4. Simulations

#### 4.1. Initial conditions

Simulations will be conducted to validate the feasibility and superiority of the proposed schemes in this part. In simulations, the maximum acceleration limit of the interceptor is 10 g 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 by the following first-order systems:

$$\frac{a\_{m\mathbf{u}}(\mathbf{s})}{a\_{m\mathbf{u}}(\mathbf{s})} = \frac{1}{\tau\_m \mathbf{s} + 1},\tag{32}$$

$$\frac{a\_{tt}(\mathbf{s})}{a\_{tq}(\mathbf{s})} = \frac{1}{\tau\_t \mathbf{s} + 1},\tag{33}$$

where the guidance commands are denoted by "atq" and "amq" and the responses are denoted by "ata" and "ama". τ<sup>t</sup> ¼ 0:5 and τ<sup>m</sup> ¼ 0:2.

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>� , q\_ <sup>0</sup> ¼ �3deg=s, Vt ¼ 500 m/s, and φ<sup>t</sup> ¼ 0 � . In accordance with the Eq. (9), δv<sup>0</sup> ¼ 70 m/s and δr<sup>0</sup> ¼ 300 m are given 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 worst-case conditions of the initial observed relative speed v<sup>0</sup> and distance r<sup>0</sup> are given.

Condition 1 (C1):

atq � � �

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

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

\_ <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 ,

obtained from Eq. (15), and γ > 0 is a constant, guarantees that s ¼ z

8 >>>><

>>>>:

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

ð Þ3

� r r

> 2 r z ð Þ2 1 z ð Þ3 1

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

<sup>b</sup><sup>k</sup> <sup>¼</sup> <sup>1</sup> γ r 2

!

!

� r r

� r r

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

!

where z

Then,

q\_ ! 0.

4. Simulations

4.1. Initial conditions

ð Þ1 <sup>2</sup> , z ð Þ2 <sup>1</sup> , and z

32 Adaptive Robust Control Systems

<sup>V</sup>\_ <sup>2</sup> ¼ � <sup>r</sup>

2 r z ð Þ2 1 z ð Þ3 1

According to Eq. (28) and \_

<sup>V</sup>\_ <sup>2</sup> <sup>≤</sup> � <sup>r</sup>

¼ � <sup>r</sup> 2 r z ð Þ2 1 z ð Þ3 1

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

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

N � 2 þ bk � �

2 � r 2 <sup>r</sup> <sup>b</sup><sup>k</sup> � <sup>k</sup> � �

h i

" #

<sup>2</sup> <sup>þ</sup> <sup>γ</sup>~k<sup>2</sup> � �

2 � r 2 r

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

<sup>r</sup> j j <sup>Δ</sup><sup>v</sup> � <sup>~</sup>vÞq\_ <sup>2</sup>: �

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

Simulations will be conducted to validate the feasibility and superiority of the proposed schemes in this part. In simulations, the maximum acceleration limit of the interceptor is 10 g

Lyapunov stability theory, we can guarantee that V<sup>2</sup> ! 0. Finally s ! 0. Since s ¼ z

ð Þ2 1 z ð Þ3 1 þ bk

!

V<sup>2</sup> ¼ 0:5 s

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

<sup>r</sup> j j <sup>v</sup> <sup>q</sup>\_2, we get

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

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

� ≤ ð Þ N � 2 þ k j j v j j q\_ , (28)

q,\_

ð Þ1

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

: (30)

<sup>q</sup>\_ <sup>þ</sup> <sup>γ</sup>~k~\_

ð Þ1 <sup>1</sup> q\_, that is,

k: (31)

jvjq\_ � atq

j j v q\_

<sup>2</sup> <sup>þ</sup> <sup>γ</sup>~k~\_ k

(29)

$$
\overline{r}\_0 = r\_0 - \delta\_{r0}, \overline{v}\_0 = v\_0 - \delta\_{t0}.\tag{34}
$$

Condition 2 (C2):

$$
\overline{r}\_0 = r\_0 + \delta\_{r0}, \overline{v}\_0 = v\_0 + \delta\_{v0}.\tag{35}
$$

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

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

$$\begin{cases} a\_{\rm ty}(t) = 0, & t \le 2s \\ a\_{\rm ty}(t-2) = -a\_{\rm ty}(t), a\_{\rm ty}(2) = 6\text{g.} \quad t > 2s \end{cases} \tag{36}$$

