Constrained adaptive neural network control of an MIMO aeroelastic system with input nonlinearities 1 2 4 5 6 7 8 9 11 12 13 14 15 16 17 18 19 20 21 Chinese Journal of Aeronautics, (2017), xxx(xx) xxx[.]
CJA 790 17 February 2017 Chinese Journal of Aeronautics, (2017), xxx(xx): xxx–xxx No of Pages 11 Chinese Society of Aeronautics and Astronautics & Beihang University Chinese Journal of Aeronautics cja@buaa.edu.cn www.sciencedirect.com FULL LENGTH ARTICLE Constrained adaptive neural network control of an MIMO aeroelastic system with input nonlinearities Gou Yiyong, Li Hongbo, Dong Xinmin *, Liu Zongcheng Aeronautics and Astronautics Engineering College, Air Force Engineering University, Xi’an 710038, China Received 20 April 2016; revised September 2016; accepted 28 November 2016 11 12 KEYWORDS 13 Aeroelastic system; Constrained control; Flutter suppression; Input nonlinearities; RBFNNs 14 15 16 17 Abstract A constrained adaptive neural network control scheme is proposed for a multi-input and multi-output (MIMO) aeroelastic system in the presence of wind gust, system uncertainties, and input nonlinearities consisting of input saturation and dead-zone In regard to the input nonlinearities, the right inverse function block of the dead-zone is added before the input nonlinearities, which simplifies the input nonlinearities into an equivalent input saturation To deal with the equivalent input saturation, an auxiliary error system is designed to compensate for the impact of the input saturation Meanwhile, uncertainties in pitch stiffness, plunge stiffness, and pitch damping are all considered, and radial basis function neural networks (RBFNNs) are applied to approximate the system uncertainties In combination with the designed auxiliary error system and the backstepping control technique, a constrained adaptive neural network controller is designed, and it is proven that all the signals in the closed-loop system are semi-globally uniformly bounded via the Lyapunov stability analysis method Finally, extensive digital simulation results demonstrate the effectiveness of the proposed control scheme towards flutter suppression in spite of the integrated effects of wind gust, system uncertainties, and input nonlinearities 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/) 18 19 Introduction 20 In the past, aeroelasticity has attracted increasing concern in aircraft design Aeroelastic systems exhibit a variety of unsta- 21 * Corresponding author E-mail addresses: gouyiyong@139.com (Y Gou), dongxinmin@139 com (X Dong) Peer review under responsibility of Editorial Committee of CJA Production and hosting by Elsevier ble phenomena as a result of the mutual interaction of structural, inertia and aerodynamic forces.1 Divergence, flutter, and limit-cycle oscillation are typical unstable phenomena which can degrade an aircraft’s flight performance, and even cause flight mission failure.1,2 Thus, a reliable and effective control strategy becomes one of the key issues in aeroelastic system control design In previous studies, researchers have analyzed the nonlinear responses of aeroelastic systems, and various control schemes have been extensively studied Based on the l method, Lind and Brenner have analyzed the unstable responses of aeroelastic systems and studied robust stability margins To study different aeroelastic phenomena, the NASA Langley Research Center has developed a benchmark http://dx.doi.org/10.1016/j.cja.2017.01.006 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: Gou Y et al Constrained adaptive neural network control of an MIMO aeroelastic system with input nonlinearities, Chin J Aeronaut (2017), http://dx.doi.org/10.1016/j.cja.2017.01.006 22 23 24 25 26 27 28 29 30 31 32 33 34 CJA 790 17 February 2017 No of Pages 11 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 active control technology (BACT) wind-tunnel model.4 For this BACT wind-tunnel model, several control laws for flutter suppression have been developed.4–6 Considering nonlinear structural stiffness, a model equipped with a single trailingedge (TE) control surface has been developed at Texas A&M University.7 Based on this model, a wide variety of control schemes have been designed.8–11 Inspired by the limited effectiveness of a single TE control surface, a wing section equipped with a leading-edge (LE) control surface and a TE control surface has been designed, and a