2.2 Formulation of the FCG

The Grünwald-Letnikov (GL) fractional differential definition to formulate the FCG is presented as

$$\sideset{\_{a}}{^{G}}{\mathop{D}}\_{t}^{\mu}f(t) = \lim\_{h \to 0} \frac{1}{h^{\mu}} \sum\_{k=0}^{\frac{t-\mu}{h}} (-1)^{k} \frac{\Gamma(\mu+1)}{k!\Gamma(\mu-k+1)} f(t-kh),\tag{4}$$

path angles of the interceptor and target; η<sup>M</sup> and η<sup>T</sup> represent their heading angles; VM and VT represent their velocities; R represents the relative distance between

Differentiating Eq. (7), and substituting Eq. (8) and Eq. (9) into it, we have

θT <sup>þ</sup> \_

For a nonlinear problem Eq. (10), classic stability analysis theories such as the Routh-Hurwitz stability criterion for linear systems cannot be applied directly.

zero in the endgame [26]. Then, the nonlinear system Eq. (10) can be simplified

From Eq. (11), the transfer function of the guidance system is obtained as

Rs <sup>þ</sup> <sup>2</sup>R\_ <sup>¼</sup> �KG

<sup>θ</sup>Mð Þ<sup>s</sup> <sup>¼</sup> �VM cosð Þ <sup>q</sup> � <sup>θ</sup><sup>M</sup>

KG <sup>¼</sup> VM cosð Þ <sup>q</sup> � <sup>θ</sup><sup>M</sup> 2 R\_ 

q s \_ðÞ¼ �KG

θM, we have

<sup>μ</sup> <sup>þ</sup> TGs <sup>þ</sup>

\_ <sup>θ</sup><sup>M</sup> <sup>¼</sup> VR VM

KGKDs

TGs � 1

Substituting Eq. (14) into Eq. (13), the characteristic equation of the fractional

VR VM

KGKP � 1 

\_

Considering the practical situation, the values of V\_ <sup>M</sup>, V\_ <sup>T</sup> and \_

Rq€ <sup>þ</sup> <sup>2</sup>R\_ <sup>q</sup>\_ <sup>≈</sup> � \_

q s \_ð Þ \_

<sup>R</sup> VM sin <sup>η</sup><sup>M</sup> � VT sin <sup>η</sup><sup>T</sup> ð Þ, (7)

q ¼ θ<sup>M</sup> þ η<sup>M</sup> ¼ θ<sup>T</sup> þ ηT: (9)

<sup>θ</sup>TVT cos <sup>q</sup>\_ � \_

θT � \_

θMVM cosð Þ q � θ<sup>M</sup> : (11)

θMð Þs : (13)

KPq\_ <sup>þ</sup> KDqs\_ <sup>μ</sup> ð Þ: (14)

TG � 1

, TG <sup>¼</sup> <sup>R</sup> 2 R\_ : <sup>θ</sup>MVM cos <sup>q</sup>\_ � \_

θ<sup>T</sup> will approach

, (12)

¼ 0: (15)

θ<sup>M</sup> : (10)

VR <sup>¼</sup> <sup>R</sup>\_ ¼ �VM cos <sup>η</sup><sup>M</sup> <sup>þ</sup> VT cos <sup>η</sup>T, (8)

them; and q is the line-of-sight angle of the interceptor. The relative motion equations are given by

Robust Guidance Algorithm against Hypersonic Targets DOI: http://dx.doi.org/10.5772/intechopen.84655

<sup>q</sup>\_ <sup>¼</sup> <sup>1</sup>

� <sup>V</sup>\_ <sup>T</sup> sin <sup>q</sup>\_ � \_

θ<sup>M</sup>

Rq€ <sup>þ</sup> <sup>2</sup>R\_ <sup>q</sup>\_ <sup>¼</sup> <sup>V</sup>\_ <sup>M</sup> sin <sup>q</sup>\_ � \_

Linearization must be done first.

2.3.1 Linearization

into a linear system:

where

109

Thus, we get

From Eq. (6), since aM <sup>¼</sup> VM \_

calculus guidance system becomes

VR VM

which extends it from integer order to fractional order.

