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[.]
Trang 17 Aeronautics and Astronautics Engineering College, Air Force Engineering University, Xi’an 710038, China
8 Received 20 April 2016; revised 2 September 2016; accepted 28 November 2016
9
12
13 Aeroelastic system;
14 Constrained control;
15 Flutter suppression;
16 Input nonlinearities;
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 nonlinear-ities, the right inverse function block of the dead-zone is added before the input nonlinearnonlinear-ities, which simplifies the input nonlinearities into an equivalent input saturation To deal with the equiv-alent 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 backstep-ping control technique, a constrained adaptive neural network controller is designed, and it is pro-ven 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/
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18
19 1 Introduction
20 In the past, aeroelasticity has attracted increasing concern in
21 aircraft design Aeroelastic systems exhibit a variety of
unsta-22
ble phenomena as a result of the mutual interaction of
struc-23
tural, inertia and aerodynamic forces.1 Divergence, flutter,
24
and limit-cycle oscillation are typical unstable phenomena
25
which can degrade an aircraft’s flight performance, and even
26
cause flight mission failure.1,2 Thus, a reliable and effective
27
control strategy becomes one of the key issues in aeroelastic
28
system control design In previous studies, researchers have
29
analyzed the nonlinear responses of aeroelastic systems, and
30
various control schemes have been extensively studied Based
31
on the l method, Lind and Brenner3have analyzed the
unsta-32
ble responses of aeroelastic systems and studied robust
stabil-33
ity margins To study different aeroelastic phenomena, the
34
NASA Langley Research Center has developed a benchmark
* 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
Chinese Society of Aeronautics and Astronautics
& Beihang University Chinese Journal of Aeronautics
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http://dx.doi.org/10.1016/j.cja.2017.01.006
Trang 235 active control technology (BACT) wind-tunnel model.4 For
36 this BACT wind-tunnel model, several control laws for flutter
37 suppression have been developed.4–6 Considering nonlinear
38 structural stiffness, a model equipped with a single
trailing-39 edge (TE) control surface has been developed at Texas A&M
40 University.7 Based on this model, a wide variety of control
41 schemes have been designed.8–11Inspired by the limited
effec-42 tiveness of a single TE control surface, a wing section equipped
43 with a leading-edge (LE) control surface and a TE control
sur-44 face has been designed, and a large number of control schemes
45 has been proposed.12–16For this wing section with
uncertain-46 ties, adaptive control has been widely used to suppress
flut-47 ter.13–15 Neural network control and adaptive control have
48 been developed in this filed and compared in control
perfor-49 mance.13With respect to external disturbance and
uncertain-50 ties, Wang et al.14 designed an output feedback adaptive
51 controller coupled with an SDU decomposition which avoids
52 the singularity problem arising from estimation of the input
54 and Singh15used an auxiliary dynamic system to compensate
55 for the input saturation and proposed a novel control scheme
56 In addition, a sliding mode control method was also applied to
