Three-dimensional adaptive finite-time guidance law for intercepting maneuvering targets

CJA 844 12 June 2017 Chinese Journal of Aeronautics, (2017), xxx(xx): xxx–xxx

No. of Pages 19

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Chinese Society of Aeronautics and Astronautics & Beihang University

Chinese Journal of Aeronautics [email protected] www.sciencedirect.com

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Three-dimensional adaptive finite-time guidance law for intercepting maneuvering targets

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Yujie Si, Shenmin Song *

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Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin 150001, China

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Received 3 November 2016; revised 7 December 2016; accepted 3 March 2017

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KEYWORDS

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Adaptive control; Finite-time guidance law; Impact angle; Sliding mode control; Three-dimensional guidance law

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Ó 2017 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).

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Abstract To intercept maneuvering targets at desired impact angles, a three-dimensional terminal guidance problem is investigated in this study. Because of a short terminal guidance time, a finitetime impact angle control guidance law is developed using the fast nonsingular terminal sliding mode control theory. However, the guidance law requires the upper bound of lumped uncertainty including target acceleration, which may not be accurately obtained. Therefore, by adopting a novel reaching law, an adaptive sliding mode guidance law is provided to release the drawback. At the same time, this method can accelerate the convergence rate and weaken the chattering phenomenon to a certain extent. In addition, another novel adaptive guidance law is also derived; this ensures systems asymptotic and finite-time stability without the knowledge of perturbations bounds. Numerical simulations have demonstrated that all the three guidance laws have effective performances and outperform the traditional terminal guidance laws.

1. Introduction To improve the impact lethality of missiles, guidance laws with terminal impact angle constraints are necessary in modern warfare.1 In the past decades, various methods have been developed in this field.2–7 In Ref.2, unlike the conventional proportional navigation guidance law (PNGL), a new guidance law with a supplementary time-varying bias was developed * Corresponding author. E-mail address: [email protected] (S. Song). Peer review under responsibility of Editorial Committee of CJA.

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to intercept targets at the desired impact angle. In Ref.3, for the problem of trajectory shaping against a stationary target, the biased pure PNGL (BPPNGL) was given. The control problem was solved indirectly by using the integrated form of the BPPNGL. In Ref.4, to intercept ground targets, a guidance law was developed for air-launched missiles using the nonlinear suboptimal method, and terminal impact angle constraints were accurately satisfied by applying this guidance law. In Refs.5,6, guidance laws were proposed against stationary targets using the optimal control method. In Ref.7, to obtain the expected interception angle for intercepting a slowmoving or stationary target, an interception angle control guidance law was formulated using a PNG-based method. During the developments of the PNGL, the optimal method, and other traditional methods, a sliding mode control (SMC) method was proposed to provide a new way for

http://dx.doi.org/10.1016/j.cja.2017.04.009 1000-9361 Ó 2017 Production and hosting by Elsevier Ltd. on behalf of Chinese Society of Aeronautics and Astronautics. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Si Y, Song S Three-dimensional adaptive finite-time guidance law for intercepting maneuvering targets, Chin J Aeronaut (2017), http://dx.doi.org/10.1016/j.cja.2017.04.009

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designing guidance laws. Because the SMC method has good robustness to external disturbances and uncertainties of systematic parameters, it has been generally used in designing guidance laws in the recent decades. In Ref.8, guidance laws were developed using the traditional SMC. In Ref.9, an adaptive nonlinear guidance law was proposed by using the SMC method so that a missile can accurately hit a target at the desired impact angle. Thus, it can be concluded that the SMC method can only guarantee the asymptotic convergence of a system. Because the terminal guidance phase lasts for only several seconds, the finite-time control theory has been extensively studied. Moreover, the finite-time stabilization of dynamic systems can contribute to both high accuracy and good performance. Notably, the terminal SMC (TSMC) is a relatively good method to solve this problem. In Refs. 10,11, finitetime sliding mode guidance laws with impact angle constraints were proposed. Guidance laws have been developed using the nonsingular TSMC (NTSMC) theory in Refs. 12,13. Using the NTSMC method, finite-time guidance laws with the desired impact angle were proposed. However, they were derived against stationary or constant-velocity targets. To intercept targets with impact angle constraints, guidance laws were proposed without exhibiting any singularity in Ref. 14. In Ref. 15, using the NTSMC theory, a strictly convergent guidance law was developed for missiles with impact angle constraints. In Ref. 16, an optimal pulsed guidance law with a time-varying weighted quadratic cost function that enables the imposing of a predetermined intercept angle was developed. However, all the guidance laws in the above literature were developed in a two-dimensional (2D) scene, which would significantly reduce the guidance precision. In practice, all the engagement scenarios of a missile intercepting targets are three-dimensional (3D) scenarios. However, most of the existing studies mainly concentrated on decoupling the 3D engagement into two mutually orthogonal 2D engagements, and their guidance laws were designed separately. This will definitely affect the guidance accuracy. Many studies also designed guidance laws in a 3D engagement. 3D sliding mode guidance laws have been reported in Refs. 17–19. However, these guidance laws were designed without considering impact angle constraints. In Refs. 20–22, some novel guidance laws with impact angle constraints were proposed using SMC for 3D engagements. However, those guidance laws were designed without considering target maneuvering. Along with high complexity of a battlefield, rapid combat rhythm, and high-tech weaponry, an interceptor should also have the ability to intercept maneuvering targets. Therefore, it is necessary to devise 3D finite-time guidance laws against maneuvering targets with impact angle constraints. In Refs.19,23,24, to intercept maneuvering targets, novel 3D finite-time convergence guidance laws have been developed using the NTSMC theory. However, all of them did not consider the impact angle constraints problem. Now, a few studies have designed 3D guidance laws simultaneously by considering the problems of finite-time convergence, terminal impact angle constraints, and maneuvering targets. Based on the summary of the abovementioned problems, the main objective of the paper is to construct finitetime guidance laws with impact angle constraints in a 3D engagement. The main contributions are as follows. (1) Based on the fast NTSMC method, a novel TSM guidance law is

Y. Si, S. Song derived. Unlike the existing NTSMC methods that decouple a 3D engagement into two mutually orthogonal 2D engagements,12–15 the guidance law is derived by using coupled 3D dynamic systems. Furthermore, the guidance law can also be applied to intercept maneuvering targets. (2) By introducing a novel reaching law designed in this study, an adaptive 3D nonsingular guidance law is developed. Compared to the existing 3D TSMC methods,20–22 the guidance law can intercept maneuvering targets without the knowledge of target information. Compared to the existing 3D TSMC guidance laws against maneuvering targets,19,23,24 this guidance law can not only successfully intercept a maneuvering target, but also intercept it from the desired angles. Moreover, the guidance law can accelerate the convergence rate and reduce the unwanted chattering level as well. (3) Another novel reaching law is derived. The adaptive guidance law designed based on this law ensures that the line-of-sight (LOS) angle and angular rate are asymptotically and finite-time stable with unknown bounded disturbances. The structure of this paper is as follows. In Section 2, nondecoupling 3D engagement dynamic systems are established. In Section 3, based on the fast NTSMC method, a novel guidance law is designed. Furthermore, two adaptive fast nonsingular terminal sliding-mode guidance laws are also derived by adopting novel reaching laws. Simulations are performed in Section 4. Conclusions are provided in Section 5.