On dividing the continuous interval [a, t] of f(t) with step h = 1, and setting n ∈ {1, 2, …, t-a}, the difference equation of the fractional differential signal of f(t) is given by

$$\frac{d^\mu f(t)}{dt^\mu} \approx f(t) + (-\mu)f(t-1) + \frac{(-\mu)(-\mu+1)}{2}f(t-2) + \dots + \frac{\Gamma(-\mu+1)}{n!\Gamma(-\mu+n+1)}f(t-n). \tag{5}$$

According to definitions of the NPDG and GL, the FCG is proposed as

$$\mathfrak{a}\_{M}(t) = K\_{P}V\_{R}(t)\mathfrak{x}(t) + K\_{D}V\_{R}(t)\frac{d^{\mu}\mathfrak{x}(t)}{dt^{\mu}},\tag{6}$$

where <sup>μ</sup> is the fractional order, <sup>d</sup><sup>μ</sup> x tð Þ dt<sup>μ</sup> is the fractional differential of x(t), and x tðÞ¼ q t \_ð Þ.

In the FCG, the future state of the GL fractional differential of q\_ depends on the previous and current states. But in the NPDG, the future state q€ only depends on the current state. It indicates that the fractional order part is a filter with the "memory" characteristic. The FCG runs like a filter, which is insensitive to the noises, and shows robustness to disturbances.

## 2.3 Stability criteria

As shown in Figure 1, the target and interceptor are located in the same plane, XOY, where M and T denote the interceptor and target; θ<sup>M</sup> and θ<sup>T</sup> represent flight

Figure 1. Planar endgame engagement geometry.

Robust Guidance Algorithm against Hypersonic Targets DOI: http://dx.doi.org/10.5772/intechopen.84655

path angles of the interceptor and target; η<sup>M</sup> and η<sup>T</sup> represent their heading angles; VM and VT represent their velocities; R represents the relative distance between them; and q is the line-of-sight angle of the interceptor.

The relative motion equations are given by

$$\dot{q} = \frac{1}{R}(V\_M \sin \eta\_M - V\_T \sin \eta\_T),\tag{7}$$

$$V\_R = \dot{R} = -V\_M \cos \eta\_M + V\_T \cos \eta\_T \tag{8}$$

$$q = \theta\_M + \eta\_M = \theta\_T + \eta\_T. \tag{9}$$

Differentiating Eq. (7), and substituting Eq. (8) and Eq. (9) into it, we have

$$R\ddot{q} + 2\dot{R}\dot{q} = \dot{V}\_M \sin\left(\dot{q} - \dot{\theta}\_M\right) - \dot{V}\_T \sin\left(\dot{q} - \dot{\theta}\_T\right) + \dot{\theta}\_T V\_T \cos\left(\dot{q} - \dot{\theta}\_T\right) - \dot{\theta}\_M V\_M \cos\left(\dot{q} - \dot{\theta}\_M\right). \tag{10}$$

#### 2.3.1 Linearization

where K is the estimation coefficient, q\_ <sup>m</sup>ð Þt is the LOS rate measured by the

^€mð Þ<sup>t</sup> . It is not easy to determine the value of <sup>K</sup>. If <sup>K</sup> is larger, the estimation will be more precise and the phase lag will be less, but the estimation will be noisier.

The Grünwald-Letnikov (GL) fractional differential definition to formulate the

On dividing the continuous interval [a, t] of f(t) with step h = 1, and setting n ∈ {1, 2, …, t-a}, the difference equation of the fractional differential signal of f(t)

According to definitions of the NPDG and GL, the FCG is proposed as

aMðÞ¼ t KPVRð Þt x tðÞþ KDVRð Þt

x tð Þ

In the FCG, the future state of the GL fractional differential of q\_ depends on the previous and current states. But in the NPDG, the future state q€ only depends on the current state. It indicates that the fractional order part is a filter with the "memory" characteristic. The FCG runs like a filter, which is insensitive to the

As shown in Figure 1, the target and interceptor are located in the same plane, XOY, where M and T denote the interceptor and target; θ<sup>M</sup> and θ<sup>T</sup> represent flight

ð Þ �<sup>1</sup> <sup>k</sup> <sup>Γ</sup>ð Þ <sup>μ</sup> <sup>þ</sup> <sup>1</sup>

<sup>2</sup> f tð Þþ � <sup>2</sup> … <sup>þ</sup>

dμ x tð Þ

dt<sup>μ</sup> is the fractional differential of x(t), and

^\_mð Þ<sup>t</sup> and

<sup>k</sup>!Γð Þ <sup>μ</sup> � <sup>k</sup> <sup>þ</sup> <sup>1</sup> f tð Þ � kh , (4)

Γð Þ �μ þ 1

dt<sup>μ</sup> , (6)

<sup>n</sup>!Γð Þ �<sup>μ</sup> <sup>þ</sup> <sup>n</sup> <sup>þ</sup> <sup>1</sup> f tð Þ � <sup>n</sup> :

(5)

seeker, q\_mð Þt and q€mð Þt are estimated by x<sup>1</sup> and x2, namely x<sup>1</sup> ¼ q

1 <sup>h</sup><sup>μ</sup> <sup>∑</sup> t�a h k¼0

which extends it from integer order to fractional order.

