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Controller Designs of a Gust Load Alleviation System for an Elastic Rectangular Wing
Cop\Tight © IFAC AlItoma til" Co ntrol in .-\ e rospace. T Sllkuha . Japa n . I!lH!l
CONTROLLER DESIGNS OF A GUST LOAD ALLEVIATION SYSTEM FOR AN ELASTIC RECTANGULAR WING A. Fujimori*, H. Ohta* and P. N. Nikiforuk** *Departlllent of A eronautical Engineering, N ago)'a U nil'enit)·, Chikllsa-kll, N ago)'a , Japan **College of Engineering, U niversity uf Saskatchewan , Saskatuun, Saskatchewan , Canada Abstract. This paper proposes two design methods of reduced-order controllers for gust load alleviation (GLA) systems of an elastic wing . and examines the con tro I performance using both simulation studies and wind-tunnel experiments. One of the methods is based on the use of the generalized Hessenberg representation
load
alleviation;
optimal
INTRODUCTION
control;
order
reduction;
robustness;
x=Ax+Bu+Bnw y=Cx+v
In recent years. many works of active control for aeroelastic systems. for example. active flutter suppression (AFS) and gust load alleviation (GLA). are reported. These control systems are required for future aircraft in the sense of the energy -efficient and high performance. A problem of aeroelastic systems is the complication of the control laws which are caused by high order of aeroelastic systems. Since the complete dynamics of the aeroelastic system is in general described by partial differential equations for both structural and aerodynamic modelings. the state-space equat i ons may require a la r ge number of states to represent such dynamics accurately. As well. an optimal control law based on the standard linear quadratic Gaussian (LQG) technique would be of the same high order as the plant. Such a controller. however. should be simple enough to implement on a flight computer and has to work well under the presence of the uncertainty of the mode ling and parameter change. Therefore. it is worthwhile to design a reduced-order controller
Oa) Ob)
x E!R n • y E!R m • u E!RR where wand v represent pro cess and measurement noises. respectively. The first method of order reduction is the one based on the generalized Hessenberg representation
This paper proposes two design methods of reduced-order con tro Ilers for gust load alleviation (GLA) systems of an elastic wing. One of the methods is based on the use of the generalized Hessenberg representation
=(A - B K
0 -
K~ C )~+ K~y
(2a) (2b)
u=-K o 'l<'
where K o and K ~ represen t the optimal regulator and Kalman fil ter gains. respectively. Equation (2) is transformed into the following GHR form (Tse. Medanic and Perkins. 1978) using the state variable transformation. z = T 'K.
ROC DESIGN BY THE GHR METHOD Consider J/ -input. m-output . n-state variable system.
Fig. 1. St r ucture of GHR.
153
A. Fujilllori. H. Ohta and P. :--J. Nikiforuk z=Fz+Gy u=Dz
(3a)
The order of the ROC n r is
(3b)
nr= Ell J j=1
where
F
=[~
I I ••
~
: ':"
0
]
G
[l 1
=
1
: '. F k-l k : Fkl ......... Fkk Gk D=[IIO"·O]. Z=[Z I T "'Zk T ]T
Singular perturbation method
(4)
. Z i
k F
IJ
E9{II"IJ.
lIl~ lI,+I.
n =
E lI, i=1
The transformed state variable z is arranged in the order depending on the observability of the state variables. F J+1J (j=I ... ·.D show the couplings from 'the upper state variables' z J (j=l. .. ·.D to 'the lower state variable' z 1+1. These couplings increase as the subscript i becomes large. Conversely. the only coupling from z 1+1 to Z J (j=l ... ·.D is through F 11+1 (see Fig. 1). Since the output u of the controller is equal to Z 1 from the structure of D in Eq. (4). F J+l j (j=I ... ·.D do not directly affect u. Therefore. if F 11+1 is small. the order reduction of the controller is possible by truncating the state variable Z 1+1 or by approximating F 11+1 Z 1+1 in terms of Z I. Two order reduction methods will be proposed in the fo 1I0wing. Order Reduction Method Norm minimization (RM1). fo rm is wri tten as
(13)
The i-I th 0=2 ... ·.k) GHR
(j)
= F
(j)
Z
(j)
+ F 11+1'
ZI+
1= F (D' Z (D + F 1+1 1+1'
G (j) y
ZI+
G 1+1' Y
(14a) 04b)
It is further assumed that F 1+11+1' is stable matrix. Then. Mul tiplying /(, ~ 1 / 11 F 1+11+ 1' 11 to Eq. 04b).