large number of control schemes has been proposed.12–16 For this wing section with uncertainties, adaptive control has been widely used to suppress flutter.13–15 Neural network control and adaptive control have been developed in this filed and compared in control performance.13 With respect to external disturbance and uncertainties, Wang et al.14 designed an output feedback adaptive controller coupled with an SDU decomposition which avoids the singularity problem arising from estimation of the input matrix Accounting on the input saturation problem, Lee and Singh15 used an auxiliary dynamic system to compensate for the input saturation and proposed a novel control scheme In addition, a sliding mode control method was also applied to flutter suppression, and Lee and Singh16 have designed a higher-order sliding mode controller which accomplished the finite-time flutter suppression of the aeroelastic system It is well known that input nonlinearities exist in a real control system, and an aeroelastic control system is no exception Both input dead-zone and saturation are considered for the uncertain aeroelastic system in this paper Input saturation and dead-zone may induce deterioration of the aeroelastic control system performance, and even make the aeroelastic control system fail Consequently, input saturation and deadzone have attracted much attention Input dead-zone could induce a zero input against a range of set values.17 An adaptive dead-zone inverse approach was proposed to tackle a system with input dead-zone.18 An adaptive fuzzy output feedback control law, which treats dead-zone inputs as system uncertainties, has been developed.19 For the input saturation problem, Chen et al.20 designed an auxiliary system, whose input was the error between the saturation input and the desired control input, to compensate for the impact of the input saturation Li et al.21 proposed an adaptive fuzzy output feedback control for output constrained nonlinear systems In general, some researchers have also studied in integrating input deadzone with saturation For uncertain multi-input and multioutput (MIMO) nonlinear systems with input nonlinearities, a robust adaptive neural network control was developed.17 Yang and Chen22 regarded input dead-zone and saturation nonlinearities as a new input saturation problem through a dead-zone inverse approach, and proposed an adaptive neural prescribed performance control law for near-space vehicles Motivated by the above discussion, a constrained adaptive neural network control scheme is proposed for an MIMO aeroelastic system with wind gust, system uncertainties, and input nonlinearities Different from the previous references, it is especially noted that uncertainties in pitch stiffness, plunge stiffness, and pitch damping are all considered Inspired by Ref 22, the right inverse function block of the dead-zone is added before the input nonlinearities, by which the input nonlinearities can be regarded as a new input saturation.22 To handle the new input saturation, an auxiliary error system is designed to compensate for the impact of the input saturation Y Gou et al Radial basis function neural networks (RBFNNs) are also applied to approximate the system uncertainties A novel constrained adaptive control law is developed by using the backstepping control technique The simulation results of the MIMO aeroelastic control system are presented to verify that the proposed control scheme can accomplish flutter suppression despite the effects of wind gust, system uncertainties, and input nonlinearities 97 Nonlinear aeroelastic model and preliminary 105 2.1 Nonlinear aeroelastic model 106 A two-degree-of-freedom (2-DOF) wing section equipped with LE and TE control surfaces is presented in Fig 1.15 The second-order differential equations signifying the dynamic of this aeroelastic system are given by13,14 _ Ia mw x a b a ca aị a_ ỵ mw xa b mt ch h_ h€ ð1Þ Mg a M ka aị ẳ ỵ ỵ Lg kh hị h L 107 where a denotes the pitch angle which is positive upward; h denotes the plunge displacement which is positive downward; Ia is the moment of inertia; mw and mt are the wing section mass and the total mass, respectively; xa is the distance between the center of mass and the elastic axis; b is the semichord of the wing; ch is the plunge damping coefficient; especially note that uncertainties in pitch stiffness, plunge stiffness, and pitch