Therefore, a fractional calculus guidance law is presented.

x<sup>2</sup> ¼ q

2.2 Formulation of the FCG

G <sup>a</sup> <sup>D</sup><sup>μ</sup>

<sup>t</sup> f tðÞ¼ lim h!0

dt<sup>μ</sup> <sup>≈</sup>f tðÞþ �ð Þ <sup>μ</sup> f tð Þþ � <sup>1</sup> ð Þ� �<sup>μ</sup> ð Þ <sup>μ</sup> <sup>þ</sup> <sup>1</sup>

where <sup>μ</sup> is the fractional order, <sup>d</sup><sup>μ</sup>

noises, and shows robustness to disturbances.

FCG is presented as

Military Engineering

is given by

x tðÞ¼ q t \_ð Þ.

Figure 1.

108

Planar endgame engagement geometry.

2.3 Stability criteria

dμ f tð Þ

For a nonlinear problem Eq. (10), classic stability analysis theories such as the Routh-Hurwitz stability criterion for linear systems cannot be applied directly. Linearization must be done first.

Considering the practical situation, the values of V\_ <sup>M</sup>, V\_ <sup>T</sup> and \_ θ<sup>T</sup> will approach zero in the endgame [26]. Then, the nonlinear system Eq. (10) can be simplified into a linear system:

$$R\ddot{q} + 2\dot{R}\dot{q} \approx -\dot{\theta}\_M V\_M \cos\left(q - \theta\_M\right). \tag{11}$$

From Eq. (11), the transfer function of the guidance system is obtained as

$$\frac{\dot{\bar{q}}(s)}{\dot{\theta}\_M(s)} = \frac{-V\_M \cos\left(q - \theta\_M\right)}{Rs + 2\dot{R}} = \frac{-K\_G}{T\_G - 1},\tag{12}$$

where

$$K\_G = \frac{V\_M \cos\left(q - \theta\_M\right)}{2|\dot{R}|}, \\ T\_G = \frac{R}{2|\dot{R}|}.$$

Thus, we get

$$
\dot{q}(s) = \frac{-K\_G}{T\_G s - 1} \dot{\theta}\_M(s). \tag{13}
$$

From Eq. (6), since aM <sup>¼</sup> VM \_ θM, we have

$$
\dot{\theta}\_M = \frac{V\_R}{V\_M} (K\_P \dot{q} + K\_D \dot{q} s^\mu). \tag{14}
$$

Substituting Eq. (14) into Eq. (13), the characteristic equation of the fractional calculus guidance system becomes

$$\left(\frac{V\_R}{V\_M} K\_G K\_D s^\mu + T\_G s + \left(\frac{V\_R}{V\_M} K\_G K\_P - 1\right) = 0.\tag{15}$$

#### 2.3.2 Stability analysis

In stability analysis of Eq. (15), the Hurwitz stability criterion is appropriate to be employed.

Lemma 1: Hurwitz stability criterion [27]

For an nth-degree polynomial characteristic equation:

$$D(\mathfrak{s}) = a\_0 \mathfrak{s}^n + a\_1 \mathfrak{s}^{n-1} + \dots + a\_{n-1} \mathfrak{s} + a\_n = \mathbf{0} \ (a\_0 > \mathbf{0}) \tag{16}$$

the necessary and sufficient stability condition, of system (16), is

$$\Delta\_1 = a\_1 > 0, \ \Delta\_2 = \begin{vmatrix} a\_1 & a\_3 \\ a\_0 & a\_2 \end{vmatrix} > 0, \ \Delta\_3 = \begin{vmatrix} a\_1 & a\_3 & a\_5 \\ a\_0 & a\_2 & a\_4 \\ 0 & a\_1 & a\_3 \end{vmatrix} > 0, \ \cdots, \ \Delta\_n > 0. \tag{17}$$

That is, the order of principal minor determinants and the main determinant of the system (16) is positive.