/(, i
1= F"(D Z (D+F" 1+11+1 Z I +1:: 1+1 y
(15)
and taking the limitation /(,-0. the steady state of Z I is obtained as
Z 1= - F" l+ll+ I - I (F"(D Z (D + 1; 1+1 y)
(6)
Letting FI,+I'F,+II+ I -I=F .. and substituting it into Eq. 04b). a ROC is obtained as
z
(j)
=[F
(j) -
F•
Z
F"(j)]
(j)
+ [G (j) - F ..1:: 1+1 ] y • (17)
ROC DESIGN BY THE NFA METHOD The other method for a ROC design is the Nyquist frequency approximation (NFA) in the frequency domain. The transfer function of a ROC will be obtained by approximating the desirable loop transfer function over the frequency region where the stability of the closed-loop system is judged. The primary objective of the NFA method is to improve the stabili ty margins.
(5)
Procedure of the NFA Method Z J E9{~J
(j=I ... ·.i-1).
From the last row of Eq. system is considered.
Z 1- 1 =
F 11'
Z 1-1 +
(5).
the following residual
We consider a single-input system represented by Eq. (1). Let the loop transfer function which consists of the plant and a n r-order strictly proper controller be L R(S)
F ,0-1) ' Z (1-1) + G ,' y
Z 1= K I Z I-I zIE9{II. lIl~n,-nl+1 Z 0-1) ~[ Z 1T ... Z I-I T]T
(6) (7)
(8)
Equa tion (7) represen ts the con tribu tion of Z I-I to Z I which is obtained in the i-th GHR transformation. Differentiating Eq. (7).
= K I(F 11' Z I-I + FO-1)' Z 0-1) + G I' y)
(9)
If the first term of the right hand side of Eq. can be approxima ted as
(9)
(0)
a ROC for Eq. (5) is obtained. where F 11 is given as
=H R(S) H (s)
(18)
where the transfer function of compensa tor H R(S) is expressed as
H R(S) =
clsr[
I/) RI (s) ... 1/) Rm(S)]
H (s)
=C (s 1 n -
transfer function of the con-
A) - I B
~1ifsr-[ I/) I (S)"'l/)m(S)]
m Z
p
W p) -
E I/) l(j W p) I/)RI(j w p) = 0 i=1 (22)
(p=l ... ·.n .. ).
The coefficients of H R(S) are in general calcula ted from Eq. (22) using the least square method. In particular, when n a + (m + 1) n r, Eq. (22) becomes a simul taneous equation and has an unique solution. With the compensator H R(S) thus obtained, the stability margin of L R(S) is examined on the Nyquist plot.
=
(2)
(21)
Now. defining the set poin ts Z p (= x p + j y P. p=l. .. ·.n ... n .. ~+(m+ Unr) at which LR(S) should be match to the desirable Nyquist plot. and using Eqs. (8) - (21). the following equations are obtained.
(11) F 11 of Eq. (1) minimizes a norm 11 KIF 11' - F 11 K I 11. The inverse matrix (K I K IT)-I exists because the rank of K I ( E 9{ l l "n l) is II I. Thus. the reduced-order system for Eq . (5) becomes
(9)
(20a) (20b)
155
Gust Load AlleYiatioll System APPLICATION TO A GUST LOAD ALLEVIATION SYSTEM Wing Model
that the gust model could be approximated by a second-order system
W..LslW(s) -
The wing model used for the experimental study consists of wing elements and an elastic beam which is made of wing elements and an aluminium with an uniform cross section as show in Fig. 2. Ten standard wing elements whose airfoil section is NACA DVLOOO18-5540 and a tip wing e lemen t which has an aileron driven by an installed electric motor were attached on the elastic beam. The frequency response of the actuator is nonlinear and depends on the ampli tude. Al though the maximum deflection angle of the aileron is 40 deg. and the maximum angular velocity is 1700 deg/s, permissible values of the root mean square (rms) of the deflection angle f3 and angular velocity ~ are assumed to be 15 deg. and 800 deg/s, respectively. Based on the experimental data of a command input u ± 10 deg, the actuator dynamics are mode led as a second -order system
=
(23)
Two accelerometors were placed on the beam near the wing tip as shown in Fig. 2, while strain gages for sensing bending and torsional strains were put on the beam near the wing root and on the center of the span, respectively. Gust Generator The flag and the generate a gust. was adjusted to mode of the wing spectra based on
grid shown in Fig. 3 were used to The peak frequency of the gust excite the first natural bending model. A study of the gust power Akaike's AIC cri terion was shown
961 4\0 STRA I N GAGE
STIIA I N GAGE
+ I ~9
J:i2 s
(24)
85.96898.6 .
