2. Pursuit-evasion game

#### 2.1. Relative motion kinematics

The interceptor's movement consists of two orthogonal channels. The pursuit-evasion game is decomposed into two 2D channels.

Figure 1 shows the geometric diagram of the interceptor-target relative motion kinematics. A Cartesian reference system is denoted by "X-O-Y." The interceptor and target are denoted by "M" and "T". The line-of-sight angle is denoted by "q". The relative distance is denoted by "r". Flight path angles of the interceptor and target are denoted by "φm" and "φt". Velocities of the interceptor and target are denoted by "Vm" and "Vt".

Endgame relative motion kinematics are given by

$$\begin{cases} \dot{r} = V\_t \cos\left(\varphi\_t - q\right) - V\_m \cos\left(\varphi\_m - q\right), \\\ r\dot{q} = V\_t \sin\left(\varphi\_t - q\right) - V\_m \sin\left(\varphi\_m - q\right). \end{cases} \tag{1}$$

r<sup>0</sup> ¼ r<sup>0</sup> þ ~r0, ∣~r0∣ ≤ δ<sup>r</sup><sup>0</sup> , v<sup>0</sup> ¼ v<sup>0</sup> þ v~0, ∣v~0∣ ≤ δ<sup>v</sup><sup>0</sup> ,

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

where initial relative speed and distance are denoted by "v0" and "r0"; observations of v<sup>0</sup> and r<sup>0</sup> are denoted by "v0" and "r0"; observation deviations of v<sup>0</sup> and r<sup>0</sup> are denoted by "v~0" and

Remark 1. v<sup>0</sup> and r<sup>0</sup> are detected by a radar on ground or aircraft carrier and are sent to the interceptor via the data link only once. δ<sup>v</sup><sup>0</sup> and δ<sup>r</sup><sup>0</sup> are treated to be maximum observation

For successfully intercepting the target, the line-of-sight angular velocity should be constrained [5, 6]. In this chapter, seeker and autopilot loops are not considered. With this premise, relative equation of the line-of-sight angular velocity q\_ is obtained as Eq. (4). However, 1=r and 2v=r in Eq. (4) are obtained as Eq. (3), which indicates relative speed v alters as r, v, q\_, atr, and amr vary, and v<sup>0</sup> and r<sup>0</sup> are preset. In accordance with the characteristic of an interceptor's engine, the thrust along the line of sight almost does not change. Moreover, the target is usually escaping orthogonally to the line of sight to increase the line-ofsight angular velocity. Although acceleration component of the target along the line-of-sight is subsistent, the relative speed does not change too much with limited energy and time. Assume that acceleration components along the line of sight of the target and interceptor are

> r\_ ¼ v, <sup>v</sup>\_ <sup>¼</sup> rq\_<sup>2</sup>:

z\_<sup>1</sup> ¼ z2, <sup>z</sup>\_<sup>2</sup> <sup>¼</sup> <sup>z</sup>1q\_<sup>2</sup>:

According to Eq. (9), initial states for Eq. (11) to get r<sup>0</sup> and v<sup>0</sup> are obtained as

(9)

29

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(10)

(11)

"~r0"; upper boundaries of v~<sup>0</sup> and ~r<sup>0</sup> are denoted by "δ<sup>v</sup><sup>0</sup> " and "δ<sup>r</sup><sup>0</sup> ".

deviation of the detector.

zero. Simplify Eq. (3) into

Define z<sup>1</sup> ¼ r and z<sup>2</sup> ¼ v. Equation (10) becomes

2.2. Kinematics simplification

Figure 1. Relative motion kinematics.