Thus, based on the Hurwitz stability criterion, the necessary and sufficient stability condition of system (15) becomes

$$a\_0 = \frac{V\_R}{V\_M} K\_G K\_D > 0,\tag{18}$$

is usually designed in the gliding or cruising phase in the near space of a hypersonic weapon before its terminal phase (i.e., before a dive attack happens); then, the interceptor-target initial position and encounter condition is designed to be a headto-head encounter. In the gliding or cruising phase in the near space of a hypersonic weapon, its velocity is relatively low (about 5 Mach), and its maneuvering amplitude cannot exceed 5 g due to the reduced aerodynamic efficiency since the atmosphere is thin in the near space, but the time instant that the hypersonic weapon starts maneuvering is flexible and adjustable for evading the interceptor's pursuit. Our preliminary studies and experiments show that it is not good for the hypersonic weapon to start maneuvering as early as possible during a pursuit-evasion game, and it is better for the hypersonic weapon to start maneuvering when the interceptor is close to it in the endgame. For the maneuvering mode of the hypersonic weapon to evade the interceptor's pursuit, the step maneuver and square maneuver are preferred to the ramp maneuver and sine maneuver since they can provide the

Based on the above analysis, the simulation parameters for a hypersonic pursuitevasion game are set as: the interceptor-target initial position and heading condition

R = 30,000 m, VT = 5 Mach, which is along the negative X-axis; VM = 5 Mach, and its initial direction is aimed at the target, that is θ<sup>M</sup> = q; the initial LOS angle q is 10°; the interceptor's maximum normal acceleration is 15 g; μ is set to the best value of 0.5 based on experience. Obviously, η<sup>M</sup> = q � θ<sup>M</sup> = 0° ∈ [�60°, 60°]. The fractional

According to authentic maneuvering characteristics of a hypersonic weapon in the gliding or cruising phase in the near space when the interceptor is close to it, its

> aT <sup>¼</sup> 5g, t <sup>∈</sup>½ Þ <sup>2</sup><sup>k</sup> <sup>þ</sup> <sup>6</sup>; <sup>2</sup><sup>k</sup> <sup>þ</sup> <sup>7</sup> <sup>s</sup>, �5g, t ∈½ Þ 2k þ 7; 2k þ 8 s,

where aT is the norm acceleration of the target, t is the time index and k ∈ N.

aT ¼ 5g, t ≥8s, (22)

(23)

hypersonic weapon the maximum evading acceleration instantly.

calculus guidance system is stable based on Theorem 1.

Robust Guidance Algorithm against Hypersonic Targets DOI: http://dx.doi.org/10.5772/intechopen.84655

The target maneuvers are shown in Figures 2 and 3.

maneuver equations are given by. Case 1: Step maneuver

Case 2: Square maneuver

Figure 2.

111

Step maneuver of the target (case 1).

is planned in a head-to-head engagement, and the initial relative distance

$$
\Delta\_1 = \mathcal{a}\_1 = T\_G > \mathbf{0},\tag{19}
$$

$$
\Delta\_2 = \begin{vmatrix} a\_1 & a\_3 \\ a\_0 & a\_2 \end{vmatrix} = \begin{vmatrix} T\_G & 0 \\ \frac{V\_R}{V\_M} K\_G K\_D & \frac{V\_R}{V\_M} K\_G K\_P - 1 \end{vmatrix} = T\_G \times \left( \frac{V\_R}{V\_M} K\_G K\_P - 1 \right) > 0. \tag{20}
$$

That is

$$\begin{cases} \frac{V\_R}{V\_M} K\_G K\_D > 0, \\ \\ T\_G > 0, \\ \frac{V\_R}{V\_M} K\_G K\_P - 1 > 0. \end{cases} \tag{21}$$

Since KP > 0 and K<sup>D</sup> > 0, KP can be preset as 4. As a consequence, we have cos(q-θM) > 0.5, that is cosη<sup>M</sup> > 0.5. It concludes

Theorem 1: When the interceptor's heading angle η<sup>M</sup> is in the range of �60° to +60°, the fractional calculus guidance system remains stable.

### 3. Numerical simulations

#### 3.1 Simulations design

For intercepting a hypersonic weapon, a space-based surveillance satellite and a ground-based X band radar or a marine X band radar should detect the target as early as possible to provide the interceptor enough time to launch from the ground or the aerial carrier. In the terminal phase of a hypersonic weapon, its velocity is too high to be intercepted. For example, the speed of a gliding entry vehicle is up to 25 Mach at maximum during a dive attack to the ground target. Thus, the interception

#### Robust Guidance Algorithm against Hypersonic Targets DOI: http://dx.doi.org/10.5772/intechopen.84655

2.3.2 Stability analysis

Military Engineering

Lemma 1: Hurwitz stability criterion [27]

D sðÞ¼ a0s

� � � � �

stability condition of system (15) becomes

� � � � ¼

VR VM

cos(q-θM) > 0.5, that is cosη<sup>M</sup> > 0.5. It concludes

3. Numerical simulations

3.1 Simulations design

110

� � � � � �

KGKD

<sup>Δ</sup><sup>1</sup> <sup>¼</sup> <sup>a</sup><sup>1</sup> <sup>&</sup>gt;0, <sup>Δ</sup><sup>2</sup> <sup>¼</sup> <sup>a</sup><sup>1</sup> <sup>a</sup><sup>3</sup>

the system (16) is positive.