57 flutter suppression, and Lee and Singh16 have designed a
58 higher-order sliding mode controller which accomplished the
59 finite-time flutter suppression of the aeroelastic system
60 It is well known that input nonlinearities exist in a real
con-61 trol system, and an aeroelastic control system is no exception
62 Both input dead-zone and saturation are considered for the
63 uncertain aeroelastic system in this paper Input saturation
64 and dead-zone may induce deterioration of the aeroelastic
con-65 trol system performance, and even make the aeroelastic
con-66 trol system fail Consequently, input saturation and
dead-67 zone have attracted much attention Input dead-zone could
68 induce a zero input against a range of set values.17An adaptive
69 dead-zone inverse approach was proposed to tackle a system
70 with input dead-zone.18 An adaptive fuzzy output feedback
71 control law, which treats dead-zone inputs as system
uncer-72 tainties, has been developed.19For the input saturation
prob-73 lem, Chen et al.20 designed an auxiliary system, whose input
74 was the error between the saturation input and the desired
con-75 trol input, to compensate for the impact of the input
satura-76 tion Li et al.21 proposed an adaptive fuzzy output feedback
77 control for output constrained nonlinear systems In general,
78 some researchers have also studied in integrating input
dead-79 zone with saturation For uncertain input and
multi-80 output (MIMO) nonlinear systems with input nonlinearities,
81 a robust adaptive neural network control was developed.17
82 Yang and Chen22 regarded input dead-zone and saturation
83 nonlinearities as a new input saturation problem through a
84 dead-zone inverse approach, and proposed an adaptive neural
85 prescribed performance control law for near-space vehicles
86 Motivated by the above discussion, a constrained adaptive
88 aeroelastic system with wind gust, system uncertainties, and
89 input nonlinearities Different from the previous references,
90 it is especially noted that uncertainties in pitch stiffness, plunge
91 stiffness, and pitch damping are all considered Inspired by
92 Ref 22, the right inverse function block of the dead-zone is
93 added before the input nonlinearities, by which the input
non-94 linearities can be regarded as a new input saturation.22To
han-95 dle the new input saturation, an auxiliary error system is
96 designed to compensate for the impact of the input saturation
97
Radial basis function neural networks (RBFNNs) are also
98
applied to approximate the system uncertainties A novel
con-99
strained adaptive control law is developed by using the
back-100
stepping control technique The simulation results of the
101
MIMO aeroelastic control system are presented to verify that
102
the proposed control scheme can accomplish flutter
suppres-103
sion despite the effects of wind gust, system uncertainties,
104
and input nonlinearities
105
2 Nonlinear aeroelastic model and preliminary
106
2.1 Nonlinear aeroelastic model
107
A two-degree-of-freedom (2-DOF) wing section equipped with
108
LE and TE control surfaces is presented in Fig 1.15 The
109
second-order differential equations signifying the dynamic of
110
this aeroelastic system are given by13,14
111
Ia mwxab
mwxab mt
€h
þ cað_aÞ 0
_h
0 khðhÞ
a h
L
Lg
113 114
where a denotes the pitch angle which is positive upward; h
115