damping are all considered, which is different from the previous references In a polynomial form, the pitch damp_ the pitch stiffness ka ðaÞ, and the plunge stiffness ing ca ðaÞ, kh ðhÞ are expressed as follows _ ẳ ca0 ỵ ca1 a_ ỵ ca2 a_ > < ca ðaÞ ð2Þ ka ðaÞ ẳ ka0 ỵ ka1 a ỵ ka2 a2 > : kh hị ẳ kh0 ỵ kh1 h ỵ kh2 h2 114 where caj , kaj and khj (j ¼ 0; 1; 2) are assumed to be unknown constants Fig Aeroelastic system with LE and TE control surfaces.15 Please cite this article in press as: Gou Y et al Constrained adaptive neural network control of an MIMO aeroelastic system with input nonlinearities, Chin J Aeronaut (2017), http://dx.doi.org/10.1016/j.cja.2017.01.006 98 99 100 101 102 103 104 108 109 110 111 113 115 116 117 118 119 120 121 122 123 124 125 127 128 129 CJA 790 17 February 2017 No of Pages 11 Constrained adaptive neural network control 130 131 132 134 135 136 137 138 139 140 141 142 143 145 146 147 148 149 150 152 153 154 155 156 157 159 160 161 162 163 164 165 166 167 168 169 171 172 173 In Eq (1), M and L represent the aerodynamic moment and lift in a quasi-steady form expressed by13 h i 1 2 _ > _ M ¼ qU b C s a ỵ h=Uị ỵ a b a=Uị m p > a-eff > > > > < þqU2 b2 Cmb-eff sp b þ qU2 b2 Cmc-eff sp c ð3Þ h i > _ > _ ỵ 12 a ba=Uị > L ẳ qU bCla sp a ỵ h=Uị > > > : ỵqU2 bClb sp b ỵ qU2 bClc sp c where q is the air density; U denotes the freestream velocity; Cla , Clb and Clc are the lift derivatives due to the pitch angle and TE and LE control surface deflections, respectively; sp is the span; a is the nondimensional distance from midchord to the elastic axis; b and c are the TE and LE control surface deflections, respectively, which are both positive downward; the effective dynamic and control moment derivatives due to a, b and c are given by13 1 > < Cma-eff ẳ 2 ỵ a Cla þ 2Cma Cmb-eff ¼ 12 þ a Clb þ 2Cmb 4ị > 1 : Cmc-eff ẳ ỵ a Clc ỵ 2Cmc Saturation function satị Fig where Cma , Cmb and Cmc are the moment derivatives due to a, b and c, respectively; and Cma can be approximately regarded to be zero.13 The moment and lift arose by wind gust can be given by14 ( Mg ¼ 12 a bLg ð5Þ qU2 bCla sp xg ts ị Lg ẳ ẳ qUbCla sp xg ts ị U where ts ẳ Ut=b, and xg ts ị denotes the disturbance velocity _ T R2 , _ h Define x1 ẳ ẵa; hT R2 , x2 ẳ ½a; and T x ¼ ½xT1 ; xT2 R4 Considering Eqs (1)–(5), the dynamics of the MIMO aeroelastic system can be described as follows > < x_ ẳ x2 x_ ẳ Fxị ỵ DFxị þ ðB þ DBÞu þ D ð6Þ > : y ¼ x1 where D is the unknown external disturbance term caused by wind gust; FðxÞ is the known state function vector; DFðxÞ is the system uncertainties including unmodeled structural nonlinearities; B is the known system control matrix; DB is the unknown system control matrix The input nonlinearity u ẳ Uvị ¼ ½b; cT which includes input saturation and deadzone can be illustrated in Fig 2.22 From Fig 3, the saturation function satðÞ can be expressed as17,22 > < vi max vi > vi max vsat i ẳ satvi ị ¼ vi vi vi vi max ð7Þ > : vi vi < vi where vi max and vi denote the known saturation values of the control input vi (i ¼ 1; 2) Fig Structural diagram of input nonlinearity UðÞ.22 Fig Dead-zone function deadðÞ From Fig 4, the dead-zone function deadðÞ can be expressed as22,23 > < kui ðvsati lui Þ vsati > lui deadvsati ị ẳ ldi vsati lui ð8Þ > : kdi ðvsati ldi Þ vsati > ldi where lui and ldi are the breakpoints of the dead-zone; kui > and kdi > are the right and left slope parameters, respectively In this paper, the control objective is to design a constrained adaptive neural network controller for the MIMO aeroelastic system in Eq (6) to ensure the output y can track the desired output signal yd by appropriately choosing design parameters Assumption 24 For 8t P t0 , the disturbance terms Di of the MIMO aeroelastic system Eq (6) satisfy jDi j pi tịgi i ẳ 1; 9ị where pi ðtÞ is the known smooth functions; and gi is the unknown bounded constants Assumption 20 For the unknown system control matrix DB of the MIMO aeroelastic system in Eq (6), there exists a known constant gDB > such that kDBk gDB Assumption 20 For the known system control matrix B of the MIMO aeroelastic system in Eq (6), there exists a known positive constant gB > such that kBk gB Please cite this article in press as: Gou Y et al Constrained adaptive neural network control of an MIMO aeroelastic system with input nonlinearities, Chin J Aeronaut (2017), http://dx.doi.org/10.1016/j.cja.2017.01.006 