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

$$\begin{cases} a\_{ly}(t) = 0, & t \le 2s \\ a\_{ly}(t) = 8\mathbf{g} \cdot \sin\left[\Im(t-2)\right]. & t > 2s \end{cases} \tag{37}$$

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

$$\begin{cases} a\_{ty}(t) = 0, \quad t \le 2s\\ a\_{ty}(t) = 8\text{g.} \quad t > 2s \end{cases} \tag{38}$$

#### 4.2. Comparisons between the OSMG and the APNG

Compare the ISMCG and SMCG with the APNG and OSMG. The actual target normal load and relative speed are considered known in the APNG; thereby, neither Condition 1 nor Condition 2 can affect the APNG. For the OSMG, it owns a simplified formulation which has robustness to target's maneuver, and it is popular in practice. Its simplified realization for online is as follows [6].

$$
\delta a\_{m\boldsymbol{\eta}} = -\mathfrak{H}\_0 \dot{\boldsymbol{\eta}} + \varepsilon \text{sgn}(\dot{\boldsymbol{\eta}}) \simeq -\mathfrak{W}\_0 \dot{\boldsymbol{\eta}} + \varepsilon \frac{\dot{\boldsymbol{\eta}}}{|\dot{\boldsymbol{\eta}}| + \Delta'}.\tag{39}
$$

where the initial observed relative speed is denoted by "v0". ε and Δ have no difference with those of the SMCG.

The expression of the APNG is obtained as [16]

$$a\_{m\eta} = \mathcal{N}' |\upsilon| \dot{\eta} + \mathcal{N}' \frac{a\_{t\eta}}{2},\tag{40}$$

Figure 3. Guidance commands and line-of-sight angular velocities in case 1 under condition 2. (a) Guidance commands

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

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Figure 4. Guidance commands and line-of-sight angular velocities in case 2 under condition 1. (a) Guidance commands

Figure 5. Guidance commands and line-of-sight angular velocities in case 2 under condition 2. (a) Guidance commands

(b) Line-of-sight angular velocities.

(b) Line-of-sight angular velocities.

(b) Line-of-sight angular velocities.

where the actual target normal load and the relative speed are denoted by "atq" and "v". An optimal value of the constant N<sup>0</sup> is 3 [16].


Table 1. Comparisons of miss distances (m).

Figure 2. Guidance commands and line-of-sight angular velocities in case 1 under condition 1. (a) Guidance commands (b) Line-of-sight angular velocities.

Adaptive Robust Guidance Scheme Based on the Sliding Mode Control in an Aircraft Pursuit-Evasion Problem http://dx.doi.org/10.5772/intechopen.72177 35

4.2. Comparisons between the OSMG and the APNG

amq ¼ �3\_

The expression of the APNG is obtained as [16]

optimal value of the constant N<sup>0</sup> is 3 [16].

Table 1. Comparisons of miss distances (m).

(b) Line-of-sight angular velocities.

online is as follows [6].

34 Adaptive Robust Control Systems

those of the SMCG.

Compare the ISMCG and SMCG with the APNG and OSMG. The actual target normal load and relative speed are considered known in the APNG; thereby, neither Condition 1 nor Condition 2 can affect the APNG. For the OSMG, it owns a simplified formulation which has robustness to target's maneuver, and it is popular in practice. Its simplified realization for

<sup>r</sup>0q\_ <sup>þ</sup> <sup>ε</sup>sgnð Þ <sup>q</sup>\_ <sup>≃</sup> � <sup>3</sup>v0q\_ <sup>þ</sup> <sup>ε</sup> <sup>q</sup>\_

<sup>∣</sup>v∣q\_ <sup>þ</sup> <sup>N</sup><sup>0</sup> atq

Condition 1 Condition 2 Condition 1 Condition 2 Condition 1 Condition 2

where the initial observed relative speed is denoted by "v0". ε and Δ have no difference with

where the actual target normal load and the relative speed are denoted by "atq" and "v". An

APNG 0.0831 0.0831 2.7448 2.7448 0.0173 0.0173 OSMG 0.0525 0.0819 0.0129 0.1148 0.0010 0.0015 ISMCG 0.0050 0.0003 0.0010 0.0019 0.0020 0.0036 SMCG 0.0289 0.0645 0.1115 0.1253 0.0010 0.0013

Figure 2. Guidance commands and line-of-sight angular velocities in case 1 under condition 1. (a) Guidance commands

amq ¼ N<sup>0</sup>

Schemes Case 1 Case 2 Case 3

<sup>∣</sup>q\_<sup>∣</sup> <sup>þ</sup> <sup>Δ</sup>, (39)

<sup>2</sup> , (40)

Figure 3. Guidance commands and line-of-sight angular velocities in case 1 under condition 2. (a) Guidance commands (b) Line-of-sight angular velocities.