where W" is the gust velocity and W is a white noise with an intensity of 17.5 m2 /s". Control System Design Analytical model. The primary natural modes of the wing are the first and second bending modes and the first torsional mode as shown in Table 1. The second bending and the first torsional modes were neglected in the synthesis of the optimal regulator, because the natural frequencies of these modes are far from that of the first bending mode, and because the bandwidth of the actuator is about 8 Hz. Therefore, these modes were considered unmode led elements in the control system design. The analytical model for GLA consists of the first bending mode of the wing, gust model, and actuator dynamics. The rational fraction approximations of the Theodorsen and the Kiissner functions are respectively included in the aerodynamics to obtain a state representation of the wing model. Accordingly, the analytical model for GLA can be expressed as Eq. 0), where x is the ten th-order state variable, and U is the first-order input. The bending acceleration and/or the bending strain are chosen as the measurement outputs for implementing controllers. Optimal regulator. To obtain an optimal regulator which has good control performance and is robust for unmodeled higher modes, a synthesis of optimal regulator using singular value is applied (Ohta and FUjimori, 1988). The controlled system G(s) has to be of low sensi tivity in the low frequeny region. This requirement can be represented in terms of the loop transfer function G (s) K c as (25) where l! represents the minimum sinular value except zero, and represents the maximum singular value. Jl(w) must satisfy O
a
A I LERON
1\0
7\0 1050
~
I
I
1015
Fig. 2. Elastic wing model.
The requirement of stability robustness is to make a[G(jw)Ko] in the high frequency region as small as possible. Let the upper bound of a mul tiple uncertainty of the model be denoted as 2 m(W). Then, the condition of stability robustness can be expressed as (Dolye and Stein, 1981) (26) Table 1 Oscillatory modes of wing model
Mode No.
Fig. 3. Gust genera tor.
Mode name
Frequency (Hz)
Bending
1.250
2
Bend i ng 2
8.375
3
Torsion
1 \. 625
156
A. Fujimori. H . O h la and P. :-I . Nikifo ruk
40
20
'0 0 2.0
-20~--~------+---~--~--~
2
1
10 20
5 W
50 100 -- - -- - -- ----~~~-~
(rod / s)
Fig. 4. Singular value plot.
..0" - - - - - 0 - - - -:'::.0-
'0
n,
Table 2 Performance of oI1timal regulator and LQG controllers Ca i n H.
Phase 11 .
h ( ols')
(d O)
( des)
( . ( u )
P (de s )
iJ
Cos t
.s
( degls)
1.9OG 411.4
Contr o l o f(
'00
Opliul regu I a tor
y=ii y
=
c
11
y= [~ .J
'00
, 180. +70. 1
1.14 3 42.92
10 . 98 159 .8
1. 41 96 4
.1 6.42
'85 . 9 ' 96.5
1. 460 190.1
1 .581 72.86
4 .88546
'6.681
' 159. . 52.2
1. 385 109. 1
12 .1 6 148.8
2.13268
tW .G2
'119. . U5.9
1.342 11 3.0
10 .59 11 2.9
2.55584
'0
n,
where w u is the crossover frequency of 1I m( w ). The crossover frequency of (J [ G (j w) K o ], W o , is related to the response speed of the system. In this paper, W c is determined f r om the settling time which is an al ternative measure of the response speed . The specification for the GLA control system of an elastic rectangular wing is given as follows. A value of JHw)=0.2
'0
Fig. 5. Control perfo r mance of the GHR method (simulation resul t). Table 3 Control I1erformance of the NFA method (simulation result) Out put
ii
(
.
ii
( o/s')
P ( deg)
Cost
Ga in 11 .