174 175 176 178 179 180 181 182 183 184 185 186 187 188 189 191 192 193 194 195 196 197 198 199 CJA 790 17 February 2017 No of Pages 11 200 201 202 204 205 206 207 208 209 210 211 213 Lemma holds Y Gou et al 25 For 8d > and 8v R, the following inequality jvj v tanhðv=dÞ kp d 10ị where kp ẳ 0:2758 Lemma 20 For the known system control matrix B with the spectral radius ðBÞ, there exists a constant Z > so that matrix B ỵ Bị ỵ ZịI is nonsingular Lemma 20 No eigenvalue of matrix A exceeds any of its norm in its absolute value, that is, kt kAk 11ị 214 where kt t ẳ 1; 2; ; nÞ are the eigenvalues of matrix A 215 2.2 Analysis of input nonlinearity 216 In this subsection, before the controller design, the characteristics of the input nonlinearity are analyzed It is well known that input nonlinear characteristics are relatively complex, so it is difficult to directly deal with the input nonlinearity problem Thus, the right inverse function deadỵ ị satisfying deadị deadỵ ị ẳ I is defined as22,26 > < vi =kui ỵ lui vi > vi ẳ deadỵ vi ị ẳ 12ị > : vi =kdi ỵ ldi vi < 217 218 219 220 221 222 224 225 226 227 228 229 230 231 232 Structural diagram of input nonlinearity satall ị.22,26 Fig and the function deadỵ ị is shown in Fig By adding the right inverse function block before the input nonlinearities, the new input nonlinearity structure diagram is shown in Fig 6, where v is the actual designed control law Base on the analysis of the characteristics of the new construction of input nonlinearity in Ref 26, ui can be described as ui ¼ satall ð vi Þ k ðv > < ui i max lui Þ vi P kui ðvi max lui Þ kdi ðvi minx ldi Þ < vi < kui vi max lui ị ẳ vi > : kdi ðvi ldi Þ vi kdi ðvi minx ldi Þ ð13Þ 234 The above equation means that the input saturation and dead-zone coupled with the right inverse function block of the dead-zone can be regarded as an equivalent input saturation 238 239 RBFNNs are considered to approximate the unknown function Fun ðxÞ By employing RBFNNs, Fun ðxÞ can be approximated to any desired accuracy over a compact set X as follows23 240 T Fun xị ẳ W wxị ỵ e 8x X R 14ị T where wxị ẳ ẵw1 xị; w2 xị; ; wfnode ðxÞ R is the basis function vector, with wq xị q ẳ 1; 2; ; fnode Þ the common Gaussian functions, and fnode P the neural networks node number; e ¼ ½e1 ; e2 T is the approximation error which satisfies jei j ei , where ei > ði ¼ 1; 2Þ Typically, the optimal weight matrix W is defined as fnode W ẳ arg minfsupkFun xị W wðxÞkg T W2Rf2 ð15Þ x2R4 where W is any weight matrix in X 241 242 243 244 246 247 248 249 250 251 252 253 255 256 Design of a constrained adaptive control scheme based on RBFNNs 257 3.1 Design of a constrained adaptive control scheme 259 In this section, the backstepping method is used to construct a constrained adaptive neural network controller for the nonlinear system in Eq (6) Define the error variables as 260 258 261 z1 ẳ x1 yd 16ị 262 263 265 z2 ¼ x2 a1 ð17Þ 266 268 z_ ẳ z2 ỵ a1 y_ d 18ị 10 for x2 in the MIMO aeroelastic The virtual control law a system in Eq (6) is designed as a10 ẳ K1 z1 19ị KT1 Right inverse function deadỵ ị 237 2.3 RBF neural networks where a1 is the virtual control law During the constrained adaptive neural network controller design, the backstepping control technique is employed and the detailed design process is described as follows Step Considering the system in Eq (6) and differentiating z1 , we obtain Fig 235 236 where ¼ K1 > is the design parameter matrix To solve the inherent problem of ‘‘explosion of complexity” due to the backstepping method, let a10 pass through a firstorder filter with a time constant matrix s to obtain a1 as27 Please cite this article in press as: Gou Y et al Constrained adaptive neural network control of an MIMO aeroelastic system with input nonlinearities, Chin J Aeronaut (2017), http://dx.doi.org/10.1016/j.cja.2017.01.006 269 270 271 272 273 274 275 277 278 279 280 282 283 284 285 286 287 CJA 790 17 February 2017 No of Pages 11 Constrained adaptive neural network control 289 290 291 292 293 295 296 297 299 300 301 302 303 304 306 a_ ỵ a1 ẳ a10 s a10 0ị ẳ a1 0ị 20ị In view of Young’s inequality,20 and invoking Lemma 1, Eq (31) can be rewritten as where s ẳ diags1 ; s2 ị > To proceed with the design of the constrained adaptive neural network control scheme, we define V_ 2 zT2 WT wxị ỵ zT2 e ỵ zT2 Fxị ỵ zT2 Bu ỵ gDB kz2 kkuk e ẳ a1 a10 where tanhz2 ị ẳ diagtanhz21 =#1 ị; tanhz22 =#2 ịị, pxị ẳ diagp1 xị; p2 xịị, W ẳ ẵkp #1 ; kp #2 T , and g ẳ ẵg1 ; g2 T , in which #1 > and #2 > From Eq (13), the control inputs u can be regarded as an input saturation problem To compensate for the impact of the input saturation, the auxiliary error system is presented as follows20 Ke e kek1 fðz2 ; u; Du; xÞe > > < e_ ẳ ỵB ỵ lIịv uị 33ị kek P r > > : kek < r ð21Þ Differentiating e and invoking Eq (20), we obtain @a10 @a10 e_ ẳ a_ a_ 10 ẳ s1e ỵ z_ x_ @x1 @z1 ¼ s1e þ Sðx1 ; z1 Þ ð22Þ where SðÞ is the sufficiently smooth function vector about P1 : x1 ; z1 Since the set P1 is compact, SðÞ has a maximum S on P1 Then, we obtain e_ s1e ỵ S 23ị ỵ zT2 tanhz2 ịpxịg ỵ kWT pxịk2 kgk2 ỵ zT2 a_ 2 ð32Þ Consider the Lyapunov function candidate 307 308 310 311 312 V1 1 ẳ zT1 z1 ỵ eTe 2 ð24Þ The derivative V1 along Eq (23) is V_ 1 ẳ zT1 z_ T_ ỵ e e zT1 z2 ỵ zT1e zT1 K1 z1 T 1 e s e ỵ e S T 1 zT1 z2 ỵ zT1 z1 ỵ eTe zT1 K1 z1 eT s1e ỵ ST S 2 T kmin ðK1 Þ z1 z1 ðkmin ðs1 Þ 1ÞeTe þ 314 315 316 318 323 324 326 327 328 329 330 332 333 334 335 336 338 339 340 341 343 ỵ ST S kWT pxịk z2 ; xị ẳ zT2 KT2 K2 z2 ỵ 2 ð25Þ Step Differentiating z2 yields z_ ẳ x_ a_ ẳ Fxị ỵ DFxị þ ðB þ DBÞu þ D a_ þ a_ V2 ẳ zT2 z2 27ị The derivative of V2 is V_ 2 ¼ zT2 z_ ẳ zT2 ẵFxị ỵ DFxị ỵ B ỵ DBịu þ D a_ ð28Þ As shown in Section 2.3, the RBFNNs will be employed to approximate the system uncertainties DFðxÞ, and the optimal approximation can be written as DFxị ẳ WT wxị ỵ e 29ị where e ¼ ½e1 ; e2 T , in which jei j ei is the approximate error and ei > i ẳ 1; 2ị Substituting Eq (29) into Eq (28) yields V_ 2 zT2 ẵWT wxị ỵ e ỵ Fxị ỵ B ỵ DBịu ỵ D a_ ð30Þ z2 ðz2 ; xÞ ð34Þ ^ is the approximation value of W; ^ where W g is the approximation value of g; and / satisfies20 ( / z ;xị _/ ẳ /2 ỵkz2 k2 k/ / kz2 k P l ð36Þ kz2 k < l ỵ ỵ gDB kz2 kkuk ỵ ỵ X jz2i jpi xịgi ỵ zT2 Bu 31ị 352 353 354 355 356 358 359 360 361 362 363 364 365 366 368 369 370 371 372 375 376 377 379 where k/ > and l > 380 3.2 Stability analysis 381 In this section, the main results will be stated, and the semiglobal boundedness of all the signals in the closed-loop system will be proven by two cases (1) kek P r Choose the Lyapunov function as follows 382 ~¼g ^ g, K1 > and K2 > where f W¼c W W , g Following from Eqs (25) and (32) and invoking Lemma 3, the time derivative of V is zT2 a_ 351 374 V_ 2 zT2 FðxÞ 350 ð35Þ 1 T f T W K1 W þ ~ g K2 ~ V ¼ V1 þ V2 þ eT e þ f g þ /2 2 2 zT2 e 349 # FðxÞ e /2 ỵ kz2 k2 where e ẳ ẵe1 ; e2 T Considering Assumptions and 2, we obtain zT2 WT wxị 348 where K2 ẳ diagK21 ; K22 Þ > Invoking Lemma and taking the input saturation into consideration, choose the control law as follows " 1 ^ T wxị tanhz2 ịpxị^ v ẳ B þ lIÞ z1 K2 ðz2 eÞ W g ð26Þ Consider the Lyapunov function candidate 319 320 322 zT1 z2 where fz2 ; u; Du; xị ẳ jzT2 BDuj ỵ 0:5l ỵ gB ị DuT Du ỵ jlzT2 ujỵ gDB kz2 kkuk, with Du ẳ u v, l ẳ gB ỵ x, x > 0; Ke ẳ diagðKe1 ; Ke2 Þ > 0; and e R2 is the state of auxiliary error system Moreover, r > is the design parameter which can be appropriately chosen to satisfy the requirement of control performance Define20 344 345 346 37ị iẳ1 Please cite this article in press as: Gou Y et al Constrained adaptive neural network control of an MIMO aeroelastic system with input nonlinearities, Chin J Aeronaut (2017), http://dx.doi.org/10.1016/j.cja.2017.01.006 383 384 385 386 387 389 390 391 392 393 CJA 790 17 February 2017 No of Pages 11 395 396 397 398 Y Gou et al V_ kmin ðK1 Þ zT1 z1 kmin s1 ị 1ịeTe ỵ zT1 z2 T ỵ S S ỵ zT2 WT wxị ỵ zT2 e ỵ zT2 Fxị ỵ zT2 Bu ỵ gDB kz2 kkuk ỵ zT2 tanhz2 ịpxịg þ kWT pðxÞk2 2 _ T_ T T ~ ~ 38ị ỵ kgk z2 a1 ỵ e e_ ỵ W K1 W ỵ ~gT K2 ~g_ ỵ //_ _ _ ~_ ¼ W c _¼c Considering W W W and ~g_ ¼ ^g_ g_ ¼ ^ g_ as well as substituting Eqs (33)–(36) into Eq (38), we obtain To ensure the closed-loop system stable, we can appropriately choose design parameters to make 2kmin ðK2 ị ỵ 2kmin K1 ị > and Ke I > The closedloop signals z1 , z2 , e, e, f W, ~ g and / are semi-globally stable, which means that all the closed-loop signals are bounded The error variable z1 asymptotically converges to a compact set Xz1 , which is defined by pffiffiffiffi Xz1 :ẳ fz1 R2 jkz1 k Eg 44ị where E ẳ V0ị ỵ CC (2) kek < r kek < r means that there does not exist input saturation, so we have v ¼ u and the control input u is bounded Thus, v is bounded The stability can be easily proven when kek < r, and the detailed process