Let the relative speed v ¼ r\_. Eq. (2) is obtained as

$$\begin{cases} \dot{v} = r\dot{q}^2 + \left[\dot{V}\_t \cos\left(\varphi\_t - q\right) - V\_t \dot{\varphi}\_t \sin\left(\varphi\_t - q\right)\right] \\ \quad - \left[\dot{V}\_m \cos\left(\varphi\_m - q\right) - V\_m \dot{\varphi}\_m \sin\left(\varphi\_m - q\right)\right], \\\ r\ddot{\eta} = -2\dot{r}\dot{\eta} + \left[\dot{V}\_t \sin\left(\varphi\_t - q\right) + V\_t \dot{\varphi}\_t \cos\left(\varphi\_t - q\right)\right] \\ \quad - \left[\dot{V}\_m \sin\left(\varphi\_m - q\right) + V\_m \dot{\varphi}\_m \cos\left(\varphi\_m - q\right)\right]. \end{cases} \tag{2}$$

For simplification, we get

$$
\dot{\upsilon} = r\dot{\eta}^2 + a\_{tr} - a\_{mr\prime} \tag{3}
$$

$$\ddot{q} = -2\frac{v}{r}\dot{q} + \frac{1}{r}a\_{t\eta} - \frac{1}{r}a\_{m\eta} \tag{4}$$

where acceleration components of the interceptor and target along the line-of-sight are denoted by "amr" and "atr"; acceleration components of the interceptor and target orthogonal to the line-of-sight are denoted by "amq"and "atq". The equations of them are formulated as

$$a\_{tr} = \dot{V}\_t \cos\left(\varphi\_t - q\right) - V\_t \dot{\varphi}\_t \sin\left(\varphi\_t - q\right),\tag{5}$$

$$a\_{mr} = \dot{V}\_m \cos\left(\varphi\_m - q\right) - V\_m \dot{\varphi}\_m \sin\left(\varphi\_m - q\right),\tag{6}$$

$$a\_{t\eta} = \dot{V}\_t \sin\left(\varphi\_t - \eta\right) + V\_t \dot{\varphi}\_t \cos\left(\varphi\_t - \eta\right),\tag{7}$$

$$a\_{m\eta} = \dot{V}\_m \sin\left(\varphi\_m - \eta\right) + V\_m \dot{\varphi}\_m \cos\left(\varphi\_m - \eta\right). \tag{8}$$

Assume the line-of-sight angular velocity is accurately observed at each instant. Initial relative speed and distance and error boundaries of them are obtained 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 29

Figure 1. Relative motion kinematics.

2. Pursuit-evasion game

28 Adaptive Robust Control Systems

2.1. Relative motion kinematics

decomposed into two 2D channels.

interceptor and target are denoted by "Vm" and "Vt".

(

Endgame relative motion kinematics are given by

Let the relative speed v ¼ r\_. Eq. (2) is obtained as

8 >>>>>><

>>>>>>:

For simplification, we get

The interceptor's movement consists of two orthogonal channels. The pursuit-evasion game is

Figure 1 shows the geometric diagram of the interceptor-target relative motion kinematics. A Cartesian reference system is denoted by "X-O-Y." The interceptor and target are denoted by "M" and "T". The line-of-sight angle is denoted by "q". The relative distance is denoted by "r". Flight path angles of the interceptor and target are denoted by "φm" and "φt". Velocities of the

> <sup>r</sup>\_ <sup>¼</sup> Vt cos <sup>φ</sup><sup>t</sup> � <sup>q</sup> � � � Vm cos <sup>φ</sup><sup>m</sup> � <sup>q</sup> � �, rq\_ <sup>¼</sup> Vt sin <sup>φ</sup><sup>t</sup> � <sup>q</sup> � � � Vm sin <sup>φ</sup><sup>m</sup> � <sup>q</sup> � �:

<sup>v</sup>\_ <sup>¼</sup> rq\_<sup>2</sup> <sup>þ</sup> <sup>V</sup>\_ <sup>t</sup> cos <sup>φ</sup><sup>t</sup> � <sup>q</sup> � � � Vtφ\_ <sup>t</sup> sin <sup>φ</sup><sup>t</sup> � <sup>q</sup> � � � � � <sup>V</sup>\_ <sup>m</sup> cos <sup>φ</sup><sup>m</sup> � <sup>q</sup> � � � Vmφ\_ <sup>m</sup> sin <sup>φ</sup><sup>m</sup> � <sup>q</sup> � � � � ,

rq€ ¼ �2r\_q\_ <sup>þ</sup> <sup>V</sup>\_ <sup>t</sup> sin <sup>φ</sup><sup>t</sup> � <sup>q</sup> � � <sup>þ</sup> Vtφ\_ <sup>t</sup> cos <sup>φ</sup><sup>t</sup> � <sup>q</sup> � � � � � <sup>V</sup>\_ <sup>m</sup> sin <sup>φ</sup><sup>m</sup> � <sup>q</sup> � � <sup>þ</sup> Vmφ\_ <sup>m</sup> cos <sup>φ</sup><sup>m</sup> � <sup>q</sup> � � � � :