<sup>Δ</sup><sup>2</sup> <sup>¼</sup> <sup>a</sup><sup>1</sup> <sup>a</sup><sup>3</sup> a<sup>0</sup> a<sup>2</sup>

� � � �

That is

For an nth-degree polynomial characteristic equation:

<sup>n</sup> <sup>þ</sup> <sup>a</sup>1<sup>s</sup>

� � � � �

a<sup>0</sup> a<sup>2</sup>

the necessary and sufficient stability condition, of system (16), is

>0, Δ<sup>3</sup> ¼

<sup>a</sup><sup>0</sup> <sup>¼</sup> VR VM

TG 0

VR VM

VR VM

8 >>>>><

>>>>>:

+60°, the fractional calculus guidance system remains stable.

TG >0, VR VM

be employed.

In stability analysis of Eq. (15), the Hurwitz stability criterion is appropriate to

� � � � � � � � �

That is, the order of principal minor determinants and the main determinant of

KGKP � 1

KGKD > 0,

KGKP � 1>0:

Since KP > 0 and K<sup>D</sup> > 0, KP can be preset as 4. As a consequence, we have

Theorem 1: When the interceptor's heading angle η<sup>M</sup> is in the range of �60° to

For intercepting a hypersonic weapon, a space-based surveillance satellite and a ground-based X band radar or a marine X band radar should detect the target as early as possible to provide the interceptor enough time to launch from the ground or the aerial carrier. In the terminal phase of a hypersonic weapon, its velocity is too high to be intercepted. For example, the speed of a gliding entry vehicle is up to 25 Mach at maximum during a dive attack to the ground target. Thus, the interception

� � � � � �

Thus, based on the Hurwitz stability criterion, the necessary and sufficient

<sup>n</sup>�<sup>1</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> an�1<sup>s</sup> <sup>þ</sup> an <sup>¼</sup> <sup>0</sup> ð Þ <sup>a</sup><sup>0</sup> <sup>&</sup>gt; <sup>0</sup> (16)

� � � � � � � � �

KGKD > 0, (18)

VR VM

KGKP � 1 � �

>0:

(20)

(21)

Δ<sup>1</sup> ¼ a<sup>1</sup> ¼ TG >0, (19)

¼ TG �

>0, ⋯ , Δ<sup>n</sup> > 0: (17)

a<sup>1</sup> a<sup>3</sup> a<sup>5</sup>

a<sup>0</sup> a<sup>2</sup> a<sup>4</sup>

0 a<sup>1</sup> a<sup>3</sup>

is usually designed in the gliding or cruising phase in the near space of a hypersonic weapon before its terminal phase (i.e., before a dive attack happens); then, the interceptor-target initial position and encounter condition is designed to be a headto-head encounter. In the gliding or cruising phase in the near space of a hypersonic weapon, its velocity is relatively low (about 5 Mach), and its maneuvering amplitude cannot exceed 5 g due to the reduced aerodynamic efficiency since the atmosphere is thin in the near space, but the time instant that the hypersonic weapon starts maneuvering is flexible and adjustable for evading the interceptor's pursuit. Our preliminary studies and experiments show that it is not good for the hypersonic weapon to start maneuvering as early as possible during a pursuit-evasion game, and it is better for the hypersonic weapon to start maneuvering when the interceptor is close to it in the endgame. For the maneuvering mode of the hypersonic weapon to evade the interceptor's pursuit, the step maneuver and square maneuver are preferred to the ramp maneuver and sine maneuver since they can provide the hypersonic weapon the maximum evading acceleration instantly.

Based on the above analysis, the simulation parameters for a hypersonic pursuitevasion game are set as: the interceptor-target initial position and heading condition is planned in a head-to-head engagement, and the initial relative distance R = 30,000 m, VT = 5 Mach, which is along the negative X-axis; VM = 5 Mach, and its initial direction is aimed at the target, that is θ<sup>M</sup> = q; the initial LOS angle q is 10°; the interceptor's maximum normal acceleration is 15 g; μ is set to the best value of 0.5 based on experience. Obviously, η<sup>M</sup> = q � θ<sup>M</sup> = 0° ∈ [�60°, 60°]. The fractional calculus guidance system is stable based on Theorem 1.

According to authentic maneuvering characteristics of a hypersonic weapon in the gliding or cruising phase in the near space when the interceptor is close to it, its maneuver equations are given by.