of proving is omitted The structure diagram of the whole control system can be seen in Fig 422 ð41Þ Example results and discussion 440 441 ð42Þ To illustrate the effectiveness of the proposed constrained adaptive neural network control scheme, the results of extensive digital simulations are given in this section For these digital simulations, the model parameters in Refs 13–15 are chosen in this study and listed in Table In Table 1, rcg is the proper distance of wing section; Icg is the center-of-wingmass moment of inertia; Icam is the center-of-total-mass moment of inertia Especially note that the pitch damping kz2 k2 z1 ;z2 ;xị /2 ỵkz2 k2 ỵ kgk2 ^_ ỵ ~gT K2 ^g_ ỵ //_ W T K1 W zT2 a_ ỵ eT e_ ỵ f g W T wðxÞ zT2 tanhðz2 ÞqðxÞ~ ðkmin ðs1 Þ 1ÞeTe zT2 f T T z2 K2 z2 e ðKe IÞe kmin ðK1 Þ 12 zT1 z1 2 _ z1 ;z2 ;xị W ỵ ~gT K2 ^g_ þ kgk þf W T K1 c þ //2 þkz þ //_ k2 ð39Þ 400 Invoking Eq (36), we obtain 401 402 /2 ðz1 ; z2 ; xị 404 405 406 /2 ỵ kz2 k2 ỵ //_ ¼ k/ /2 ð40Þ Substituting Eq (40) into Eq (39) yields _ V zT2 K2 z2 eT ðKe IÞe kmin ðK1 Þ 12 zT1 z1 W T wðxÞ zT tanhðz2 Þ ðkmin ðs1 Þ 1ÞeTe zT f 2 417 418 419 zT2 K2 z2 ỵ gDB kz2 kkuk ỵ zT2 BDu ỵ zT2 e T ỵ kW pxịk 43ị where C ẳ 2kmin K2 ị þ 2kmin ðK1 Þ 1; 2kmin ðKe IÞ; > > < 2-1 2-2 ; ; 2k / kmax ðK1 Þ kmax ðK2 Þ > > 2 k2 : ỵ -2 kgk C ẳ kgk2 ỵ -1 kW 2 V_ ðkmin ðK1 Þ 12ÞzT1 z1 ðkmin ðs1 Þ 1ÞeTe lzT2 u W T wxị zT tanhz2 ịpxị~g ỵzT K2 e zT f V_ zT2 K2 z2 eT ðKe IÞe kmin ðK1 Þ zT1 z1 2 f kgk k Wk -1 kW k2 ỵ kmin s1 ị 1ịeTe ỵ 2 -2 k~ gk2 -2 kgk2 ỵ k/ /2 CV ỵ C 2 _ W ỵ ~gT K2 ^g_ ỵ kgk2 k/ /2 pxị~g ỵ f W T K1 c 421 423 424 425 426 427 428 429 431 432 433 434 435 436 437 438 439 408 409 410 412 413 414 415 ( The adaptive laws of c W and ^g are designed as _ T c c W ẳ K1 wxịz2 -1 Wị ^g_ ẳ K1 pxị tanhz2 ịz2 -2 ^gị where -1 > and -2 > Substituting Eq (42) into Eq (41), we obtain Fig Structural diagram of whole control system Please cite this article in press as: Gou Y et al Constrained adaptive neural network control of an MIMO aeroelastic system with input nonlinearities, Chin J Aeronaut (2017), http://dx.doi.org/10.1016/j.cja.2017.01.006 442 443 444 445 446 447 448 CJA 790 17 February 2017 No of Pages 11 Constrained adaptive neural network control Model parameters.13–15 Table Parameter Value q (kg/m3) a b (m) rcg xa sp (m) ch (kg/s) mw (kg) mt (kg) Icam (kgm2) Icg (kgm2) Ia 1.225 0.6719 0.1905 b0:0998 ỵ aị rcg =b 0.5945 27.43 4.340 15.57 0.04697 0.04342 Icam ỵ Icg ỵ mw r2cg 6.757 3.774 0.1566 0.6719 0.1005 12:77 ỵ 53:47a ỵ 1003a2 2844 ỵ 255:99h2 Cla Clb Clc Cma Cmb Cmc ka ðaÞ ðN m=radÞ kh ðhÞ ðN=mÞ Fig 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 Real part of eigenvalues in open-loop system _ is considered, and the initial state values are chosen as ca aị _ _ a0ị ẳ 11:4 , h0ị ẳ 0:05 m, a0ị ẳ 0 ị=s and h0ị ẳ m=s Firstly, it is essential to analyze the stability property of the open-loop system and the pitch damping coefficient is _ ¼ 0:036 kg m2 =s Fig shows that the stability propca ðaÞ erty of the linearized model varies with the freestream velocity U, and it is found that the linearized model has a pair of purely imaginary eigenvalues at the critical velocity Uc ¼ 11:3 m=s, which means that the flutter speed for the linearized system is approximately Uc ¼ 11:3 m=s Considering different freestream velocities, deeper research on the dynamic behaviors of the aeroelastic system is undertaken The pitch and plunge phase diagrams of the aeroelastic system at different freestream velocities are presented in Fig 9, which shows that the freestream velocity apparently affects the limit cycle oscillation (LCO) feature and the system doesn’t exhibit an LCO phenomenon at a freestream velocity of 0:5Uc In terms of frequency and amplitude, from Fig 10, the LCO frequency spectra illustrate the effects on the aeroelastic system at different freestream velocities Fig Aeroelastic system phase diagrams at different freestream velocities Fig 10 Aeroelastic system LCO frequency spectra at different freestream velocities Please cite this article in press as: Gou Y et al Constrained adaptive neural network control of an MIMO aeroelastic system with input nonlinearities, Chin J Aeronaut (2017), http://dx.doi.org/10.1016/j.cja.2017.01.006 CJA 790 17 February 2017 No of Pages 11 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 Y Gou et al In the closed-loop simulation study, the design parameters are chosen as