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2. Problem statement

130

Fig. 1 shows the 3D homing guidance geometry. OI XI YI ZI represents the inertial reference frame, and its origin is the missile launching point. OM XM YM ZM and OT XT YT ZT represent the missile and target velocity coordinate systems, respectively. R denotes the LOS distance. In this geometry, the missile velocity is denoted by Vm , and its direction is defined by hm and /m with respect to the LOS frame. Moreover, it is assumed that the missile flies at a constant speed in this study. The target flies at a constant speed Vt , and its direction is denoted by ht and /t . hL and /L denote the LOS angles. Then, the 3D engagement dynamic systems can be expressed as follows:25

131

R_ ¼ ðq cos ht cos /t  cos hm cos /m ÞVm

ð1Þ

Rh_ L ¼ ðq sin ht  sin hm ÞVm

ð2Þ

/_ L R cos hL ¼ ðq cos ht sin /t  cos hm sin /m ÞVm

ð3Þ

Fig. 1

105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129

132 133 134 135 136 137 138 139 140 141 142 144 145 147 148

Geometry in a 3D space.

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Three-dimensional adaptive finite-time guidance law 151 153 154

156 157 159

azm _ h_ m ¼  /L sin hL sin /m  h_ L cos /m Vm aym /_ m ¼ þ /_ L sin hL cos /m tan hm  h_ L sin /m Vm cos hm  tan hm  /_ L cos hL azt h_ t ¼  /_ L sin hL sin /t  h_ L cos /t qVm

160

/_ t ¼ 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177

ayt þ /_ L sin hL cos /t tan ht  h_ L sin /t tan ht qVm cos ht  /_ L cos hL

3 ð4Þ

ð5Þ ð6Þ

ð7Þ

where aym and azm are the lateral accelerations of the missile in the yaw and pitch directions, respectively. Similarly, ayt and azt are the target accelerations, and q ¼ VVmt . In the study, the main objective is to design 3D finite-time guidance laws so that the target can be successfully intercepted at the desired angles, i.e., to design guidance laws azm and aym so that h_ L and /_ L converge to zero within finite time, while the LOS angles hL and /L converge to the desired values in finite time. The main results are described in the following sections. _ hL , /L , h_ L , In the study, it is assumed that the signals R, R, _/L , hm , and / can be measured. Let hLf and / be the desired m Lf LOS angles. Let x1 and x2 be the LOS angle errors, which are defined by x1 ¼ hL  hLf and x2 ¼ /L  /Lf . To facilitate the design, the following lemmas and assumptions are considered for further application.

Assumption 2 29. Assume that the missile intercepting the target by impact (‘‘hit-to-kill”) occurs when R ¼ R0 – 0, but belongs to the interval ½Rmin ; Rmax  ¼ ½0:1; 0:25 m. In this study, guidance laws with impact angle constraints are designed by using nonlinear engagement dynamics, as shown in Eqs. (1)–(7). Moreover, guidance laws are also developed for intercepting maneuvering targets. Second-order dynamic systems between the control input and the LOS angles can be obtained by differentiating Eqs. (2) and (3) as cos ht cos hm 2R_ h_ L € azt  azm  /_ 2L cos hL sin hL  hL ¼ R R R € L ¼ cos /t ayt  sin ht sin /t azt þ sin hm sin /m azm / R cos hL R cos hL R cos hL cos /m 2R_ /_ L  aym þ 2/_ L h_ L tan hL  R cos hL R

2 F¼4

cos ht R

179 180 181 183 184 185 186 187

Lemma 1 26. Consider the nonlinear system x_ ¼ fðx; tÞ; x 2 Rn . Assume the existence of a continuous and positive definite function VðxÞ, _ VðxÞ 6 lVðxÞ  kVa ðxÞ

ð8Þ

where l, k > 0, and 0 < a < 1 are constants. xðt0 Þ ¼ x0 , in which t0 is the initial time. Then, the time of system states arriving at the equilibrium point T satisfies the following inequality:

189

1 lV1a ðx0 Þ þ k ln T6 lð1  aÞ k

190

That is, the system states are finite-time convergent.

ð9Þ

196

Lemma 2 27. Consider the system x_ ¼ fðx; tÞ; x 2 Rn . If VðxÞ is a continuous and positive definite function (defined on U 2 Rn ), _ and VðxÞ þ fVs ðxÞ is negative semidefinite on U 2 Rn for s 2 ð0; 1Þ and f 2 Rþ , then an area U0 2 Rn exists so that any VðxÞ starting from U0 2 Rn can reach VðxÞ  0 in finite time. Moreover, if Tr is the time needed to reach VðxÞ  0, then

197

ðx0 Þ Tr 6 Vfð1sÞ , where Vðx0 Þ is the initial value of VðxÞ.

191 192 193 194 195

1s

199

Lemma 3 28. For bi 2 R; i ¼ 1; 2;    ; n, if q is a real number and 0 < q < 1, then the following inequality can be satisfied:

200 202

ðjb1 j þ jb2 j þ    þ jbn jÞq 6 jb1 jq þ jb2 jq þ    þ jbn jq

198

203 204 205 206

ð10Þ

Assumption 1. Assume that the target accelerations, ayt and azt , are bounded and satisfy jayt j 6 a1 , jazt j 6 a2 , for all t P 0, where a1 and a2 are the upper bounds of the target accelerations.

__ azt  /_ 2L cos hL sin hL  2RRhL

__ t sin /t ayt  sinRhcos azt þ 2/_ L h_ L tan hL  2RR/L hL " #    cosRhm 0 azm B ¼ sin hm sin /m ; u ¼ /m aym  Rcos R cos h cos h cos /t R cos hL

208 209 210 211 212 213 214 215 216 218 219

ð12Þ

In the notational form, the dynamics, expressed using Eqs. (11) and (12), can also be rewritten as " # € hL ¼ F þ Bu ð13Þ €L /

L

178

ð11Þ

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3

221 222 223 224

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5;

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L

Note that because u is multiplied by matrix B, the LOS angles can be controlled only if R – 0, and the angles hm and /m satisfy hm ; /m –  ðp=2Þ. Because the missile intercepting the target by impact occurs when R ¼ R0 – 0 with Assumption 2, R P R0 during the terminal guidance phase. Hence, acceleration u can be used to control under the assumptions of hm –  ðp=2Þ and /m –  ðp=2Þ.