<sup>2</sup> <sup>þ</sup> atr � amr, (3)

atr <sup>¼</sup> <sup>V</sup>\_ <sup>t</sup> cos <sup>φ</sup><sup>t</sup> � <sup>q</sup> � � � Vtφ\_ <sup>t</sup> sin <sup>φ</sup><sup>t</sup> � <sup>q</sup> � �, (5)

amr <sup>¼</sup> <sup>V</sup>\_ <sup>m</sup> cos <sup>φ</sup><sup>m</sup> � <sup>q</sup> � � � Vmφ\_ <sup>m</sup> sin <sup>φ</sup><sup>m</sup> � <sup>q</sup> � �, (6)

atq <sup>¼</sup> <sup>V</sup>\_ <sup>t</sup> sin <sup>φ</sup><sup>t</sup> � <sup>q</sup> � � <sup>þ</sup> Vtφ\_ <sup>t</sup> cos <sup>φ</sup><sup>t</sup> � <sup>q</sup> � �, (7)

amq <sup>¼</sup> <sup>V</sup>\_ <sup>m</sup> sin <sup>φ</sup><sup>m</sup> � <sup>q</sup> � � <sup>þ</sup> Vmφ\_ <sup>m</sup> cos <sup>φ</sup><sup>m</sup> � <sup>q</sup> � �: (8)

amq, (4)

v\_ ¼ rq\_

v r q\_ þ 1 r atq � <sup>1</sup> r

where acceleration components of the interceptor and target along the line-of-sight are denoted by "amr" and "atr"; acceleration components of the interceptor and target orthogonal to the line-of-sight are denoted by "amq"and "atq". The equations of them are formulated as

Assume the line-of-sight angular velocity is accurately observed at each instant. Initial relative

q€ ¼ �2

speed and distance and error boundaries of them are obtained as

(1)

(2)

$$\begin{cases} \begin{aligned} r\_0 &= \overline{r}\_0 + \check{r}\_{0\prime} & |\check{r}\_0| \le \delta\_{r\_{0\prime}} \\ \upsilon\_0 &= \overline{\upsilon}\_0 + \check{\upsilon}\_{0\prime} & |\check{\upsilon}\_0| \le \delta\_{\upsilon\_{0\prime}} \end{aligned} \end{cases} \tag{9}$$

where initial relative speed and distance are denoted by "v0" and "r0"; observations of v<sup>0</sup> and r<sup>0</sup> are denoted by "v0" and "r0"; observation deviations of v<sup>0</sup> and r<sup>0</sup> are denoted by "v~0" and "~r0"; upper boundaries of v~<sup>0</sup> and ~r<sup>0</sup> are denoted by "δ<sup>v</sup><sup>0</sup> " and "δ<sup>r</sup><sup>0</sup> ".

Remark 1. v<sup>0</sup> and r<sup>0</sup> are detected by a radar on ground or aircraft carrier and are sent to the interceptor via the data link only once. δ<sup>v</sup><sup>0</sup> and δ<sup>r</sup><sup>0</sup> are treated to be maximum observation deviation of the detector.