Phase 11 .
(d B)
(dex)
(. ( u )
NII I
+\ 5. 16
'19.1 '72.2
1. 395 140. 9
9.965 \04. 4
3.2619 1
NII2
' 13.36
'60.8 +74.9
1.399 14 1.2
10.23 108 . 4
3.35128
NI 13
'1 2.34
'50 .1 +76 . 1
1.4 18 146.2
10.49 11 2.4
3.6 1363
N11 4
'16.01
' 111 . ' 66. 7
1.4 37 156.8
9.633 99.33
3.7752 1
N2 11
+\ 4 .52
' 180. ' 39.2
1. 138 24.76
13. 44 131. 5
I. 42509
N212
'1 2 .83
' 180. ' 35.6
1. 134 19.30
13.53 129.6
1. 45224
N2 13
'1 2.58
, 180. ' 34.0
1.1 32 15.5 1
13.6 1 127. 0
1. 46044
N2 14
+1 1.64
'180. +31.2
1. 131 12.71
13.68 123.6
1.50280
N215
+11 .2 1
' 180. +28.9
1. 131 10.54
13. 76 11 8. 7
\. 53486
Nallle
y
P (de'gl,)
The control performance of the closed-loop system with full-order LQG compensators is also shown in Table 2. Figure 5 illustrates the control performance of the controllers designed .Py the GHR method. Comparing the rms values of hand £ b in this table, the controlle r s with the bending strain output show larger reduction of their rms values than those with the acceleration output, although the control efforts, that Is, the rms values of fJ and iJ, increase. These values are far below the prescribed bounds, fJ = 15 deg and iJ = 800 deg/s. This shows that the GLA systems with a bending strain output are superior to the ones with an acceleration output. The first or second order
Gust Load Alleviation System con tro llers are obtained in almost every simulation case. They show nearly as good control performance as the LQG compensa tors. Table 3 shows the performance of the Roes designed by the NFA method. The order of the controllers Is set to be n .=1 In this design. In the acceleration output cases, the Roes have large negative phase margin, and are superior to the ROCs designed by the GHR method. This is due to the fact that the frequency responses of these Roes are designed so as to approximate that of the optimal regulator. In the bending strain output, the Roes which have nearly the same control performance as the optimal regulator are obtained by the NFA method. The rms values of {J and iJ of these Roes are generally larger than those by the GHR, which leads to better con tro 1 performance.
157
achieved. For all experimental cases, a satisfactory reduction of bending strain was achieved, bu t the acceleration responses were no t as much reduced as expected. Unlike the analytical results, lower order controllers among the Roes showed better control performance. A reason for this discrepancy may be due to its digitalization of the control laws, because the computation time becomes longer as the dimension of the controller becomes higher. Thus, the time for activating the
~
'"VI 0·02
i
:-C
0·01 0
Wind-Tunnel Test The block diagram of the measurement and control systems used for the wind-tunnel test is shown in Fig. 6. The output signals from the accelerometors or strain gages were sampled using an A/D converter, and the control law was calculated using a microprocessor PC-9801m2. The pulse signals to the aileron deflection angle were then output through a % board. At the same time, a digital spectrum analyzer was used to observe the input and output signals, and the accelerometors, strain gages and aileron angle signals were recorded.
VI
""... w
~ OFF
V ON
400 0
...VI
" .,
The wind-tunnel test was carried out using the ROCs derived in the previous section under the condition that the wing is initially oscillating at the peak frequency of 1.25 Hz. The frequency of the first bending mode of the wing model is shown in Fig. 7. Under this condition, the rms bending moment is about 369 kgmm at the root of the wing. The power spectrum obtained using the second -order controller is also shown in this figure. It is seen that the first bending mode is reduced to less than half of its control-off value and that the second bending and the first torsional modes are no t excited. The time history is shown in Fig. 8, where the amplitudes of the acceleration and bending strain are effectively reduced due to the active control of the aileron and that the amplitude of the torsional strain remains almost unchanged.