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3. Design of guidance laws

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3.1. Design of a fast nonsingular guidance law

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In this study, the main objective is to design guidance laws so that a missile can intercept maneuvering targets not only with the miss distance, but also at the desired LOS angles  minimum  hLf . To achieve this objective, the design of an appropriate /Lf sliding mode variable is discussed first. The LOS angle error is     x1 hL  hLf defined by x ¼ ¼ , and Eq. (14) is selected as x2 /L  /Lf the sliding mode manifold.   s1 ¼ x_ þ ux þ bfðxÞ ð14Þ S¼ s2

239



fðx1 Þ fðxÞ ¼ fðx2 Þ  fðxi Þ ¼



231 232 233 234 235 236

240 241

242 243

244 245 246

248 249

ð15Þ

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r1 xi þ r2 x2i signðxi Þ jxi j < g jxi jr signðxi Þ otherwise

i ¼ 1; 2

ð16Þ

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255 257

r1 ¼ ð2  rÞgr1

ð17Þ

Proof. Consider the Lyapunov function candidate as

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270

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r2 ¼ ðr  1Þgr2

where r, g, b, and u are positive constants that need to be determined, and 0 < r < 1. The derivative of S can be expressed as _ € þ ux_ þ bfðxÞ S_ ¼ x " # " # €hL h_ _ þ u L þ bfðxÞ ¼ € /L /_ L

where

"

h_ L G¼u /_ L " _ fðxÞ ¼

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_ iÞ ¼ fðx

277

 M¼

279 280

" N¼

282 283 284 285 286 287 289

290 291

292 293 294 295

ð19Þ

# _ þ bfðxÞ

m1 m2

ð20Þ

# ð21Þ

r1 x_ i þ 2r2 xi x_ i signðxi Þ jxi j < g i ¼ 1; 2 rjxi jr1 x_ i otherwise "

 ¼

cos ht R cos /t R cos hL

ayt 

#

azt sin ht sin /t R cos hL

/_ 2L cos hL sin hL  2RRhL __

__

þ2/_ L h_ L tan hL  2RR/L

ð22Þ

azt

ð23Þ

# ð24Þ

298 299 300 301 302 303

304 305 307

311

The time derivative of V1 along with Eqs. (1)–(7) results in

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V_ 1 ¼ ST S_ 6 kSkkMk  hkSk  kST S

ð28Þ

6 kSkðh  mÞ  kST S pffiffiffi 1 6  2ðh  mÞV21  2kV1

315

According to Lemma 1, inequality Eq. (28) indicates that the finite-time convergence of the sliding mode surface is available. Thus, conclusion (i) is proven. h

The fast nonsingular terminal sliding mode guidance law (FNTSMGL) of a missile intercepting maneuvering targets with impact angle constraints can be expressed using the following equation with Assumptions 1 and 2: u1 ¼ B1 ðN þ G þ kS þ HsignðSÞÞ ð25Þ   h 0 ; h1 , and h2 are positive constants, where H ¼ 1 0 h2 h ¼ minðh1 ; h2 Þ, m ¼ kMk, and m < h . Theorem 1. Considering the systems in Eqs. (1)–(7), if the external disturbance M is bounded, then kMk 6 m. If Eq. (14) is selected as the sliding mode surface and Eq. (25) is selected as the guidance law, the following conclusions can be satisfied:

(i) The sliding manifold S converges to zero in finite time. (ii) The LOS tracking error angles hL  hLf and /L  /Lf converge to the regions jhL  hLf j < g and j/L  /Lf j < g in finite time, where g is a small positive constant. (iii) The LOS angular rates h_ L and /_ L converge to the regions jh_ L j < g1 and j/_ L j < g1 in finite time, in which

g1 ¼ ug þ bgr

Case I. If jxi j P gði ¼ 1; 2Þ, consider another Lyapunov function candidate as

316 317 318

ð26Þ

320 321

322 323

1 V2 ¼ x2i 2

ð29Þ

When S ¼ 0, the derivation of V2 can be written as V_ 2 ¼ xi x_ i ¼ xi ðuxi þ bjxi jr signðxi ÞÞ

325 326 327

1þr

6 ux2i  bðx2i Þ 2 1þr

296

297

ð27Þ

319

_ 1Þ fðx _fðx2 Þ



1 V1 ¼ ST S 2

¼ ST M  kST S  ST HsignðsÞ

¼ F þ Bu þ G ¼ M þ N þ Bu þ G

271

276

ð18Þ

1þr

¼ 2uV2  2 2 bV22

Because 0 < 1þr < 1, xi converges to the region jxi j < g in finite 2 time, i.e., the LOS tracking error angles hL  hLf and /L  /Lf converge to the regions jhL  hLf j < g and j/L  /Lf j < g in finite time, respectively. Case II. If jxi j < gði ¼ 1; 2Þ and because si ¼ x_ i þ uxi þ bfðxi Þ ¼ 0 we can obtain x_ i ¼ uxi  bfðxi Þ jx_ i j ¼ juxi þ bfðxi Þj 6 juxi j þ jbfðxi Þj 6 ug þ bgr ¼ g1 Then, x_ i converges to the region jx_ i j < g1 in finite time, i.e., the LOS rates h_ L and /_ L converge to the regions jh_ L j < g1 and j/_ L j < g1 , respectively, in finite time. Thus, Conclusions (ii) and (iii) are proven. Therefore, the conclusions of Theorem 1 can be easily obtained. Remark 1. By adding the proportional term x to the sliding mode surface in Eq. (14), the guidance laws designed in this study can significantly accelerate the convergence rate towards the manifold compared to the traditional terminal sliding mode guidance laws according to Ref. 30.

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Three-dimensional adaptive finite-time guidance law 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371

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Remark 2. From Theorem 1, the guidance law in Eq. (25) ensures that the LOS angular rates h_ L and /_ L converge to a small region, which is defined using Eq. (26). From Eq. (26), we can conclude that g1 is related to u, g, b, and r. When g ¼ 0:05 and u ¼ 0:1, the curves of g1 are illustrated in Fig. 2. Fig. 2 shows that the value of g1 decreases with a decrease in b and decreases with an increase in r. Therefore, a reasonable convergence region of the LOS angular rates can be obtained by adjusting the parameters. Now, we have proposed a 3D finite-time convergent guidance law. In Theorem 1, it is assumed that kMk 6 m. However, the upper bound of M including ayt , azt , ht , and /t cannot be measured or estimated accurately. In addition, the exponential reaching law is used in the guidance law of Eq. (25). The exponential reaching law has the advantage of a fast convergence rate, but has the disadvantage in suppressing chattering. To solve these problems, new reaching laws will be introduced in Section 3.2.