denotes the plunge displacement which is positive downward;
116
Ia is the moment of inertia; mw and mt are the wing section
117
mass and the total mass, respectively; xa is the distance
118
between the center of mass and the elastic axis; b is the
semi-119
chord of the wing; ch is the plunge damping coefficient;
espe-120
cially note that uncertainties in pitch stiffness, plunge stiffness,
121
and pitch damping are all considered, which is different from
122
the previous references In a polynomial form, the pitch
damp-123
ing cað_aÞ, the pitch stiffness kaðaÞ, and the plunge stiffness
124
khðhÞ are expressed as follows
125
cað_aÞ ¼ ca0þ ca1_a þ ca2_a2
kaðaÞ ¼ ka0þ ka1aþ ka2a2
khðhÞ ¼ kh þ khhþ khh2
8
>
127 128
where caj, kajand khj(j¼ 0; 1; 2) are assumed to be unknown
129
constants
Fig 1 Aeroelastic system with LE and TE control surfaces.15
Trang 3130 In Eq.(1), M and L represent the aerodynamic moment and
131 lift in a quasi-steady form expressed by13
132
M¼ qU2b2Cma-effsp aþ ð _h=UÞ þ 1
2 a
bð_a=UÞ
þqU2b2Cmb-effspbþ qU2b2Cmc-effspc
L¼ qU2bCl asp aþ ð _h=UÞ þ 1
2 a
bð_a=UÞ
þqU2
bCl bspbþ qU2
bCl cspc
8
>
>
<
>
>
:
ð3Þ
134
135 where q is the air density; U denotes the freestream velocity;
136 Cl a, Cl b and Cl c are the lift derivatives due to the pitch angle
137 and TE and LE control surface deflections, respectively; sp is
138 the span; a is the nondimensional distance from midchord to
139 the elastic axis; b and c are the TE and LE control surface
140 deflections, respectively, which are both positive downward;
141 the effective dynamic and control moment derivatives due to
142 a, b and c are given by13
143
Cma-eff¼ 1
2þ a
Cl aþ 2Cm a
Cmb-eff¼1þ a
Clbþ 2Cm b
Cmc-eff¼ 1
2þ a
Cl cþ 2Cm c
8
>
145
146 where Cm a, Cm band Cm care the moment derivatives due to a, b
147 and c, respectively; and Cm a can be approximately regarded to
148 be zero.13The moment and lift arose by wind gust can be given
149 by14
150
Mg¼1 a
bLg
Lg¼qU 2 bClas p x g ðt s Þ
U ¼ qUbCl aspxgðtsÞ
(
ð5Þ
152
153 where ts¼ Ut=b, and xgðtsÞ denotes the disturbance velocity
154 Define x1¼ ½a; hT2 R2, x2¼ ½_a; _hT2 R2, and
155 x ¼ ½xT
1; xT
2T
2 R4
Considering Eqs.(1)–(5), the dynamics of
156 the MIMO aeroelastic system can be described as follows
157
_x1¼ x2
_x2¼ FðxÞ þ DFðxÞ þ ðB þ DBÞu þ D
y ¼ x1
8
>
159
160 whereD is the unknown external disturbance term caused by
161 wind gust;FðxÞ is the known state function vector; DFðxÞ is
162 the system uncertainties including unmodeled structural
non-163 linearities; B is the known system control matrix; DB is the
165 u ¼ UðvÞ ¼ ½b; cT
which includes input saturation and
dead-166 zone can be illustrated inFig 2.22
167 FromFig 3, the saturation function satðÞ can be expressed
168 as17,22
169
vsat i¼ satðviÞ ¼
vi max vi> vi max
vi vimin6 vi6 vi max
vimin vi< vi min
8
>
171
172 where vimax and vimin denote the known saturation values of
173 the control input vi (i¼ 1; 2)
174
From Fig 4, the dead-zone function deadðÞ can be
175
expressed as22,23
176
deadðvsatiÞ ¼
kuiðvsati luiÞ vsati> lui
kdi ðvsati ldiÞ vsati> ldi
8
>
178 179
where luiand ldiare the breakpoints of the dead-zone; kui> 0
180
and kdi> 0 are the right and left slope parameters,
181
respectively
182
In this paper, the control objective is to design a
con-183
strained adaptive neural network controller for the MIMO
184
aeroelastic system in Eq.(6)to ensure the outputy can track
185
the desired output signalydby appropriately choosing design