800
w
30
t~FF ON
15 0
0·2
0·5
1
2
5
10
20
Frequency (Hz ) Fig. 7. Power spectra of the bending acceleration and bending and torsional strains (6 m/s).
Figure 9 illustrates the reduction ratiO 1/ of the bending moment at the wing root W the deflection angle of the ai leron {J. It is shown in this figure that gust load alleviation of 10.3 - 60.8 % is
COMPUTER A C E G
POTENTIOMETER. B: STRAIN GAGE ACCELEROMETER. D: DC MOTOR AMPLIFIER. F : DATA RECORDER SPECTRUM ANALYZER, H: SERVOPACK
Fig. 6. Signal flow of the wind-tunnel test.
o
20
40 Time (sec)
Fig. 8. Time history (6 m/g).
60
80
A. Fujimori . H. Ohta and P. N. Nikiforuk
158
100
0
"
t
.,. 50
0
y.h
15
€,
Ch
6 ,1'
..
0.
Ou tpu t
~
Na.e
h (lis' )
I: ,
I:
(/./ )
( /./ )
2 . 194
481 . 6
18 .05
NIII HI12 NI13 NI14
1. 651 I. 551 1.69 1 I. 808
273. 3 284.9 302 . 7 33 1. 9
N211 N212 N21 3 N214 N215
1.336 I. 235 1.23 1 I. 259 I. 215
162. 9 167 . 9 167 . 1 171.1 174.7
t
fJ
~
(deg)
(96 )
17 .3 1 17 . 82 16. 68 16 .09
12.40 12.12 12.42 12 . 05
43 . 3 40.8 37 . 1 31.1
17 . 82 17.31 16 .09 17 . 88 14.17
14.85 14.81 14. 98 14.92 14.76
66 . 2 65 . 1 65.3 64. 5 63.7
"'-
~ n,
Table 4 Control 2erformance of the NFA method (ex2erimen tal resulO
Control off
10
ii
10
n,
10
1:,
Flg.9. Control performance of the GHR method (experimen tal resulO . .10. '
10 ~If)
;;-
E
:.c
designed wind speed. respectively. Controllers with the bending strain output can achieve GLA effect of 30.6 % in the best case.
5 0
CONCLUDING REMARKS
-;n 80
'3 oD
w
40 0 15
tI1
~~
.
tU
10 5
0·2
0·5
2
5
10
20
Frequency (Hz)
Fig. 10. Power spectra of the bending acceleration and bending and torsional strains (9 m/s). aileron becomes shorter in one sampling period. Furthermore. the aileron movement may not be appropriate when the computation time is longer. This is observed in some cases where the torsional strain is increased in spite of lesser amounts of aileron activation. Controllers obtained by the NFA method show a good control performance in the experiment like in the simulation. The best reduction ratio of the bending moment at the wing root is 66.2 % as shown in Table 4. A GLA experiment in a higher wind velocity. U=9 m/so was carried out using controllers which show good control performance in the designed wind velocity U=6 m/so This experiment is a verification of the robustness for the parameter perturbation of the controlled system. The peak frequency of the wing oscillation increases from 1.25 Hz to 1.75 Hz as shown in Fig. 10. The open-loop magnitudes of the acceleration and the bending strain spectra decrease to 49 96 and 17 96 of the values in the
This paper proposed two design methods of reduced-order controllers for the gust load alleviation systems of a cantilevered elastic rectangular wing. and examined the performance of the designed controllers by both simulation study and Wind-tunnel experiments. The Ist- or the 2nd-order controllers was designed by these methods. and they showed as good performance as the LQG compensator. The difference of the results between the simulation and the experiment is due to that the calculation time of the control law is not taken into account and approximate expressions of the unsteady aerodynamics lead to the uncertainty of the mode ling. Nevertheless. the designed reduced-order controller showed the gust load alleViation effect of 66.2 96 in the best case. Furthermore. the ROCs designed by the GHR and the NFA methods are robust for the parameter variation of the controlled system. because they show the control effect at the wind velocity of 50 96 increase of the design poin 1.
ACKNOWLEDGEMENT Part of this research was supported by the Natural Sciences and Engineering Research Council of Canada under Gran ts No. A-5625 and A-IOSO. and part by Ishida Founda tion .