Case 1: Step maneuver

$$\mathfrak{a}\_T = \mathfrak{F}\mathfrak{g}, \quad t \ge \mathbf{8}\mathfrak{s},\tag{22}$$

Case 2: Square maneuver

$$a\_T = \begin{cases} \text{5g}, & t \in [2k+6, 2k+7)\text{s}, \\ -\text{5g}, & t \in [2k+7, 2k+8)\text{s}, \end{cases} \tag{23}$$

where aT is the norm acceleration of the target, t is the time index and k ∈ N. The target maneuvers are shown in Figures 2 and 3.

Figure 2. Step maneuver of the target (case 1).

Figure 3. Square maneuver of the target (case 2).

### 3.2 Interception accuracy

The trajectories, line-of-sight rates and guidance commands of the interceptor and target are shown in Figures 4–9. From Figures 4 and 5, since the velocities of the interceptor and target are hypersonic (5 Mach), the amplitude of the target maneuvers is 5 g which cannot change the velocities and trajectories of the target a lot in a limited endgame time. Thus, there is no big difference between the trajectories of the target between Figures 4 and 5. From Figures 6 and 7, the line-of-sight rates constrained by the FCG are much smaller than those constrained by the NPDG. And the line-of-sight rates of the NPDG are always non-convergent. From Figures 8 and 9, the guidance commands of the FCG are much smoother than those of the NPDG, which are more appropriate for the interceptor's autopilot to track. The reason is that the NPDG uses a nonlinear tracking differentiator Eq. (3) to estimate q€. In Eq. (3), K is the coefficient of the estimator. The larger the K is, the more precise the estimation is and the less the phase lag is, but the noisier the

estimation is. Comparing Figure 9 with Figure 8, the guidance command of the NPDG in case 2 is noisier than that in case 1, which means the target maneuver of case 2 is more challenging to the NPDG than that of case 1. It is also validated by the results in Table 1 that the miss distance of the NPDG in case 2 is larger than that of the NPDG in case 1. However, the target maneuver of case 2 has little influence on the interception accuracy of the FCG, since the miss distance of the FCG in case 2 is even smaller than that of the FCG in case 1, which indicates the superiority of

Numerical results are demonstrated in Table 1. The FCG has the minimum miss

distance under different scenarios. In case 1, the miss distance of the FCG is 0.0322 m, which is 91% less than that of the NPDG (0.3406 m). In case 2, the miss

distance of the FCG is 0.0294 m, which is 93% less than that of the NPDG

the FCG.

Figure 6.

Line-of-sight rates (case 1).

Figure 5.

Trajectories of the interceptor and target (case 2).

Robust Guidance Algorithm against Hypersonic Targets DOI: http://dx.doi.org/10.5772/intechopen.84655

(0.4151 m).

113

Figure 4. Trajectories of the interceptor and target (case 1).

### Robust Guidance Algorithm against Hypersonic Targets DOI: http://dx.doi.org/10.5772/intechopen.84655

#### Figure 5. Trajectories of the interceptor and target (case 2).

3.2 Interception accuracy

Square maneuver of the target (case 2).

Figure 3.

Military Engineering

Figure 4.

112

Trajectories of the interceptor and target (case 1).

The trajectories, line-of-sight rates and guidance commands of the interceptor and target are shown in Figures 4–9. From Figures 4 and 5, since the velocities of the interceptor and target are hypersonic (5 Mach), the amplitude of the target maneuvers is 5 g which cannot change the velocities and trajectories of the target a lot in a limited endgame time. Thus, there is no big difference between the trajectories of the target between Figures 4 and 5. From Figures 6 and 7, the line-of-sight rates constrained by the FCG are much smaller than those constrained by the NPDG. And the line-of-sight rates of the NPDG are always non-convergent. From Figures 8 and 9, the guidance commands of the FCG are much smoother than those of the NPDG, which are more appropriate for the interceptor's autopilot to track. The reason is that the NPDG uses a nonlinear tracking differentiator Eq. (3) to estimate q€. In Eq. (3), K is the coefficient of the estimator. The larger the K is, the more precise the estimation is and the less the phase lag is, but the noisier the

Figure 6. Line-of-sight rates (case 1).

estimation is. Comparing Figure 9 with Figure 8, the guidance command of the NPDG in case 2 is noisier than that in case 1, which means the target maneuver of case 2 is more challenging to the NPDG than that of case 1. It is also validated by the results in Table 1 that the miss distance of the NPDG in case 2 is larger than that of the NPDG in case 1. However, the target maneuver of case 2 has little influence on the interception accuracy of the FCG, since the miss distance of the FCG in case 2 is even smaller than that of the FCG in case 1, which indicates the superiority of the FCG.