186
parameters
187
Assumption 1 24For8t P t0, the disturbance terms Diof the
188
MIMO aeroelastic system Eq.(6)satisfy
189
192
where piðtÞ is the known smooth functions; and gi is the
193
unknown bounded constants
194
Assumption 2 20For the unknown system control matrixDB
195
of the MIMO aeroelastic system in Eq (6), there exists a
196
known constant gDB> 0 such that kDBk 6 gDB
197
Assumption 3 20For the known system control matrix B of
198
the MIMO aeroelastic system in Eq.(6), there exists a known
199
positive constant gB> 0 such that kBk 6 gB
Fig 2 Structural diagram of input nonlinearityUðÞ.22
Fig 3 Saturation function satðÞ
Fig 4 Dead-zone function deadðÞ
Trang 4200 Lemma 1 25 For8d > 0 and 8v 2 R, the following inequality
202
204
205 where kp¼ 0:2758
206 Lemma 2 20 For the known system control matrix B with the
207 spectral radius ðBÞ, there exists a constant Z > 0 so that
208 matrixB þ ð ðBÞ þ ZÞI is nonsingular
209 Lemma 3 20 No eigenvalue of matrixA exceeds any of its norm
210 in its absolute value, that is,
211
213
214 wherektðt ¼ 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
character-217 istics of the input nonlinearity are analyzed It is well known
218 that input nonlinear characteristics are relatively complex, so
219 it is difficult to directly deal with the input nonlinearity
prob-220 lem Thus, the right inverse function deadþðÞ satisfying
221 deadðÞdeadþðÞ ¼ I is defined as22,26
222
deadþðviÞ ¼
vi=kui þ lui vi> 0
vi=kdi þ ldi vi< 0
8
>
224
225 and the function deadþðÞ is shown in Fig 5 By adding the
226 right inverse function block before the input nonlinearities,
227 the new input nonlinearity structure diagram is shown in
228 Fig 6, wherev is the actual designed control law
229 Base on the analysis of the characteristics of the new
con-230 struction of input nonlinearity in Ref.26, ui can be described
231 as
232
ui¼ satallðviÞ
¼
kuiðvi max luiÞ viP kuiðvi max luiÞ
vi kdiðvi minx ldiÞ < vi< kuiðvi max luiÞ
kdiðvi min ldiÞ vi 6 kdiðvi minx ldiÞ
8
>
>
ð13Þ
234
235
The above equation means that the input saturation and
236
dead-zone coupled with the right inverse function block of
237
the dead-zone can be regarded as an equivalent input
238
saturation
239
2.3 RBF neural networks
240
RBFNNs are considered to approximate the unknown
func-241
tion FunðxÞ By employing RBFNNs, FunðxÞ can be
approxi-242
243
follows23
244
247
where wðxÞ ¼ ½w1ðxÞ; w2ðxÞ; ; wfnodeðxÞT
2 Rfnode
is the basis
248
function vector, with wqðxÞ ðq ¼ 1; 2; ; fnodeÞ the common
249
Gaussian functions, and fnodeP 2 the neural networks node
250
number;e ¼ ½e1; e2T
is the approximation error which satisfies
251
jeij 6 e
i, where ei > 0 ði ¼ 1; 2Þ Typically, the optimal weight
252
matrixWis defined as
253
W¼ arg min
W2R f2 fsup
x2R 4
255 256
whereW is any weight matrix in X
257
3 Design of a constrained adaptive control scheme based on
258
RBFNNs
259
3.1 Design of a constrained adaptive control scheme
260
In this section, the backstepping method is used to construct a
261
constrained adaptive neural network controller for the
nonlin-262
ear system in Eq.(6) Define the error variables as
263
266
269
wherea1is the virtual control law
270
During the constrained adaptive neural network controller
271
design, the backstepping control technique is employed and
272
the detailed design process is described as follows
273
Step 1 Considering the system in Eq.(6)and differentiating
274
z1, we obtain
