > 0.\tag{24}$$

Next, the following is obtained:

$$\frac{z\_1^{(2)}}{z\_1^{(3)}} - \frac{r}{\overline{r}} \ge 0.\tag{25}$$

Since N > 2, then

$$N - 2 + \frac{z\_1^{(2)}}{z\_1^{(3)}} - \frac{r}{\overline{r}} > 0. \tag{26}$$

Because <sup>ε</sup>sgnð Þ� <sup>q</sup>\_ atq <sup>&</sup>gt; 0, j j <sup>Δ</sup><sup>v</sup> � <sup>~</sup><sup>v</sup> <sup>&</sup>gt; 0, <sup>r</sup> <sup>&</sup>gt; 0, and <sup>r</sup> <sup>&</sup>gt; 0, we get <sup>V</sup>\_ <sup>1</sup> <sup>&</sup>lt; 0. Using Lyapunov stability theory, we can guarantee that V<sup>1</sup> ! 0. Finally s ! 0. Since s ¼ z ð Þ1 <sup>1</sup> q\_, that is, q\_ ! 0.

Remark 2. The "sgn" function in Eq. (17) is replaced by the following function to suppress the jitter:

$$\mathfrak{a}\_{m\boldsymbol{\eta}} = \left[ \left( N - \frac{z\_1^{(2)}}{z\_1^{(3)}} \right) \middle| z\_2^{(1)} \middle| + 2|\Delta \boldsymbol{\nu}| \right] \dot{\boldsymbol{\eta}} + \varepsilon \frac{\dot{\boldsymbol{\eta}}}{|\dot{\boldsymbol{\eta}}| + \Delta} \,. \tag{27}$$

where Δ is a tiny positive constant.

#### 3.2. Improved guidance scheme based on the SMCG

ε in Eq. (27) or (17) is unchanged, which indicates that an unchanged upper boundary of atq is employed to ensure the sliding mode's subsistence. By this means, the guidance command's jitter might exist in the vicinity of the sliding mode although "sgn" is already replaced in Eq. (27). For smoothing the command, the better way is to use the adaptive approach to dynamically estimate ε [23, 24].

Proposition 1. An unchanged constant k > 0 exists, so that

$$|a\_{t\eta}| \le (N - 2 + k)|\overline{v}||\dot{q}|.\tag{28}$$

for verifying the performance of the interceptor with a rather constrained maneuverability. Assume that the target is less agile than the interceptor. Control systems of them are expressed

τms þ 1

Adaptive Robust Guidance Scheme Based on the Sliding Mode Control in an Aircraft Pursuit-Evasion Problem

τts þ 1

. In accordance with the Eq. (9), δv<sup>0</sup> ¼ 70 m/s and δr<sup>0</sup> ¼ 300 m are given

r<sup>0</sup> ¼ r<sup>0</sup> � δr0, v<sup>0</sup> ¼ v<sup>0</sup> � δv0: (34)

r<sup>0</sup> ¼ r<sup>0</sup> þ δr0, v<sup>0</sup> ¼ v<sup>0</sup> þ δv0: (35)

where the guidance commands are denoted by "atq" and "amq" and the responses are denoted

as upper boundaries of j~vj and j~rj. In Eqs. (27) and (29), N = 3, ε = 8 g, Δ = 0.0001, and γ = 125. Two

Following maneuver modes of the target, including case 1, case 2, and case 3, are used to test the performance of the proposed schemes. Assume that the interceptor is detected by the

> atyðÞ¼ t 0, t ≤ 2s atyð Þ¼� t � 2 atyð Þt , atyð Þ¼ 2 6g: t > 2s

atyðÞ¼ t 0, t ≤ 2s atyðÞ¼ t 8g � sin 3½ � ð Þ t � 2 : t > 2s

> atyðÞ¼ t 0, t ≤ 2s atyðÞ¼ t 8g: t > 2s

worst-case conditions of the initial observed relative speed v<sup>0</sup> and distance r<sup>0</sup> are given.

, (32)

http://dx.doi.org/10.5772/intechopen.72177

, (33)

, q\_

<sup>0</sup> ¼ �3deg=s,

33

(36)

(37)

(38)

amað Þs amqð Þ<sup>s</sup> <sup>¼</sup> <sup>1</sup>

atað Þs atqð Þ<sup>s</sup> <sup>¼</sup> <sup>1</sup>

Initial conditions are preset to <sup>r</sup><sup>0</sup> <sup>¼</sup> 3000 m, <sup>v</sup><sup>0</sup> <sup>¼</sup> <sup>r</sup>\_<sup>0</sup> ¼ �350 m/s, <sup>q</sup><sup>0</sup> <sup>¼</sup> <sup>10</sup>�

by the following first-order systems:

by "ata" and "ama". τ<sup>t</sup> ¼ 0:5 and τ<sup>m</sup> ¼ 0:2.