r ¼ 104 , l ¼ 0:1, K1 ẳ diag20; 20ị, l ẳ 1:3, fnode ẳ 12, Ke ẳ diag10; 10ị, K2 ẳ diag5; 5ị, k/ ¼ 10, -1 ¼ -2 ¼ 0:0001, K2 ¼ diagð0:2Þ1212 , K2 ẳ diag0:2; 0:2ị, e ẳ ẵ0:02; 0:02T , #1 ẳ #2 ẳ 0:1, pxị ẳ diag1; 1ị, DB ¼ 0:1B, lui ¼ 2:29 , ldi ¼ 2:29 , kui ẳ kdi ẳ and yd ẳ ẵ0; 0T The maximum control surface deflection is set to be 17:7 For the purpose of examining the effectiveness of the proposed constrained adaptive neural network control scheme at different freestream velocities, simulations at three different freestream velocities Uc , 1:5Uc and 2Uc are undertaken The results are presented in Fig 11, which shows that the closedloop system is stable despite different freestream velocities, and for a higher freestream velocity, the responses are quicker To examine that the LCOs can be suppressed, the aeroelastic system at a freestream velocity of 12 m/s is held in an open loop for 10 s and then the loop is closed In Fig 12, we can observe that the pitch LCO is suppressed in about s and the plunge LCO is suppressed in about s; in terms of control surface, the TE control surface deflection converges to zero in less than s, and the LE control surface deflection converges to zero in about s To verify the applicability and robustness of the aeroelastic control system, based on four types of wind gust, four sets of simulations are done as follows (1) Constrained control for sinusoidal gust, U ¼ 12 m=s The mathematical model of sinusoidal gust is given by14 496 498 499 xg ts ị ẳ x0 sin 6pbts Hðts Þ U Fig 12 Constrained control, controller active at t = 10 s 45ị where x0 ẳ 0:07 m=s and HðÞ denotes the unit step function Fig 11 Constrained control at different freestream velocities Fig 13 Constrained control for sinusoidal gust, U ¼ 12 m=s Please cite this article in press as: Gou Y et al Constrained adaptive neural network control of an MIMO aeroelastic system with input nonlinearities, Chin J Aeronaut (2017), http://dx.doi.org/10.1016/j.cja.2017.01.006 CJA 790 17 February 2017 No of Pages 11 Constrained adaptive neural network control 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 524 Under the sinusoidal gust with a freestream velocity of 12 m/s, the closed-loop responses of the system are given in Fig 13, which shows that the pitch angle converges to zero, and the plunge displacement doesn’t converge to zero; however, the perturbation in the plunge displacement is not significant, which can be accepted The TE and LE control surfaces always deflect with small angles and are in phase with the sinusoidal gust, which is essential for compensating the adverse effect of the persistent and periodic sinusoidal gust (2) Constrained control for random gust, U ¼ 12 m=s The random gust can be generated by passing a white noise with a unit variance through a transfer function Gsị ẳ 106 =s ỵ 10ị, and simulations are undertaken using a freestream velocity of 12 m/s under the effect of the random gust.15 The response results are shown in Fig 14 We can observe that the pitch and plunge displacements converge to zero in about s and the TE control surface converges to zero in less than s, but the LE control surface perturbs a little after convergence in that the random gust obtains the random and uncertain properties (3) Constrained control for triangular gust, U ¼ 12 m=s For the triangular gust, one has14 ts sG xg ts ị ẳ 2x0 Hts ị H ts sG ts sG ð46Þ Hðts sG Þ H ts þ 2x0 sG Figure 14 Constrained control for random gust, U ¼ 12 m=s where x0 ¼ 0:7 m=s, sG ¼ UtG =b, tG ¼ 0:5 s In the presence of the triangular gust above, simulations are undertaken with U ¼ 12 m=s Fig 15 shows the results that the pitch and plunge displacements become stable in no more than 1.5 s and the deflections of both control surfaces tend to quickly in about s (4) Constrained control for exponential gust, U ¼ 12 m=s For the exponential gust, the mathematical model can be described as15 0:25ts xg ts ị ẳ Hts ịx0 e Þ ð47Þ where x0 ¼ 0:04 m=s In the presence of the exponential gust above, simulations are undertaken with U ¼ 12 m=s as in Case (3) The simulation results are presented in Fig 16 We can note that the pitch and plunge displacements and the deflections of both LE and TE control surfaces all converge to zero in about s, which verifies the exponential gust rejection capability of the designed controller To investigate the effectiveness of the proposed constrained adaptive neural