372

3.2. Design of adaptive fast nonsingular guidance laws

373

In the sliding mode method, the control performance can be improved by selecting an appropriate reaching law that can accelerate the convergence rate and weaken the chattering phenomenon. Among the traditional reaching laws, the constant reaching law not only suffers from a slow convergence speed, but also causes a large chattering phenomenon to the systems.31 The exponential reaching law significantly accelerates the convergence rate because of the presence of a constant term.32 The power reaching law can weaken the chattering phenomenon to a certain extent.33 Specific forms are shown below. Gao34 proposed the concept of reaching law and designed the power reaching law as

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s_ ¼ hjsja signðsÞ

ð30Þ

where h and a are positive coefficients. The exponential reaching law was proposed in Ref.35, and the specific function is s_ ¼ hsignðsÞ  ks

where k are positive coefficients. This type of reaching law was adopted in u1 . It has the merit of a fast convergence rate, but it has the shortcoming of suppressing chattering.To overcome the drawbacks of the exponential reaching law and simultaneously maintain its advantages, we introduce a new reaching law by combining an integral adaptation term with an exponential term as s_ ¼ ks  ðay þ NðsÞÞsignðsÞ

ð32Þ

395 396 397 398 399 400 401 402 404 405 407

y_ ¼ ajsj; yð0Þ > 0

408 p

NðsÞ ¼ c0 ðec1 jsj  c2 Þ

410

where k > 0, a > 0, c0 > 0, c1 > 0, 1 > c2 > 0, and p > 0. ðay þ NðsÞÞsignðsÞ can suppress chattering, and ks can accelerate convergence. By selecting the novel reaching law in Eq. (32) in this section, we can obtain S_ ¼ kS  Qðay þ NðSÞÞ

ð33Þ

where c0 , c00 , and c1 are positive constants; 1 > c2 > 0 and a > 1.     y_ 1 ajs1 j y_ ¼ ¼ ; yi ð0Þ > 0ði ¼ 1; 2Þ y_ 2 ajs2 j " NðSÞ ¼

# p c0 ðec1 js1 j  c2 Þ ; p c00 ðec1 js2 j  c2 Þ



0 signðs1 Þ Q¼ 0 signðs2 Þ

411 412 413 414 415 416 418 419 420 421

423 424



426

Then, a new finite-time adaptive guidance law for a missile intercepting maneuvering targets with impact angle constraints can be defined as: 1

u2 ¼ B ½N þ G þ kS þ Qðay þ NðSÞÞ

ð34Þ

427 428 429 430 432 433

Theorem 2. Considering the systems in Eqs. (1)–(7), suppose that the external disturbance M is bounded. When Eq. (14) is selected as the sliding mode surface and Eq. (34) as the guidance law, the following conclusions are satisfied.

ð31Þ

434 435 436 437 438

(i) The sliding manifold S converges to zero in finite time. (ii) The LOS tracking error angles hL  hLf and /L  /Lf converge to the regions jhL  hLf j < g and j/L  /Lf j < g in finite time, where g is a small positive constant. (iii) The LOS angular rates h_ L and /_ L converge to the regions jh_ L j < g1 and j/_ L j < g1 in finite time, in which

g1 ¼ ug þ bg

r

 Proof. Assuming that jm1 j 6 e1 and jm2 j 6 e2 , then e ¼

439 440 441 442 443 444 445

ð35Þ

446 447 449

 e1 . e2

450

The Lyapunov function candidate can be expressed as

451 452

Fig. 2

Curves of g1 and r with different values of b.

1 1 V3 ¼ ST S þ ðe  yÞT ðe  yÞ 2 2

ð36Þ

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Y. Si, S. Song

The time derivative of V3 along with Eqs. (1)–(7) results in V_ 3 ¼ S S_  ðe  yÞ y_ T

T



¼ ST ½M  kS  Qðay þ NðSÞÞ  ðe  yÞT

ajs1 j ajs2 j



  2 X ajs1 j jsi jei  ST Qðay þ NðSÞÞ  ðe  yÞT 6 kST S þ ajs2 j i¼1 6 kST S þ

2 2 X X jsi jei  ST QNðSÞ  jsi jaei i¼1

i¼1

2 X ¼ k1 ST S  ST QNðSÞ  jsi jei ða  1Þ i¼1

458 459 460 461 462

463 464 465 467 468 469 470

60 From the above inequality, V3 ðtÞ 6 V3 ð0Þ, indicating that V3 ðtÞ is bounded. Therefore, it can be concluded that si and ei  yi ði ¼ 1; 2Þ are all bounded, i.e., yi is bounded. Then, a positive  constant di exists so that yi 6 di and jmi j 6 di . d d ¼ 1 . In the case S – 0, consider another Lyapunov funcd2 tion as 1 1 V4 ¼ ST S þ ðd  yÞT ðd  yÞ 2 2l

ð37Þ

where 0 < l < 1. The time derivative of V4 along with Eqs. (1)–(7) results in 1 V_ 4 ¼ ST S_  ðd  yÞT y_ l 1 ¼ ST ½M  kS  Qðay þ NðSÞÞ  ðd  yÞT y_ l 2 X 1 jsi jdi  ST Qðay þ NðSÞÞ  ðd  yÞT y_ 6 kST S þ l i¼1 2 2 X X 1 T jsi jdi  ST QNðSÞ  jsi jayi  ðd  yÞT y_ ¼ kS S þ l i¼1 i¼1 2 2 X X T T jsi jdi  jsi jayi 6 kS S  S QNðSÞ þ  X 2 2 X a a  jsi jaðdi  yi Þ  jsi jðdi  yi Þ l i¼1 i¼1 2 X ¼ kST S  ST QNðSÞ  ða  1Þ jsi jdi i¼1



i¼1

i¼1

473 474 475 476 477 478

Remark 3. The integration term y in the guidance law of Eq. (34) can compensate a perturbation with unknown bounds and force the sliding variable to converge to the sliding surface in finite time. The exponential term NðSÞ is a sufficiently high gain. When the state is far away from the sliding surface, the time of the compensating phase can be very short, i.e., this term can accelerate the system’s response to perturbations. When S ! 0, the integration term gradually slows down until it stops growing. Moreover, the value of the exponential term decreases rapidly until it disappears at the sliding surface. On reaching the sliding surface, the overall gain can be reduced. In other word, this method can reduce the unwanted chattering level. As a result, when c2 is closer to 1, the reaching law has a better performance. Notably, c2 cannot be equal to 1; otherwise, the inequality Eq. (38) may not satisfy Lemma 1.