Figure 4. Guidance commands and line-of-sight angular velocities in case 2 under condition 1. (a) Guidance commands (b) Line-of-sight angular velocities.

Figure 5. Guidance commands and line-of-sight angular velocities in case 2 under condition 2. (a) Guidance commands (b) Line-of-sight angular velocities.

Figure 6. Guidance commands and line-of-sight angular velocities in case 3 under condition 1. (a) Guidance commands (b) Line-of-sight angular velocities.

The missing distance is illustrated in Table 1, and the guidance commands and the line-ofsight angular velocities are shown in Figures 2–7. As shown in Table 1, these four guidance schemes all accomplish the interception task with the constraint ∣amq∣ ≤ 10 g. In case 2, APNG's miss distances are comparatively greater, because Eq. (40) is deduced assuming atq is unchanged [17]. Nevertheless, the target might have a complicated maneuvering kind of escape. For the APNG, there is a greater miss distance to case 2, rather than to case 1 and case 3. It indicates the ANPG's limitations on intercepting unconventional maneuvering targets. Table 1 also illustrates that the ISMCG owns the smallest miss distance in case 1 and case 2 for complicated types of target maneuvers, and the SMCG behaves like the OSMG. In case 3 for step maneuver targets, the miss distances of ISMGC, SMCG, and OSMG are small, and there is a little difference in performance of them.

From Figures 2–7, the APNG is not very appropriate to intercept complicated maneuvering targets because line-of-sight angular velocities of the APNG are greater. As a whole, the plots of the OSMG and the SMCG have little difference between each other. Their line-of-sight angular velocities are very small before the end of case 1 and case 2. Nevertheless, although continuous functions are employed to take the place of the "sgn" functions in the OSMG and the SMCG, guidance commands of the OSMCG and the SMCG all have jitters, which are

Figure 8. δatq and ∣atq∣ of the ISMCG. (a) case 1 (C1); (b) case 1 (C2); (c) case 2 (C1); (d) case 2 (C2); (e) case 3 (C1); and (f)

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

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37

case 3 (C2).

Figure 7. Guidance commands and line-of-sight angular velocities in case 3 under condition 2. (a) Guidance commands (b) Line-of-sight angular velocities.

Adaptive Robust Guidance Scheme Based on the Sliding Mode Control in an Aircraft Pursuit-Evasion Problem http://dx.doi.org/10.5772/intechopen.72177 37

The missing distance is illustrated in Table 1, and the guidance commands and the line-ofsight angular velocities are shown in Figures 2–7. As shown in Table 1, these four guidance schemes all accomplish the interception task with the constraint ∣amq∣ ≤ 10 g. In case 2, APNG's miss distances are comparatively greater, because Eq. (40) is deduced assuming atq is unchanged [17]. Nevertheless, the target might have a complicated maneuvering kind of escape. For the APNG, there is a greater miss distance to case 2, rather than to case 1 and case 3. It indicates the ANPG's limitations on intercepting unconventional maneuvering targets. Table 1 also illustrates that the ISMCG owns the smallest miss distance in case 1 and case 2 for complicated types of target maneuvers, and the SMCG behaves like the OSMG. In case 3 for step maneuver targets, the miss distances of ISMGC, SMCG, and OSMG are small, and there is

Figure 7. Guidance commands and line-of-sight angular velocities in case 3 under condition 2. (a) Guidance commands

Figure 6. Guidance commands and line-of-sight angular velocities in case 3 under condition 1. (a) Guidance commands

a little difference in performance of them.

(b) Line-of-sight angular velocities.

(b) Line-of-sight angular velocities.

36 Adaptive Robust Control Systems

Figure 8. δatq and ∣atq∣ of the ISMCG. (a) case 1 (C1); (b) case 1 (C2); (c) case 2 (C1); (d) case 2 (C2); (e) case 3 (C1); and (f) case 3 (C2).