275
278
The virtual control lawa10forx2 in the MIMO aeroelastic
279
system in Eq.(6)is designed as
280
283
whereKT
1 ¼ K1> 0 is the design parameter matrix
284
To solve the inherent problem of ‘‘explosion of complexity”
285
due to the backstepping method, leta10pass through a
first-286
order filter with a time constant matrixs to obtaina1 as27
287 Fig 5 Right inverse function deadþðÞ
Fig 6 Structural diagram of input nonlinearity satallðÞ.22,26
Trang 5a10ð0Þ ¼a1ð0Þ
ð20Þ
289
290 wheres ¼ diagðs1; s2Þ > 0
291 To proceed with the design of the constrained adaptive
neu-292 ral network control scheme, we define
293
295
296 Differentiatinge and invoking Eq.(20), we obtain
297
_e ¼ _a1 _a10¼ s1e þ @a10
@x1 _x1 @a10
@z1 _z1
299
300 where SðÞ is the sufficiently smooth function vector about
301 P1: x1; z1 Since the set P1 is compact,SðÞ has a maximum
302 S on P1
304
306
308
V1¼1
2zT
1z1þ1
310
311 The derivative V1 along Eq.(23)is
312
_V
1¼ zT
1_z1þeT_e 6 zT
1z2þ zT
1e zT
1K1z1eTs1e þ eTS
6 zT
1z2þ 1
2zT1z1þeTe zT
1K1z1eTs1e þ1
2STS
6 kminðK1Þ 1
2
zT
1z1 ðkminðs1Þ 1ÞeTe
þ zT
1z2þ 1
314
315 Step 2 Differentiatingz2 yields
316
_z2¼ _x2 _a1¼ FðxÞ þ DFðxÞ þ ðB þ DBÞu þ D _a1 ð26Þ
318
320
V2¼1
2zT
322
323 The derivative of V2is
324
_V
2¼ zT
2_z2¼ zT
2½FðxÞ þ DFðxÞ þ ðB þ DBÞu þ D _a1 ð28Þ
326
327 As shown in Section2.3, the RBFNNs will be employed to
328 approximate the system uncertaintiesDFðxÞ, and the optimal
329 approximation can be written as
330
332
333 wheree ¼ ½e1; e2T
, in whichjeij 6 e
i is the approximate error
334 and ei > 0 ði ¼ 1; 2Þ
335 Substituting Eq.(29)into Eq.(28)yields
336
_V
26 zT
2½WTwðxÞ þ eþ FðxÞ þ ðB þ DBÞu þ D _a1 ð30Þ
338
339 wheree¼ ½e
1; e
2T
341
_V
2 6 zT
2WTwðxÞ þ zT
2eþ zT
2FðxÞ zT
2_a1þ zT
2Bu
þ gDBkz2kkuk þX
2
i¼1
343
344
In view of Young’s inequality,20and invoking Lemma 1,
345
Eq.(31)can be rewritten as
346
_V
26 zT
2WTwðxÞ þ zT
2eþ zT
2FðxÞ þ zT
2Bu þ gDBkz2kkuk
þ zT
2tanhðz2ÞpðxÞg þkWTpðxÞk
2
2
2 zT
2_a1 ð32Þ 348
349
where tanhðz2Þ ¼ diagðtanhðz21=#1Þ; tanhðz22=#2ÞÞ, pðxÞ ¼
350
diagðp1ðxÞ; p2ðxÞÞ, W ¼ ½kp#1; kp#2T
, and g ¼ ½g1; g2T
, in
351
which#1> 0 and #2> 0
352
From Eq.(13), the control inputsu can be regarded as an
353
input saturation problem To compensate for the impact of
354
the input saturation, the auxiliary error system is presented
355
as follows20
356
_e ¼
Kee 1 kek 2fðz2; u; Du; xÞe
8
>
358 359
where fðz2; u; Du; xÞ ¼ jzT
2BDuj þ 0:5ðl þ gBÞ2DuTDu þ jlzT
2ujþ
360
gDBkz2kkuk, with Du ¼ u v, l¼ gBþ x, x> 0;
361
Ke¼ diagðKe1; Ke2Þ > 0; and e 2 R2 is the state of auxiliary
362
error system Moreover, r> 0 is the design parameter which
363
can be appropriately chosen to satisfy the requirement of
con-364
trol performance
365
Define20
366
ðz2; xÞ ¼1
2zT
2KT
2K2z2þkW
TpðxÞk2
369
whereK2¼ diagðK21; K22Þ > 0
370
Invoking Lemma 2 and taking the input saturation into
371
consideration, choose the control law as follows
372
v ¼ ðB þ lIÞ1 z1 K2ðz2 eÞ ^WTwðxÞ tanhðz2ÞpðxÞ^g
"
þ_a1 z2 ðz2; xÞ /2þ kz2k2 FðxÞ e
#
ð35Þ
374 375
where ^W is the approximation value of W; ^g is the
approxima-376
tion value ofg; and / satisfies20
377
_/ ¼ // ðz2 þkz2;xÞ2 k 2 k// kz2k P l
(
ð36Þ
379 380
where k/> 0 and l > 0
381
3.2 Stability analysis
382
In this section, the main results will be stated, and the
semi-383
global boundedness of all the signals in the closed-loop system
384
will be proven by two cases
385
(1)kek P r
386
Choose the Lyapunov function as follows