�

target in 2 s and then the target begins to escape.

Case 1: Square maneuver in the direction of the axis Y.

Case 2: Sine maneuver in the direction of the axis Y.

Case 3: Step maneuver in the direction of the axis Y.

Vt ¼ 500 m/s, and φ<sup>t</sup> ¼ 0

Condition 1 (C1):

Condition 2 (C2):

where upper boundaries' estimations of atq are formulated by ð Þ N � 2 þ k ∣v∣q\_.

Theorem 2. An improved sliding mode control-based guidance (ISMCG) scheme described by

$$\begin{cases} a\_{mq} = \left[ \left( N - \frac{z\_1^{(2)}}{z\_1^{(3)}} + \hat{k} \right) \Big| z\_2^{(1)} \Big| + 2|\Lambda \nu| \right] \dot{q}, \\\ \dot{\hat{k}} = \frac{1}{\mathcal{V}} \frac{\overline{r}^2}{r} |\overline{\nu}| \dot{q}^2, \end{cases} \tag{29}$$

where z ð Þ1 <sup>2</sup> , z ð Þ2 <sup>1</sup> , and z ð Þ3 <sup>1</sup> are deduced from Eq. (11) with Eqs. (12–14), N > 2 is an integer, j j Δv is obtained from Eq. (15), and γ > 0 is a constant, guarantees that s ¼ z ð Þ1 <sup>1</sup> q\_ is driven to 0.

Proof. Define <sup>~</sup><sup>k</sup> <sup>¼</sup> <sup>k</sup> � <sup>b</sup><sup>k</sup> and the Lyapunov function:

$$V\_2 = 0.5 \left(\mathbf{s}^2 + \gamma \tilde{\mathbf{k}}^2\right). \tag{30}$$

Then,

$$\dot{V}\_2 = -\frac{\overline{r}^2}{r} \left( \frac{z\_1^{(2)}}{z\_1^{(3)}} - \frac{r}{\overline{r}} \right) |\overline{v}| \dot{q}^2 - 2\frac{\overline{r}^2}{r} (|\Delta v| - \ddot{v}) \dot{q}^2 - \frac{\overline{r}^2}{r} \left[ \left( N - 2 + \hat{k} \right) |\overline{v}| \dot{q} - a\_{\text{tq}} \right] \dot{q} + \gamma \ddot{k} \dot{\hat{k}}.\tag{31}$$

According to Eq. (28) and \_ <sup>b</sup><sup>k</sup> <sup>¼</sup> <sup>1</sup> γ r 2 <sup>r</sup> j j <sup>v</sup> <sup>q</sup>\_2, we get

$$\begin{split} \dot{V}\_{2} &\leq -\frac{\overline{r}^{2}}{r} \left( \frac{z\_{1}^{(2)}}{z\_{1}^{(3)}} - \frac{r}{\overline{r}} \right) |\overline{v}| \dot{q}^{2} - 2\frac{\overline{r}^{2}}{r} (|\Delta v| - \widetilde{v}) \dot{q}^{2} - \frac{\overline{r}^{2}}{r} \left( \widehat{k} - k \right) |\overline{v}| \dot{q}^{2} + \gamma \dot{k} \dot{\overline{k}}, \\ &= -\frac{\overline{r}^{2}}{r} \left( \frac{z\_{1}^{(2)}}{z\_{1}^{(3)}} - \frac{r}{\overline{r}} \right) |\overline{v}| \dot{q}^{2} - 2\frac{\overline{r}^{2}}{r} \left( |\Delta v| - \widetilde{v} \right) \dot{q}^{2}. \end{split}$$

In accordance with condition (25), because j j <sup>Δ</sup><sup>v</sup> � <sup>~</sup><sup>v</sup> <sup>&</sup>gt; 0, <sup>r</sup> <sup>&</sup>gt; 0, and <sup>r</sup> <sup>&</sup>gt; 0, we get <sup>V</sup>\_ <sup>2</sup> <sup>&</sup>lt; 0. Using Lyapunov stability theory, we can guarantee that V<sup>2</sup> ! 0. Finally s ! 0. Since s ¼ z ð Þ1 <sup>1</sup> q\_, that is, q\_ ! 0.

Remark 3. r is employed to take the place of the real r in Eq. (29).