network control law against the system uncertainties, we consider the pitch stiffness ka aị ẳ 6:833ỵ _ ẳ 9:967a ỵ 667:685a2 N m=rad, the pitch damping ca ðaÞ 0:029 kg m2 =s, and the plunge stiffness kh hị ẳ 2800 ỵ 280h2 N=m, which are different from those in Table In addition, simulations are undertaken under the effect of a triangular gust and the freestream velocity is Fig 15 Constrained control for triangular gust, U ¼ 12 m=s Please cite this article in press as: Gou Y et al Constrained adaptive neural network control of an MIMO aeroelastic system with input nonlinearities, Chin J Aeronaut (2017), http://dx.doi.org/10.1016/j.cja.2017.01.006 525 526 527 528 529 530 531 532 533 534 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 CJA 790 17 February 2017 10 No of Pages 11 Y Gou et al 12 m/s The response results are presented in Fig 17, which shows that the closed-loop system can still tend to stable in about s in spite of the system uncertainties Taking the failure of the control surface deflection into consideration, simulations are done under the effect of a triangular gust and using a freestream velocity of 12 m/s Figs 18 and 19 show the results with only the TE or LE control surface working From Fig 18, we can note that the closed-loop system can still tend to stable in about s despite the LE control surface failure In Fig 18, we can observe that the controller fails to accomplish the flutter suppression only with the TE control surface deflecting In accordance with Figs 17 and 18, we can conclude this control method can also be applied Fig 16 Constrained control for exponential gust, U ¼ 12 m=s Fig 18 Constrained control with LE control surface failure, U ¼ 12 m=s Fig 17 Constrained control against system uncertainties, U ¼ 12 m=s Fig 19 Constrained control with TE control surface failure, U ¼ 12 m=s Please cite this article in press as: Gou Y et al Constrained adaptive neural network control of an MIMO aeroelastic system with input nonlinearities, Chin J Aeronaut (2017), http://dx.doi.org/10.1016/j.cja.2017.01.006 558 559 560 561 562 563 564 565 566 567 568 569 570 CJA 790 17 February 2017 Constrained adaptive neural network control 571 572 573 574 576 575 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 to the aeroelastic system with only the TE control surface, and the aerodynamic efficiency of the TE control surface is higher compared with that of the LE control surface Conclusions (1) An effective constrained adaptive neural network control scheme has been developed for an MIMO aeroelastic system with wind gust, system uncertainties, and input nonlinearities (2) In order to handle the system uncertainties, RBFNNs have been employed to approximate the system uncertainties effectively, and simulation results demonstrate the effectiveness of the proposed control scheme against the system uncertainties (3) To deal with the input nonlinearities, the right inverse function block of the dead-zone is added before the input nonlinearities, and the input nonlinearities can be treated as a single input saturation nonlinearity Moreover, an auxiliary error system is designed to compensate for the impact of the input saturation (4) By using the Lyapunov stability theory and the backstepping control technique, all signals of the closedloop system based on the proposed constrained adaptive neural network control scheme are semi-globally uniformly bounded (5) Digital simulation results illustrate the effectiveness of the 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680 681 682 683 684 685 686 687 688 689 690 Please cite this article in press as: Gou Y et al Constrained adaptive neural network control of an MIMO aeroelastic system with input nonlinearities, Chin J Aeronaut (2017), http://dx.doi.org/10.1016/j.cja.2017.01.006 ... Structural diagram of whole control system Please cite this article in press as: Gou Y et al Constrained adaptive neural network control of an MIMO aeroelastic system with input nonlinearities, ... 15 Constrained control for triangular gust, U ¼ 12 m=s Please cite this article in press as: Gou Y et al Constrained adaptive neural network control of an MIMO aeroelastic system with input nonlinearities, ... filter with a time constant matrix s to obtain a1 as27 Please cite this article in press as: Gou Y et al Constrained adaptive neural network control of an MIMO aeroelastic system with input nonlinearities,