REFERENCES Dolye. J.C •• and G. Stein (19S1). Mul tivariable Feedback Design : Concepts for a Classical /Modern SyntheSiS. IEEE Trans. AutomatiC Control. AC-26. 1. 4-16. Ohta. H• • and A. Fujimori (1988). A Synthesis of Robust Optimal Regulators Using Singular Value with Application to Gust Load Alleviation. AIAA Pa2er. 88-4114-CP. 519-52S. Tse. E.C.Y •• J.V. Medanic. and W. R. Perkins <197S). Generalized Hessenberg Transformat i on for Reduced-Order Modeling of Large-Scale Systems. Int. J . Control. 27. 493-512.
CONTROLLER DESIGNS OF A GUST LOAD ALLEVIATION SYSTEM FOR AN ELASTIC RECTANGULAR WING A. Fujimori*, H. Ohta* and P. N. Nikiforuk** *Departlllent of A eronautical Engineering, N ago)'a U nil'enit)·, Chikllsa-kll, N ago)'a , Japan **College of Engineering, U niversity uf Saskatchewan , Saskatuun, Saskatchewan , Canada Abstract. This paper proposes two design methods of reduced-order controllers for gust load alleviation (GLA) systems of an elastic wing . and examines the con tro I performance using both simulation studies and wind-tunnel experiments. One of the methods is based on the use of the generalized Hessenberg representation
load
alleviation;
optimal
INTRODUCTION
control;
order
reduction;
robustness;
x=Ax+Bu+Bnw y=Cx+v
In recent years. many works of active control for aeroelastic systems. for example. active flutter suppression (AFS) and gust load alleviation (GLA). are reported. These control systems are required for future aircraft in the sense of the energy -efficient and high performance. A problem of aeroelastic systems is the complication of the control laws which are caused by high order of aeroelastic systems. Since the complete dynamics of the aeroelastic system is in general described by partial differential equations for both structural and aerodynamic modelings. the state-space equat i ons may require a la r ge number of states to represent such dynamics accurately. As well. an optimal control law based on the standard linear quadratic Gaussian (LQG) technique would be of the same high order as the plant. Such a controller. however. should be simple enough to implement on a flight computer and has to work well under the presence of the uncertainty of the mode ling and parameter change. Therefore. it is worthwhile to design a reduced-order controller
Oa) Ob)
x E!R n • y E!R m • u E!RR where wand v represent pro cess and measurement noises. respectively. The first method of order reduction is the one based on the generalized Hessenberg representation
This paper proposes two design methods of reduced-order con tro Ilers for gust load alleviation (GLA) systems of an elastic wing. One of the methods is based on the use of the generalized Hessenberg representation
=(A - B K
0 -
K~ C )~+ K~y
(2a) (2b)
u=-K o 'l<'
where K o and K ~ represen t the optimal regulator and Kalman fil ter gains. respectively. Equation (2) is transformed into the following GHR form (Tse. Medanic and Perkins. 1978) using the state variable transformation. z = T 'K.
ROC DESIGN BY THE GHR METHOD Consider J/ -input. m-output . n-state variable system.
Fig. 1. St r ucture of GHR.
153
A. Fujilllori. H. Ohta and P. :--J. Nikiforuk z=Fz+Gy u=Dz
(3a)
The order of the ROC n r is
(3b)
nr= Ell J j=1
where
F
=[~
I I ••
~
: ':"
0
]
G
[l 1
=
1
: '. F k-l k : Fkl ......... Fkk Gk D=[IIO"·O]. Z=[Z I T "'Zk T ]T
Singular perturbation method
(4)
. Z i
k F
IJ
E9{II"IJ.
lIl~ lI,+I.