From Figures 2–7, the APNG is not very appropriate to intercept complicated maneuvering targets because line-of-sight angular velocities of the APNG are greater. As a whole, the plots of the OSMG and the SMCG have little difference between each other. Their line-of-sight angular velocities are very small before the end of case 1 and case 2. Nevertheless, although continuous functions are employed to take the place of the "sgn" functions in the OSMG and the SMCG, guidance commands of the OSMCG and the SMCG all have jitters, which are detrimental to aero fins. Line-of-sight angular velocities of the ISMCG are less than those of the APNG with the actual target's acceleration. From Figures 2(a), 3(a), 4(a), and 5(a), in case 1 and case 2, ISMCG's guidance commands are smoother than others, which are appropriate for continuous aero surfaces to track. From Figures 2(b), 3(b), 4(b), and 5(b), because the ISMCG uses an adaptive estimation to identify atq's upper boundary, line-of-sight angular velocities of the ISMCG are not as moderate as those of the SMCG; however, from Table 1, line-of-sight angular velocities in the endgame of the ISMCG are less than those of the SMCG and the OSMG in case 1 and case 2. From Figures 6 and 7, the guidance commands and line-of-sight angular velocities of OSMG, SMCG, and ISMCG have little differences and are superior to those of the APNG in case 3.

Grant Nos. 51475472 and 61403396, and Beijing Youth Top-notch Talent Support Program under

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

\*, Yuan Tian<sup>3</sup>

[1] Faruqi F. Differential Game Theory with Applications to Missiles and Autonomous

[2] Ben-Asher J, Speyer J. Game in Aerospace: Homing Missile Guidance. Handbook of

[3] Khalid S, Abrar S. A low-complexity interacting multiple model filter for maneuvering target tracking. AEU-International Journal of Electronics and Communications. 2017;73:157-164 [4] Jauffret C, Pérez A, Pillon D. Observability: Range-only versus bearings-only target motion analysis when the observer maneuvers smoothly. IEEE Transactions on Aero-

[5] Zhou D, Mu C, Xu W. Adaptive sliding-mode guidance of a homing missile. Journal of

[6] Zhou D, Mu C, Ling Q. and Xu W. Optimal Sliding-mode guidance of a homing-missile. In: Kamen EW, Cassandras C, editors. 38th IEEE Conference on Decision and Control;

[7] Hou Z, Su M, Wang Y. and Liu L. A fuzzy optimal sliding-mode guidance for intercepting problem. In: Camisani F, editor. IFAC World Congress; Cape Town. IFAC;

[8] Wang Z. Adaptive smooth second-order sliding mode control method with application to missile guidance. Transactions of the Institute of Measurement and Control. 2017;39(6):

, Chen Bai4

, Zhang Ren<sup>5</sup>

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

, Guangqi Wang<sup>1</sup>

,

39

Grant No. 2017000026833ZK23.

, Yongjun Zheng<sup>1</sup>

5 Beihang University, Beijing, China

, Yuan Ren<sup>2</sup>

1 College of Engineering, China Agricultural University, Beijing, China

\*Address all correspondence to: renyuan\_823@aliyun.com

3 Beijing Institute of Space Long March Vehicle, Beijing, China

4 Beijing Institute of Aerospace System Engineering, Beijing, China

Systems Guidance. Hoboken: John Wiley & Sons; 2017

Guidance, Control, and Dynamics. 1999;22(4):589-594

Dynamic Game Theory. Basel: Springer; 2017

space and Electronic Systems. 2017

Phoenix, USA: IEEE; 1999. pp. 5131-5136

2014. pp. 3401-3406

848-860

2 Space Engineering University, Beijing, China

Author details

Nannan Du<sup>1</sup> and Yu Tan<sup>1</sup>

Jian Chen<sup>1</sup>

References

From Figure 8, it illustrates the δatq identified by the ISMCG in three cases under two conditions. Compared with ∣atq∣, in the initial 2 s, δatq is larger since q\_ and bk are larger, and then, the tracking error decreases since the ISMCG restrains the line-of-sight angular velocity. Because δatq is not the estimation of atq, tracking phases are considered and tracking errors are not concerned. Tracking phases reflect that estimations lag behind the actual target maneuver; thereby, it decides whether the compensation bk is timely and can influence the guidance precision. With tracking phases under consideration, δatq mostly tracks ∣atq∣ with a tiny time delay. In fact, for the step maneuver target in case 3, from Figure 8(e, f), δatq tracks ∣atq∣ well. As shown in Table 1, small tracking phases obtain small miss distances.