479

We have designed a guidance law that can deal with external disturbances. However, there is still a problem: the guidance laws in Eqs. (25) and (34) cannot guarantee that the states of the systems converge to zero. To deal with this problem, based on Theorem 2 and inspired by Refs.36,37, another new reaching law is introduced by adding k1 -signðsÞ to the reaching law in Eq. (32) as

494

s_ ¼ ks  ðk1 - þ ay þ NðsÞÞsignðsÞ ( -¼

ð39Þ

480 481 482 483 484 485 486 487 488 489 490 491 492 493

495 496 497 498 499 500 501 503 504

1 jsj jsj

–0

0jsj ¼ 0

506 507 509

y_ ¼ ajsj; yð0Þ > 0

510 p

NðsÞ ¼ c0 ðec1 jsj  c2 Þ

512

where k > 0, k1 > 0, a > 0, c0 > 0, c1 > 0, 1 P c2 > 0, and p > 0. ðay þ NðsÞÞsignðsÞ can suppress chattering, and ks can accelerate convergence. Moreover, k1 -signðsÞ ensures that the systems are asymptotically and finite-time stable. By selecting the novel reaching law in Eq. (39), we can obtain

X 2 a a  jsi jðdi  yi Þ l i¼1  X 2 a T a jsi jðdi  yi Þ 6 S QNðSÞ  l i¼1  p p 6  min c0 ðec1 js1 j c2 Þ; ðc00 ec1 js2 j  c2 Þ kSk a kSk  akSk kd  yk  l  1 pffiffiffi  1 T 2 p p 6  2 min c0 ðec1 js1 j  c2 Þ; ðc00 ec1 js2 j  c2 Þ S S 2 12   pffiffiffiffiffiffi a 1  2l kSk  akSk ðd  yÞT ðd  yÞ l 2l

p ffiffiffi  p p 6  min 2 min c0 ðec1 js1 j  c2 Þ; ðc00 ec1 js2 j  c2 Þ ;    pffiffiffiffiffiffi a 1 1  a kSk V24 ¼ rV24 2l l 472

l always exists so that la  a > 0, i.e., r > 0. According to Lemma 1, V4 converges to zero in finite time. Furthermore, S also converges to zero in finite time. Thus, Conclusion (i) is proven. The rest of the steps are the same as those of Theorem 1. Therefore, the conclusions of Theorem 2 can be easily obtained. h

S_ ¼ kS  k1 -signðSÞ  Qðay þ NðSÞÞ ( -¼

ð40Þ

513 514 515 516 517 518 519 521 522

1 kSk kSk

–0

0kSk ¼ 0 

y_ 1 y_ ¼ y_ 2



" NðSÞ ¼

524



 ajs1 j ¼ ; yi ð0Þ > 0; i ¼ 1; 2 ajs2 j #

  p 0 signðs1 Þ c0 ðec1 js1 j  c2 Þ ; Q ¼ p 0 signðs2 Þ c00 ðec1 js2 j  c2 Þ

ð38Þ

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525

527 528

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531 532 533 534

Then, a new finite-time adaptive guidance law for a missile intercepting maneuvering targets with impact angle constraints can be defined as u3 ¼ B1 ðN þ G þ kS þ k2 x þ k1 -signðSÞ þ Qðay þ NðSÞÞÞ ð41Þ

536 537 538

1 V_ 6 ¼ ST S_  ðd  yÞT y_ þ k2 xT x l   signðSÞ  Qðay þ NðSÞÞ ¼ ST M  kS  k1 kSk 1  ðd  yÞT y_  k2 ST x þ k2 xT x_ l

where k, k1 , a, c0 , c00 , and c1 are positive constants, 1 P c2 > 0, and a > 1.

6 kST S  k1 ST

539

540 541 542 543 544

1  ðd  yÞT y_  k2 ST x þ k2 xT x_ l

Theorem 3. Considering the systems in Eqs. (1)–(7), suppose that the external disturbance M is bounded. Both the LOS angular rate and LOS angle error can converge to zero while being asymptotically and finite-time stable, if Eq. (14) is selected as the sliding mode surface and Eq. (41) as the guidance law.

¼ kST S  k1 ST

545

 546

Proof. Consider the Lyapunov function candidate as

550 551 552

1 1 k2 V5 ¼ ST S þ ðe  yÞT ðe  yÞ þ xT x 2 2 2

ð42Þ

2 X signðSÞ  ST QNðSÞ þ jsi jdi kSk i¼1  X 2 2 2 X X a a  jsi jayi  jsi jaðdi  yi Þ  jsi jðdi  yi Þ l i¼1 i¼1 i¼1

V_ 5 ¼ ST S_  ðe  yÞT y_ þ k2 xT x_ h i ¼ ST M  kS  k1 signðSÞ  Qðay þ NðSÞÞ kSk "

ðe  yÞ

ajs1 j

 k2 ðux þ bfðxÞÞT x

#

¼ kST S  k1 ST

 k2 S x þ k2 xT x_ T

ajs2 j

ðe  yÞ

T

ajs1 j ajs2 j



6 k1 ST

T

 k2 ðx_ þ ux þ bfðxÞÞ x þ k2 xT x_

6 kST S  k1 ST signðSÞ þ kSk

2 X

2 X

i¼1

i¼1

jsi jei  ST QNðSÞ 

jsi jaei

 ST QNðSÞ  ¼ kST S  k1 ST signðSÞ kSk k2 ðux þ bfðxÞÞT x 6 0

560

561 562

ð45Þ

t0

V6 ðtÞ  0; 8t P t



570 571 572

ð46Þ

Because V6 ðtÞ P 0 and V6 ðtÞ is a monotone decreasing function, we have ð47Þ

where

556

559

ð44Þ

Integrating V_ 6 6 k1 from t0 to t yields Z t Z t V_ 6 dt 6 V6 ðt0 Þ  V6 ðtÞ 6 k1 dt ¼ k1 ðt  t0 Þ

V6 ðtÞ  V6 ðt0 Þ 6 k1 ðt  t0 Þ:

2 X jsi jei ða  1Þ

555

558

6 k1

Then

i¼1

557

signðSÞ kSk

t0

k2 ðux þ bfðxÞÞT x

554

X 2 a a jsi jðdi  yi Þ  k2 ðux þ bfðxÞÞT x l i¼1

i¼1

#

2 X signðSÞ  ST QNðSÞ  ða  1Þ jsi jdi kSk i¼1



2 X þ jsi jei  ST Qðay þ NðSÞÞ 6 kST S  k1 ST signðSÞ kSk

"

2 X 1 jsi jayi  ðd  yÞT y_ l i¼1

¼ kST S  k1 ST

The time derivative of V5 along with Eqs. (1)–(7) results in

T

2 signðSÞ X þ jsi jdi  ST QNðSÞ kSk i¼1

 k2 ðx_ þ ux þ bfðxÞÞT x þ k2 xT x_

547 549

2 signðSÞ X þ jsi jdi  ST Qðay þ NðSÞÞ kSk i¼1

V6 ðt0 Þ k1

From the above inequality, V5 ðtÞ 6 V5 ð0Þ, indicating that V5 ðtÞ is bounded. Therefore, it can be concluded that si and ei  yi ði ¼ 1; 2Þ are all bounded, i.e., yi is bounded. Then, a positive constant di exists so that yi 6 di and jmi j 6 di .   d d ¼ 1 . For the case S – 0, another Lyapunov function d2 can be considered as

t ¼ t0 þ

1 1 k2 V6 ¼ ST S þ ðd  yÞT ðd  yÞ þ xT x 2 2l 2

Remark 4. In Theorem 3, the guidance law u3 ensures that the LOS angular rate and LOS angle error converge to zero while being asymptotically and finite-time stable. Compared to u2 , c2 can be equal to 1 by adopting the newly designed reaching law

Because V6 ðtÞ  0, the sliding modes S and x converge to 0 when t P t . Then, according to Eq. (14), x_ converges to 0 in finite time. In addition, V_ 6 6 k1 ensures that the system is asymptotically and finite-time stable. Therefore, the conclusions of Theorem 3 can be easily obtained. h

574 575 576 578 579 580 581 583 584 585 587 588 589 590 591 592

563 565 566 567

ð43Þ

where 0 < l < 1. The time derivative of V4 along with Eqs. (1)–(7) results in

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8 597 598 599

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Y. Si, S. Song

in Eq. (39). Moreover, the time of system states arriving at the equilibrium point can be accurately calculated using Theorem 3, but can only be estimated using Theorems 1 and 2. Remark 5. In fact, the conclusions of Theorem 3 can only be obtained theoretically. The main reason is as follows. The value of the guidance law increases when the sliding surface approaches to 0, but the capacity of dynamic actuators is limited in practice. To avoid similar problems, the traditional approach often uses signðsÞ instead of signðsÞ , where @ is a small kskþ@ ksk positive constant. However, to some extent, the convergence precision is sacrificed. Generally speaking, there is a trade-off between practicality and good performance. Remark 6. Note that the guidance laws proposed in this paper are discontinuous because of the presence of a signum function, leading to the undesired chattering phenomenon. To weaken this phenomenon, a continuous saturation function satðsÞ is given as Eq. (48). This function is used to approximate the signum function, i.e., Eq. (48) can be used to replace the signum function.   satðs1 Þ satðSÞ ¼ ð48Þ satðs2 Þ

621

8 > < 1; si > h satðsi Þ ¼ si =h; jsi j 6 h i ¼ 1; 2 > : 1; si < h

622

4. Simulation results

623

In this part, numerical simulations are carried out to verify the performance of the designed guidance laws. To illustrate the effectiveness of the designed guidance laws, the two cases shown below were selected for different target accelerations.

619

624 625 626 627

628 629

ð49Þ

Case 1. ayt ¼ azt ¼ 19:6 cosð2tÞ m=s2 . Case 2. azt ¼ 19:6 m=s2 and ayt ¼ 19:6 m=s2 .

630 631 632 633 634

636

To analyze the superiority of the proposed guidance laws, the PNGL and NTSMGL22 were compared. The PNGL can be expressed as " # N1 R_ h_ L ð50Þ nc ¼ N2 R_ /_ L

Table 1

Initial conditions for missile and target.

Dataset

Dataset 1

Dataset 2

Rð0Þ ðmÞ hL(0) (°) /L(0) (°) hm(0) (°) /m(0) (°) Vm ðm=sÞ ht(0) (°) /t(0) (°) Vt ðm=sÞ

12,000 30 0 10 10 850 20 180 510

9000 20 60 25 30 600 10 160 300

The nonsingular terminal sliding mode surface is given as   r1 1 ¼ x þ x_ p=q r¼ ð51Þ b r2 where b > 0 is a constant. p > 0 and q > 0 are odd constants, following the condition of p>q The corresponding NTSMGL is given as   bq u0 ¼ B1 N þ x_ 2p=q þ k0 r þ H0 sgnðrÞ p

ð52Þ

637 638

640 641 642 643 645 646 647

ð53Þ

Two sets of initial engagement parameters are considered, and the details are given in Table 1. The capacity of dynamic actuators is limited in practice; therefore, the maximum lateral accelerations are assumed to be limited as  aM max signðaM Þ if jaM j P aM max aM ¼ aM if jaM j < aM max where aMmax ¼ 25g, i.e., the maximum allowable value of the missile acceleration is assumed to be 25g, and g is the acceleration of gravity (g ¼ 9:8 m=s2 ). The parameters in the guidance law u0 are selected as follows: b ¼ 2, p ¼ 9, q ¼ 7, k0 ¼ 0:5, u ¼ 0:5, and   1:5 0 H0 ¼ . The parameters N1 and N2 in the PNGL 0 1:5 are selected as N1 ¼ 7 and N2 ¼ 7, respectively.

649 650 651 652 653 654 655

657 658 659 660 661 662

663 664

4.1. Simulation results under u1

665

The parameters of the first finite-time controller u1 are selected as follows: g ¼ 0:025, r ¼ 0:2, b ¼ 0:7, k ¼ 1, and   1:138 0 H¼ . The parameter in the saturation func0 1:382 tion is selected as h ¼ 0:04. For Case 1, When ayt ¼ azt ¼ 19:6 cosð2tÞ m=s2 and the initial engagement parameters are considered as Dataset 1 in Table 1, the simulation results under u1 are shown in Fig. 3. Fig. 3(a) shows the relative movement trajectories under three different guidance laws, and the missile can intercept the target successfully for Case 1, even though the flight paths of the missile are different. Fig. 3(b) clearly shows that both u1 and u0 ensure the convergences of hL and /L to the desired values. However, the PNGL cannot guarantee the convergence of the LOS angles. In addition, clearly the LOS angles under u1 have a faster convergence speed and better performance than those under u0 . Fig. 3(c) shows the curves of h_ L and /_ L . Similarly, still a good convergence performance was observed under u1 . As shown in Fig. 3(d), under the three different guidance laws, accelerations of the missile are in reasonable scales. For u1 and u0 , an acceleration saturation phenomenon exists, which lasts for about 4 s. Further, clearly the accelerations of the missile produced under the PNGL are smaller than those under u1 or u0 . However, a large acceleration of the missile under u1 and u0 ensures that the LOS angles and angular rates have preferable convergence performances. For Case 2, Fig. 4 shows the simulation results under u1 with azt ¼ ayt ¼ 19:6 m=s2 , and the initial engagement parameters are considered as Dataset 2 in Table 1. Fig. 4(a) clearly