387
V¼ V
1þ V
2þ1
2eTe þ1
2fTK1fW þ1
2~gTK2~g þ1
2/
2 ð37Þ 389
390
where fW ¼ cW W,~g ¼ ^g g, K1> 0 and K2> 0
391
Following from Eqs.(25) and (32)and invokingLemma 3,
392
the time derivative of V is
393
Trang 6_V 6 kminðK1Þ 1
2
zT
1z1 ðkminðs1Þ 1ÞeTe þ zT
1z2
þ1
2STS þ zT
2WTwðxÞ þ zT
2eþ zT
2FðxÞ þ zT
2Bu
þ gDBkz2kkuk þ zT
2tanhðz2ÞpðxÞg þ1
2kW
TpðxÞk2
þ 1
2kgk
2 zT
2_a1þ eT_e þ ~WTK1 _~W þ ~gTK2_~g þ / _/ ð38Þ
395
396 Considering _~W ¼ _cW _W¼ _cW and _~g ¼ _^g _g ¼ _^g as well
397 as substituting Eqs.(33)–(36)into Eq.(38), we obtain
398
_V 6 ðkminðK1Þ 1
2ÞzT
1z1 ðkminðs1Þ 1ÞeTe lzT
2u
þzT
2K2e zT
2fTwðxÞ zT
2tanhðz2ÞpðxÞ~g
zT
2K2z2þ gDBkz2kkuk þ zT
2BDu þ zT
2e
þkW T pðxÞk 2
2 kz 2 k 2 ðz 1 ;z 2 ;xÞ
/2þkz 2 k 2 þkgk 2
2
zT
2_a1þ eT_e þ fWTK1 _^W þ ~gTK2_^g þ / _/
6 ðkminðs1Þ 1ÞeTe zT
2fTwðxÞ zT
2tanhðz2ÞqðxÞ~g
zT
2K2z2 eTðKe IÞe k minðK1Þ 1
zT
1z1
þfWTK1W þ ~g_c TK2_^g þkgk 2
2 þ/ 2 ðz 1 ;z 2 ;xÞ / 2 þkz 2 k 2 þ / _/
ð39Þ
400
402
/2 ðz1; z2; xÞ
404
405 Substituting Eq.(40)into Eq.(39)yields
406
_V 6 zT
2K2z2 eTðKe IÞe kminðK1Þ 1
2
zT
1z1
ðkminðs1Þ 1ÞeTe zT
2fTwðxÞ zT
2tanhðz2Þ pðxÞ~g þ fWTK1W þ ~g_c TK2_^g þkgk 2
2 k//2
ð41Þ
408
409 The adaptive laws of cW and ^g are designed as
410
_c
W ¼ K1
1 ðwðxÞzT
2 -1cWÞ _^g ¼ K1
2 ðpðxÞ tanhðz2Þz2 -2^gÞ
(
ð42Þ
412
413 where -1> 0 and -2> 0
414 Substituting Eq.(42)into Eq.(41), we obtain
415
_V 6 zT
2K2z2 eTðKe IÞe kminðK1Þ 1
2
zT
1z1
ðkminðs1Þ 1ÞeTe þkgk
2
2 -1kfWk2
2 þ-1kWk2
2
-2k~gk2
2 þ-2kgk2
418
where
419
C¼ min 2kð minðK2Þ þ 2kminðK1Þ 1; 2kminðKe IÞ;
2- 1
k max ðK 1 Þ; 2- 2
k max ðK 2 Þ; 2k/
C¼kgk 2
2 þ- 1 kW k 2
2 þ- 2 kgk 2
2
8
>
>
421 422
To ensure the closed-loop system stable, we can
appropri-423
424
2kminðK2Þ þ 2kminðK1Þ 1 > 0 and Ke I > 0 The
closed-425
loop signals z1,z2,e, e, fW, ~g and / are semi-globally stable,
426
which means that all the closed-loop signals are bounded
427
The error variablez1 asymptotically converges to a compact
428
set Xz 1, which is defined by
429
Xz1:¼ fz12 R2jkz1k 6pffiffiffiffiE
432
where E¼ 2 Vð0Þ þ C
433
(2) kek < r
434
kek < r means that there does not exist input saturation, so
435
we havev ¼ u and the control input u is bounded Thus, v is
436
bounded The stability can be easily proven when kek < r,
437
and the detailed process of proving is omitted
438
The structure diagram of the whole control system can be
439
seen inFig 7
440
4 Example results and discussion
441
To illustrate the effectiveness of the proposed constrained
442
adaptive neural network control scheme, the results of
exten-443
sive digital simulations are given in this section For these
dig-444
ital simulations, the model parameters in Refs 13–15 are
445
chosen in this study and listed in Table 1 InTable 1, rcg is
446
the proper distance of wing section; Icg is the
center-of-wing-447
mass moment of inertia; Icam is the center-of-total-mass
448
moment of inertia Especially note that the pitch damping
Fig 7 Structural diagram of whole control system
Trang 7449 cað_aÞ is considered, and the initial state values are chosen as
450 að0Þ ¼ 11:4, hð0Þ ¼ 0:05 m, _að0Þ ¼ 0ðÞ=s and _hð0Þ ¼ 0 m=s
451 Firstly, it is essential to analyze the stability property of the
453 cað_aÞ ¼ 0:036 kg m2=s.Fig 8 shows that the stability
prop-454 erty of the linearized model varies with the freestream velocity
455 U, and it is found that the linearized model has a pair of purely
456 imaginary eigenvalues at the critical velocity Uc¼ 11:3 m=s,