Numerical results are demonstrated in Table 1. The FCG has the minimum miss distance under different scenarios. In case 1, the miss distance of the FCG is 0.0322 m, which is 91% less than that of the NPDG (0.3406 m). In case 2, the miss distance of the FCG is 0.0294 m, which is 93% less than that of the NPDG (0.4151 m).

Figure 7. Line-of-sight rates (case 2).

Figure 8. Guidance commands (case 1).

### 3.3 Stability

In case 1, when pre-setting the simulation parameters, if the initial flight path angle θ<sup>M</sup> is set as 40°, 70° and 75°, and other parameters remain unchanged, obviously, the heading angle η<sup>M</sup> = qθM, will be 30°, 60° and 65°, respectively. The stabilities of the fractional calculus guidance system with the FCG can be analyzed based on Theorem 1.

3.4 Robustness

Figure 10.

115

Table 1.

Figure 9.

Guidance commands (case 2).

Performance evaluation of guidance laws.

Robust Guidance Algorithm against Hypersonic Targets DOI: http://dx.doi.org/10.5772/intechopen.84655

In case 1, three white noises are added into q\_ to run 50 groups of the Monte Carlo simulations, including the amplitudes of 0.5°/s, 1.5°/s and 2.5°/s. The total number of tests is 50. The miss distance distributions of the NPDG and the FCG with a noise

Guidance law Case 1: miss distance (m) Case 2: miss distance (m)

NPDG 0.3406 0.4151 FCG 0.0322 0.0294

From Figures 13, 15 and 17, it can be seen that the miss distances of the NPDG obviously increase as the noise increases. Similarly, from Figures 14, 16 and 18, the

of 0.5°/s, 1.5°/s and 2.5°/s are shown in Figures 13–18.

Trajectories of the interceptor and target (No. 1 η<sup>M</sup> = 30°).

As shown in Figures 10–12, when the heading angle ηM belongs to the closed interval [60°, 60°], the interceptor can hit and kill the target; when the heading angle ηM is beyond the closed interval [60°, 60°], the interception mission fails.

Simulation results are compared and summarized in Table 2. The miss distances increase as the heading angle goes beyond the closed interval [60°, 60°]; when the heading angle η<sup>M</sup> is 60°, it is a critical condition. The experimental results in Table 2 validate the conclusion of Theorem 1.

Robust Guidance Algorithm against Hypersonic Targets DOI: http://dx.doi.org/10.5772/intechopen.84655

#### Figure 9.

Guidance commands (case 2).


#### Table 1.

Performance evaluation of guidance laws.

Figure 10.

3.3 Stability

Guidance commands (case 1).

Figure 8.

114

Figure 7.

Line-of-sight rates (case 2).

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analyzed based on Theorem 1.

Table 2 validate the conclusion of Theorem 1.

In case 1, when pre-setting the simulation parameters, if the initial flight path

obviously, the heading angle η<sup>M</sup> = qθM, will be 30°, 60° and 65°, respectively. The stabilities of the fractional calculus guidance system with the FCG can be

As shown in Figures 10–12, when the heading angle ηM belongs to the closed interval [60°, 60°], the interceptor can hit and kill the target; when the heading angle ηM is beyond the closed interval [60°, 60°], the interception mission fails. Simulation results are compared and summarized in Table 2. The miss distances increase as the heading angle goes beyond the closed interval [60°, 60°]; when the heading angle η<sup>M</sup> is 60°, it is a critical condition. The experimental results in

angle θ<sup>M</sup> is set as 40°, 70° and 75°, and other parameters remain unchanged,

Trajectories of the interceptor and target (No. 1 η<sup>M</sup> = 30°).

#### 3.4 Robustness

In case 1, three white noises are added into q\_ to run 50 groups of the Monte Carlo simulations, including the amplitudes of 0.5°/s, 1.5°/s and 2.5°/s. The total number of tests is 50. The miss distance distributions of the NPDG and the FCG with a noise of 0.5°/s, 1.5°/s and 2.5°/s are shown in Figures 13–18.

From Figures 13, 15 and 17, it can be seen that the miss distances of the NPDG obviously increase as the noise increases. Similarly, from Figures 14, 16 and 18, the

Figure 11. Trajectories of the interceptor and target (No. 2 η<sup>M</sup> = 60°).

Figure 12. Trajectories of the interceptor and target (No. 3 η<sup>M</sup> = 65°).


FCG are always smaller than those of the NPDG, which indicates the stronger

the FCG has a better robustness to the guidance noises.