#### 2.2. Kinematics simplification

For successfully intercepting the target, the line-of-sight angular velocity should be constrained [5, 6]. In this chapter, seeker and autopilot loops are not considered. With this premise, relative equation of the line-of-sight angular velocity q\_ is obtained as Eq. (4). However, 1=r and 2v=r in Eq. (4) are obtained as Eq. (3), which indicates relative speed v alters as r, v, q\_, atr, and amr vary, and v<sup>0</sup> and r<sup>0</sup> are preset. In accordance with the characteristic of an interceptor's engine, the thrust along the line of sight almost does not change. Moreover, the target is usually escaping orthogonally to the line of sight to increase the line-ofsight angular velocity. Although acceleration component of the target along the line-of-sight is subsistent, the relative speed does not change too much with limited energy and time. Assume that acceleration components along the line of sight of the target and interceptor are zero. Simplify Eq. (3) into

$$\begin{cases} \dot{r} = v, \\ \dot{v} = r\dot{q}^2. \end{cases} \tag{10}$$

Define z<sup>1</sup> ¼ r and z<sup>2</sup> ¼ v. Equation (10) becomes

$$\begin{cases} \dot{z}\_1 = z\_2, \\ \dot{z}\_2 = z\_1 \dot{q}^2. \end{cases} \tag{11}$$

According to Eq. (9), initial states for Eq. (11) to get r<sup>0</sup> and v<sup>0</sup> are obtained as

$$z\_1^{(1)}(t\_0) = \overline{r}\_{0\prime} \ z\_2^{(1)}(t\_0) = \overline{v}\_{0\prime} \tag{12}$$

<sup>s</sup>\_ <sup>¼</sup> <sup>r</sup> r

<sup>V</sup>\_ <sup>1</sup> ¼ � <sup>r</sup>

Next, the following is obtained:

where Δ is a tiny positive constant.

dynamically estimate ε [23, 24].

Since N > 2, then

2 r

Then,

{ � N � 2 þ

N � 2 þ

z ð Þ2 1 z ð Þ3 1

!

z ð Þ2 1 z ð Þ3 1

!

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

� r r

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

j j v q\_ <sup>2</sup> � <sup>2</sup> r 2

> 0 < z ð Þ3

z ð Þ2

z ð Þ2 1 z ð Þ3 1

N � 2 þ

ð Þ2 1 z ð Þ3 1

!

stability theory, we can guarantee that V<sup>1</sup> ! 0. Finally s ! 0. Since s ¼ z

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

3.2. Improved guidance scheme based on the SMCG

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

� r r

z ð Þ2 1 z ð Þ3 1

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

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

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

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

" #

� r r

<sup>r</sup> ðj j <sup>Δ</sup><sup>v</sup> � <sup>~</sup>vÞq\_

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

� r r

j j <sup>v</sup> <sup>q</sup>\_ � <sup>2</sup> j j <sup>Δ</sup><sup>v</sup> � <sup>~</sup>vÞq\_ � <sup>ε</sup>sgnð Þ� <sup>q</sup>\_ atq � �}: � (21)

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31

<sup>1</sup> ≤ r: (23)

<sup>1</sup> ≥ r > 0: (24)

≥ 0: (25)

> 0: (26)

ð Þ1

<sup>q</sup>\_ <sup>þ</sup> <sup>ε</sup> <sup>q</sup>\_

<sup>1</sup> q\_, that is, q\_ ! 0.

<sup>∣</sup>q\_<sup>∣</sup> <sup>þ</sup> <sup>Δ</sup>, (27)

<sup>ε</sup>sgnð Þ� <sup>q</sup>\_ atq � �q\_: (22)

2 � r 2 r

$$z\_1^{(2)}(t\_0) = \overline{r}\_0 + \delta\_{r0\prime} \ z\_2^{(2)}(t\_0) = \overline{v}\_0 + \delta\_{r0\prime} \tag{13}$$

$$z\_1^{(3)}(t\_0) = \overline{r}\_0 - \delta\_{r0} \cdot z\_2^{(3)}(t\_0) = \overline{v}\_0 - \delta\_{v0} \cdot \tag{14}$$

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

$$\begin{cases} |\Delta r| = \max \left\{ z\_1^{(2)} - z\_1^{(1)}, z\_1^{(1)} - z\_1^{(3)} \right\}, \\ |\Delta v| = \max \left\{ z\_2^{(2)} - z\_2^{(1)}, z\_2^{(1)} - z\_2^{(3)} \right\}. \end{cases} \tag{15}$$