457 which means that the flutter speed for the linearized system
458 is approximately Uc¼ 11:3 m=s Considering different
free-459 stream velocities, deeper research on the dynamic behaviors
460 of the aeroelastic system is undertaken The pitch and plunge
461 phase diagrams of the aeroelastic system at different
free-462 stream velocities are presented in Fig 9, which shows that
463 the freestream velocity apparently affects the limit cycle
oscil-464 lation (LCO) feature and the system doesn’t exhibit an LCO
465 phenomenon at a freestream velocity of 0:5Uc In terms of
fre-466 quency and amplitude, fromFig 10, the LCO frequency
spec-467 tra illustrate the effects on the aeroelastic system at different
468 freestream velocities
Table 1 Model parameters.13–15
I cg (kgm 2
cg
kaðaÞ ðN m=radÞ 12 :77 þ 53:47a þ 1003a 2
Fig 8 Real part of eigenvalues in open-loop system Fig 9 Aeroelastic system phase diagrams at different freestream
velocities
Fig 10 Aeroelastic system LCO frequency spectra at different freestream velocities
Trang 8469 In the closed-loop simulation study, the design parameters
470 are chosen as r¼ 104, l¼ 0:1, K1¼ diagð20; 20Þ, l ¼ 1:3,
471 fnode¼ 12, Ke¼ diagð10; 10Þ, K2¼ diagð5; 5Þ, k/¼ 10,
472 -1¼ -2¼ 0:0001, K2¼ diagð0:2Þ1212, K2¼ diagð0:2; 0:2Þ,
473 e¼ ½0:02; 0:02T
, #1¼ #2¼ 0:1, pðxÞ ¼ diagð1; 1Þ, DB ¼
474 0:1B, lui¼ 2:29, ldi¼ 2:29, kui¼ kdi¼ 1 and yd¼ ½0; 0T
475 The maximum control surface deflection is set to be 17:7
476 For the purpose of examining the effectiveness of the
pro-477 posed constrained adaptive neural network control scheme at
478 different freestream velocities, simulations at three different
479 freestream velocities Uc, 1:5Uc and 2Uc are undertaken The
480 results are presented inFig 11, which shows that the
closed-481 loop system is stable despite different freestream velocities,
482 and for a higher freestream velocity, the responses are quicker
483 To examine that the LCOs can be suppressed, the aeroelastic
484 system at a freestream velocity of 12 m/s is held in an open
485 loop for 10 s and then the loop is closed InFig 12, we can
486 observe that the pitch LCO is suppressed in about 5 s and
487 the plunge LCO is suppressed in about 1 s; in terms of control
488 surface, the TE control surface deflection converges to zero in
489 less than 6 s, and the LE control surface deflection converges
490 to zero in about 2 s
491 To verify the applicability and robustness of the aeroelastic
492 control system, based on four types of wind gust, four sets of
493 simulations are done as follows
494 (1) Constrained control for sinusoidal gust, U¼ 12 m=s
495 The mathematical model of sinusoidal gust is given by14
496
xgðtsÞ ¼ x0sin 6pbts
U
498
499 where x0¼ 0:07 m=s and HðÞ denotes the unit step function
Fig 11 Constrained control at different freestream velocities
Fig 12 Constrained control, controller active at t = 10 s
Fig 13 Constrained control for sinusoidal gust, U¼ 12 m=s
Trang 9500 Under the sinusoidal gust with a freestream velocity of
501 12 m/s, the closed-loop responses of the system are given in
502 Fig 13, which shows that the pitch angle converges to zero,
503 and the plunge displacement doesn’t converge to zero;
how-504 ever, the perturbation in the plunge displacement is not
signif-505 icant, which can be accepted The TE and LE control surfaces
506 always deflect with small angles and are in phase with the