Miss distance distribution of the FCG with a noise of 0.5°/s.

Miss distance distribution of the NPDG with a noise of 0.5°/s.

Robust Guidance Algorithm against Hypersonic Targets DOI: http://dx.doi.org/10.5772/intechopen.84655

Statistical results are indicated in Table 3. Obviously, compared with the NPDG,

To summarize the interception accuracy and robustness experiments, a conclusion can be drawn. The unique filtering properties of the fractional calculus guidance law make its interception accuracy and robustness better. For intercepting a hypersonic weapon, introducing the differential signal of the line-of-sight rate as the guidance information can effectively suppress the target maneuvers, and it has a good robustness, which can make it a feasible guidance strategy. The specifications

robustness of the FCG.

Figure 14.

Figure 13.

are as follows:

117

Table 2. Stability analysis.

miss distances of the FCG slightly increase as the noise increases. These phenomena indicate the effect of noise impacting on the miss distances of both the NPDG and the FCG. Moreover, comparing Figure 14 with Figure 13, comparing Figure 16 with Figure 15, and comparing Figure 18 with Figure 17, the miss distances of the

Robust Guidance Algorithm against Hypersonic Targets DOI: http://dx.doi.org/10.5772/intechopen.84655

Figure 13. Miss distance distribution of the NPDG with a noise of 0.5°/s.

Figure 14. Miss distance distribution of the FCG with a noise of 0.5°/s.

FCG are always smaller than those of the NPDG, which indicates the stronger robustness of the FCG.

Statistical results are indicated in Table 3. Obviously, compared with the NPDG, the FCG has a better robustness to the guidance noises.

To summarize the interception accuracy and robustness experiments, a conclusion can be drawn. The unique filtering properties of the fractional calculus guidance law make its interception accuracy and robustness better. For intercepting a hypersonic weapon, introducing the differential signal of the line-of-sight rate as the guidance information can effectively suppress the target maneuvers, and it has a good robustness, which can make it a feasible guidance strategy. The specifications are as follows:

miss distances of the FCG slightly increase as the noise increases. These phenomena indicate the effect of noise impacting on the miss distances of both the NPDG and the FCG. Moreover, comparing Figure 14 with Figure 13, comparing Figure 16 with Figure 15, and comparing Figure 18 with Figure 17, the miss distances of the

No. Heading angle η<sup>M</sup> (°) Stability Miss distance (m) 30 Stable 0.1060 60 Stable 8.9125 65 Unstable 820.7977

Figure 11.

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Figure 12.

Table 2. Stability analysis.

116

Trajectories of the interceptor and target (No. 2 η<sup>M</sup> = 60°).

Trajectories of the interceptor and target (No. 3 η<sup>M</sup> = 65°).

Figure 17.

Figure 18.

Table 3.

119

Miss distance distribution of the NPDG with a noise of 2.5°/s.

Robust Guidance Algorithm against Hypersonic Targets DOI: http://dx.doi.org/10.5772/intechopen.84655

Miss distance distribution of the FCG with a noise of 2.5°/s.

Statistical results of the miss distances under noisy conditions.

Noise (°/s) Guidance law Expectation (m) Variance (m) 0.5 FCG 0.0396 6.7768e004 0.5 NPDG 0.3322 0.0014 1.5 FCG 0.0786 0.0036 1.5 NPDG 0.5842 0.0274 2.5 FCG 0.1457 0.0091 2.5 NPDG 1.0092 0.2044

Figure 15. Miss distance distribution of the NPDG with a noise of 1.5°/s.

Figure 16. Miss distance distribution of the FCG with a noise of 1.5°/s.


Robust Guidance Algorithm against Hypersonic Targets DOI: http://dx.doi.org/10.5772/intechopen.84655

Figure 17. Miss distance distribution of the NPDG with a noise of 2.5°/s.

Figure 18. Miss distance distribution of the FCG with a noise of 2.5°/s.


Table 3. Statistical results of the miss distances under noisy conditions.

1. The FCG can improve the guidance accuracy. Compared with the NPDG, it has a better feasibility, since the NPDG requires the measurement of q€, while this angular acceleration usually cannot be directly measured by the interceptor's

2. The robustness of the FCG is better than that of the NPDG. The FCG using the fractional differential of q\_ improves the precision of the estimation. The filtering capability of the fractional order part in the FCG provides good stability to the system in a hypersonic pursuit-evasion game under noisy

seeker.

Figure 16.

Figure 15.

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Miss distance distribution of the NPDG with a noise of 1.5°/s.

Miss distance distribution of the FCG with a noise of 1.5°/s.

conditions.

118