666

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Fig. 3

694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711

9

Responses under u1 with Case 1.

shows that the target is maneuvering at a new way, and the missile can still intercept it successfully under three different guidance laws. Fig. 4(b) shows the curves of hL and /L . The LOS angles hL and /L converge to the desired values under both u1 and u0 . Fig. 4(c) shows the curves of h_ L and /_ L . Figs. 4 (b) and (c) clearly show that the dynamic systems have the best convergence performances under u1 . Fig. 4(d) shows the missile acceleration profiles. The missile acceleration curves shown in Fig. 4(d) converge to a constant within 5 s. In conclusion, the control strategy u1 also performs very well when the target is maneuvering in different ways. Furthermore, in Ref.22, the designed guidance law was simulated and verified only for nonmaneuvering targets. In this paper, simulation experiments were carried out for targets with different accelerations. The simulation results show that the performance of guidance laws obtained in this study is obviously better than that of the guidance law designed in Ref.22. This also validates that the guidance laws designed in this study have a

faster convergence speed and stronger robustness than the PNGL and the guidance law reported in Ref.22. To further verify the effectiveness of the guidance laws, simulation results for different desired impact angles, but with the same initial LOS angles, are presented. The initial conditions are given as Dataset 1 in Table 1. Let the desired impact angles hLf be 5 , 15 , 25 , 35 , 45 , and 60 . Let the desired impact angles /Lf be 25 , 15 , 5 , 5 , 15 , and 30 . Fig. 5 shows the trajectories of the missile and the target and the curves of the LOS angle hL . Fig. 5(a) shows that the missile can intercept the target successfully for different desired impact angles. Fig. 5(b) shows that when /Lf ¼ 5 and hLf ¼ 5 , 15 , 25 , 35 , 45 , and 60 , hL can converge to different desired angles precisely. However, the convergence time of hL is significantly affected by different desired impact angles. Using the same parameters, the greater the difference between the desired and initial angles is, the slower the convergence rate of hL is. That is because the value of the adaptive term increases with

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Fig. 4

Fig. 5

Responses under u1 with Case 2.

Responses under u1 with Dataset 1 when /L ¼ 5 and hL ¼ 5 ; 15 ; 25 ; 35 ; 45 ; 60 .

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Fig. 6

Responses under u1 with Dataset 1 when hL ¼ 35 and /L ¼ 25 ; 15 ; 5 ; 5 ; 15 ; 30 .

739

time. Therefore, it requires much more time to compensate the large errors between the initial and desired values. As shown in Fig. 5(b), the optimal desired LOS angles of the proposed guidance law u1 approach hLf ¼ ½5 ; 60 . Fig. 6 shows the responses under u1 with Dataset 1 when hLf ¼ 35 and /Lf ¼ 25 , 15 , 5 , 5 , 15 , and 30 . /L also converges to different desired angles precisely, even though the convergence time is different. In addition, the optimal desired LOS angles of the proposed guidance law u1 approach /Lf ¼ ½25 ; 30 .

740

4.2. Simulation results under guidance law u2

730 731 732 733 734 735 736 737 738

741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768

11

Similar to Section 4.1, the PNGL in Eq. (50) and the guidance law u0 are still selected for comparison. To validate u2 , the selected parameters are g ¼ 0:025, r ¼ 0:2, b ¼ 0:7, k ¼ 1, a ¼ 2, c0 ¼ 4:849, c00 ¼ 5:259, c1 ¼ 0:1, c2 ¼ 0:99, and p ¼ 0:1. The parameter in the saturation function is selected as h ¼ 0:04. For Case 1, Fig. 7 shows the simulation results under the guidance law u2 with ayt ¼ azt ¼ 19:6 cosð2tÞ m=s2 , and the initial engagement parameters are considered as Dataset 1 in Table 1. Fig. 7(a) shows the relative movement trajectories under u2 , u0 , and the PNGL. Clearly, the missile can intercept the target successfully under u2 . Fig. 7(b)–(c) show the curves of the LOS angles and angular rates. Both the guidance laws u0 or u2 ensure the convergence of the LOS angles to the expected values, and the LOS angular rates converge to zero. Under u2 , u0 , and the PNGL, Fig. 7(d) shows the curves of the missile accelerations, which are within reasonable bounds. Fig. 7(e) shows the adaptive value curves. Clearly, the dynamic systems under u2 have a faster convergence speed and better performance than those under u0 or the PNGL. For Case 2, Fig. 8 shows the simulation results under the second guidance law u2 with azt ¼ ayt ¼ 19:6 m=s2 , and the initial engagement parameters are considered as Dataset 2 in Table 1. Fig. 8(a) shows the relative movement trajectories under three different guidance laws. Fig. 8(b)–(c) show the curves of hL , /L , h_ L , and /_ L . Fig. 8(d) shows the missile acceleration profiles. Fig. 8(e) shows the adaptive value curves. The comparison results are the same as those described in the pre-

vious section. Therefore, the remaining analyses are omitted here. According to the above analyses, u2 has a better performance than the PNGL and the NTSMGL. Fig. 9 shows the responses under u2 with Dataset 1 when /Lf ¼ 5 and hLf ¼ 5 , 15 , 25 , 35 , 45 , and 60 . Fig. 10 shows the responses under u2 with Dataset 1 when hLf ¼ 35 and /Lf ¼ 25 , 15 , 5 , 5 , 15 , and 30 . The situations are similar to those shown in Figs. 5 and 6, thus not repeated here. To further prove the superiority of the chattering elimination of u2 , the sliding mode guidance law u1 is selected for comparison. Notably, the signum function is no longer replaced by a saturation function in this part. Fig. 11(a) and (b) show the comparison results of u1 and u2 including the curves of sliding mode surfaces. The sliding mode surface profiles shows a severe chattering phenomenon under u1 , i.e., the undesired chattering is reduced effectively by the guidance law u2 . The comparison results between u1 and u2 show that on one hand, the integration term y in u2 can compensate a perturbation with unknown bounds and force the sliding variable to converge to the sliding surface in finite time. On the other hand, the magnitude of y can be adjusted according to the value of the sliding mode surface, reducing the unwanted chattering level. This is sufficient to validate the good performance and superiority of u2 .