sinu-507 soidal gust, which is essential for compensating the adverse
508 effect of the persistent and periodic sinusoidal gust
509 (2) Constrained control for random gust, U¼ 12 m=s
510 The random gust can be generated by passing a white noise
512 GðsÞ ¼ 7 106=ðs þ 10Þ, and simulations are undertaken
513 using a freestream velocity of 12 m/s under the effect of the
514 random gust.15 The response results are shown in Fig 14
515 We can observe that the pitch and plunge displacements
con-516 verge to zero in about 1 s and the TE control surface converges
517 to zero in less than 2 s, but the LE control surface perturbs a
518 little after convergence in that the random gust obtains the
519 random and uncertain properties
520 (3) Constrained control for triangular gust, U¼ 12 m=s
521 For the triangular gust, one has14
522
xgðtsÞ ¼ 2x0
ts
sG
HðtsÞ H tssG
2
þ 2x0
ts
sG 1
Hðts sGÞ H tssG
2
ð46Þ
524
525
where x0¼ 0:7 m=s, sG¼ UtG=b, tG¼ 0:5 s
526
In the presence of the triangular gust above, simulations are
527
undertaken with U¼ 12 m=s.Fig 15 shows the results that
528
the pitch and plunge displacements become stable in no more
529
than 1.5 s and the deflections of both control surfaces tend to 0
530
quickly in about 2 s
531
(4) Constrained control for exponential gust, U¼ 12 m=s
532
For the exponential gust, the mathematical model can be
533
described as15
534
537
where x0¼ 0:04 m=s
538
In the presence of the exponential gust above, simulations
539
are undertaken with U¼ 12 m=s as in Case (3) The simulation
540
results are presented inFig 16 We can note that the pitch and
541
plunge displacements and the deflections of both LE and TE
542
control surfaces all converge to zero in about 2 s, which verifies
543
the exponential gust rejection capability of the designed
544
controller
545
To investigate the effectiveness of the proposed constrained
546
adaptive neural network control law against the system
uncer-547
tainties, we consider the pitch stiffness kaðaÞ ¼ 6:833þ
548
9:967a þ 667:685a2 N m=rad, the pitch damping cað_aÞ ¼
549
550
khðhÞ ¼ 2800 þ 280h2
N=m, which are different from those in
551
Table 1 In addition, simulations are undertaken under the
552
effect of a triangular gust and the freestream velocity is
Figure 14 Constrained control for random gust, U¼ 12 m=s Fig 15 Constrained control for triangular gust, U¼ 12 m=s
Trang 1012 m/s The response results are presented in Fig 17, which
559
shows that the closed-loop system can still tend to stable in
560
about 3 s in spite of the system uncertainties
561
Taking the failure of the control surface deflection into
con-562
sideration, simulations are done under the effect of a
triangu-563
lar gust and using a freestream velocity of 12 m/s.Figs 18 and
564
19 show the results with only the TE or LE control surface
565
working FromFig 18, we can note that the closed-loop
sys-566
tem can still tend to stable in about 3 s despite the LE control
567
surface failure InFig 18, we can observe that the controller
568
fails to accomplish the flutter suppression only with the TE
569
control surface deflecting In accordance with Figs 17 and
570
18, we can conclude this control method can also be applied
Fig 16 Constrained control for exponential gust, U¼ 12 m=s
Fig 17 Constrained control against system uncertainties,
U¼ 12 m=s
Fig 18 Constrained control with LE control surface failure,
U¼ 12 m=s
Fig 19 Constrained control with TE control surface failure,
U¼ 12 m=s