769

4.3. Simulation results under guidance law u3

794

Similar to Sections 4.1 and 4.2, the PNGL and u0 are selected for comparison to verify the effectiveness of u3 . The selected parameters in u3 are as follows: g ¼ 0:25, r ¼ 0:2, b ¼ 7, k ¼ 0:5, k1 ¼ 0:1, k2 ¼ 0:3, a ¼ 2, c0 ¼ 4:849, c00 ¼ 5:259, c1 ¼ 0:1, c2 ¼ 1, and p ¼ 0:1. The parameter in the saturation function is selected as h ¼ 0:05. To reduce chatsignðsÞ tering, when ksk < 0:01, kskþ0:01 is used to replace signðsÞ . ksk For Case 1, Fig. 12 shows the simulation results under u3 for Case 1, and the initial engagement parameters are considered as Dataset 1 in Table 1. Fig. 12(a) shows the relative movement trajectories under u3 , u0 , and the PNGL, and the missile can hit the target successfully under the three different guidance laws. The relative movement curves under u3 are similar to

795

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770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793

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Y. Si, S. Song

Fig. 7

Responses under u2 with Case 1.

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Fig. 8

13

Responses under u2 with Case 2.

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Fig. 9

Fig. 10

Responses under u2 with Dataset 1 when /Lf ¼ 5 and hLf ¼ 5 ; 15 ; 25 ; 35 ; 45 ; 60 .

Responses under u2 with Dataset 1 when hLf ¼ 35 and /Lf ¼ 25 ; 15 ; 5 ; 5 ; 15 ; 30 .

Fig. 11 808 809 810 811 812

Comparison between u1 and u2 with azt ¼ ayt ¼ 20g.

those under u0 , but are quite different from those under the PNGL. Fig. 12(b) and (c) show the curves of the LOS angles and angular rates. Clearly, the guidance systems have a faster convergence speed under u3 than those under u0 and the PNGL. Fig. 12(d) shows the curves of the missile accelerations under

the three different guidance laws, which are within reasonable bounds. Fig. 12(e) shows the adaptive value curves. From the above analysis, clearly the dynamic systems under u3 have a faster convergence speed and better performance than those under u0 or the PNGL.

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Fig. 12

15

Responses under u3 with Case 1.

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Fig. 13

Responses under u3 with Case 2.

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Fig. 14

Fig. 15

Table 2

818 819 820 821 822 823 824 825 826

17

Comparison between u1 and u3 with azt ¼ ayt ¼ 20g.

Effect of gain on the performance of the guidance law.

Interception time, miss distance, and LOS angle error.

Guidance laws

Case 1

Case 2

Interception time (s)

Miss distance (m)

Error of hL ð Þ

Error of /L ð Þ

Interception time (s)

Miss distance (m)

Error of hL ð Þ

Error of /L ð Þ

PNGL NTSMGL u1 u2 u3

9.159 9.335 9.353 9.345 9.361

0.1597 0.0830 0.0022 0.0107 0.0444

4.8391 0.0992 0.0611 0.0561 0.0584

3.9930 0.2484 0.0841 0.0776 0.0812

11.109 11.289 11.359 11.364 11.383

0.1057 0.3036 0.0927 0.0233 0.0115

5.2086 0.7790 0.0815 0.0873 0.0772

9.5773 0.9643 0.0743 0.0774 0.0713

For Case 2, Fig. 13 shows the simulation results under u3 with azt ¼ ayt ¼ 19:6 m=s2 , and the initial engagement parameters are considered as Dataset 2 in Table 1. Fig. 13(a) shows the relative movement trajectories under three different guidance laws. Figs. 13(b) and 13(c) show the curves of hL , /L , h_ L , and /_ L . Fig. 13(d) shows the missile acceleration profiles. Fig. 13(e) shows the adaptive value curves. The comparison results are the same as described in the previous sections. Therefore, the remaining analyses are omitted here.

According to the above analysis, the proposed adaptive guidance law u3 has a better performance than those of the PNGL and the NTSMGL. The simulation results of different desired impact angles under u3 are similar to those shown in Figs. 5 and 6, thus not repeated here. Similar to Section 4.2, u1 is still selected to compare with u3 . Notably, the signum function is no longer replaced by the saturation function in this part. Fig. 14(a) and (b) show the comparison results between u1 and u3 . The sliding mode

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surface profiles show a severe chattering phenomenon under u1 . In u3 , weakening of undesired chattering was caused by adding Qðay þ NðSÞÞ to the reaching law. This is sufficient to demonstrate the good performance and superiority of u3 . In addition, the effect of k1 on the value of the guidance law is analyzed by simulation, and the results are shown in Fig. 15. Fig. 15(a) shows that k1 is too large and would cause an overshoot phenomenon. Fig. 15(b) shows that the duration of the saturation phenomenon would increase with an increase in the numerical value of k1 . Therefore, a route to obtain a better convergence performance is by adjusting k1 . Table 2 shows the precise data including interception time, miss distance, and LOS angle error. Although the guidance laws are different, the interception time is similar. Further, the comparison between different guidance laws under the two cases shows that the miss distances and LOS angle errors derived under u1 , u2 , and u3 are much less than those derived under u0 and the PNGL. Therefore, u1 , u2 , and u3 ensure the guidance accuracy and success rate of the interception. Moreover, a high guidance precision can be achieved even in the case of an existing strong target maneuver.

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For a missile intercepting maneuvering targets in a 3D engagement scene, this study has proposed three novel fastconvergence guidance laws with terminal impact angle constraints. To intercept maneuvering targets, all the three guidance laws designed in this study ensure that the LOS angular rate and LOS angle are finite-time stabilized. The details can be summarized as follows:

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(1) The first guidance law is developed based on the fast nonsingular terminal sliding mode control theory. This guidance law can compensate target acceleration with a known upper bound. (2) Based on the newly designed reaching law, an adaptive sliding mode guidance law is provided. Besides, this guidance law can deal with the problem of target acceleration with unknown bounds. (3) Based on another newly designed reaching law, the third guidance law is developed. This guidance law can ensure that the LOS angular rate and LOS angle are asymptotically and finite-time stable. (4) The theoretical proofs and simulation results have verified that the three guidance laws designed in this study are effective and superior.

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Acknowledgements

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This study was co-supported by the National Natural Science Foundation of China (No. 61333003) and the China Aerospace Science and Technology Innovation Foundation (No. JZ20160008).

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