n =
E lI, i=1
The transformed state variable z is arranged in the order depending on the observability of the state variables. F J+1J (j=I ... ·.D show the couplings from 'the upper state variables' z J (j=l. .. ·.D to 'the lower state variable' z 1+1. These couplings increase as the subscript i becomes large. Conversely. the only coupling from z 1+1 to Z J (j=l ... ·.D is through F 11+1 (see Fig. 1). Since the output u of the controller is equal to Z 1 from the structure of D in Eq. (4). F J+l j (j=I ... ·.D do not directly affect u. Therefore. if F 11+1 is small. the order reduction of the controller is possible by truncating the state variable Z 1+1 or by approximating F 11+1 Z 1+1 in terms of Z I. Two order reduction methods will be proposed in the fo 1I0wing. Order Reduction Method Norm minimization (RM1). fo rm is wri tten as
(13)
The i-I th 0=2 ... ·.k) GHR
(j)
= F
(j)
Z
(j)
+ F 11+1'
ZI+
1= F (D' Z (D + F 1+1 1+1'
G (j) y
ZI+
G 1+1' Y
(14a) 04b)
It is further assumed that F 1+11+1' is stable matrix. Then. Mul tiplying /(, ~ 1 / 11 F 1+11+ 1' 11 to Eq. 04b).
/(, i
1= F"(D Z (D+F" 1+11+1 Z I +1:: 1+1 y
(15)
and taking the limitation /(,-0. the steady state of Z I is obtained as
Z 1= - F" l+ll+ I - I (F"(D Z (D + 1; 1+1 y)
(6)
Letting FI,+I'F,+II+ I -I=F .. and substituting it into Eq. 04b). a ROC is obtained as
z
(j)
=[F
(j) -
F•
Z
F"(j)]
(j)
+ [G (j) - F ..1:: 1+1 ] y • (17)
ROC DESIGN BY THE NFA METHOD The other method for a ROC design is the Nyquist frequency approximation (NFA) in the frequency domain. The transfer function of a ROC will be obtained by approximating the desirable loop transfer function over the frequency region where the stability of the closed-loop system is judged. The primary objective of the NFA method is to improve the stabili ty margins.
(5)
Procedure of the NFA Method Z J E9{~J
(j=I ... ·.i-1).
From the last row of Eq. system is considered.
Z 1- 1 =
F 11'
Z 1-1 +
(5).
the following residual
We consider a single-input system represented by Eq. (1). Let the loop transfer function which consists of the plant and a n r-order strictly proper controller be L R(S)
F ,0-1) ' Z (1-1) + G ,' y
Z 1= K I Z I-I zIE9{II. lIl~n,-nl+1 Z 0-1) ~[ Z 1T ... Z I-I T]T
(6) (7)
(8)
Equa tion (7) represen ts the con tribu tion of Z I-I to Z I which is obtained in the i-th GHR transformation. Differentiating Eq. (7).
= K I(F 11' Z I-I + FO-1)' Z 0-1) + G I' y)
(9)
If the first term of the right hand side of Eq. can be approxima ted as
(9)
(0)
a ROC for Eq. (5) is obtained. where F 11 is given as
=H R(S) H (s)
(18)
where the transfer function of compensa tor H R(S) is expressed as
H R(S) =
clsr[
I/) RI (s) ... 1/) Rm(S)]
H (s)
=C (s 1 n -
transfer function of the con-
A) - I B
~1ifsr-[ I/) I (S)"'l/)m(S)]
m Z
p
W p) -
E I/) l(j W p) I/)RI(j w p) = 0 i=1 (22)
(p=l ... ·.n .. ).
The coefficients of H R(S) are in general calcula ted from Eq. (22) using the least square method. In particular, when n a + (m + 1) n r, Eq. (22) becomes a simul taneous equation and has an unique solution. With the compensator H R(S) thus obtained, the stability margin of L R(S) is examined on the Nyquist plot.
=
(2)
(21)
Now. defining the set poin ts Z p (= x p + j y P. p=l. .. ·.n ... n .. ~+(m+ Unr) at which LR(S) should be match to the desirable Nyquist plot. and using Eqs. (8) - (21). the following equations are obtained.
(11) F 11 of Eq. (1) minimizes a norm 11 KIF 11' - F 11 K I 11. The inverse matrix (K I K IT)-I exists because the rank of K I ( E 9{ l l "n l) is II I. Thus. the reduced-order system for Eq . (5) becomes
(9)
(20a) (20b)
155
Gust Load AlleYiatioll System APPLICATION TO A GUST LOAD ALLEVIATION SYSTEM Wing Model
that the gust model could be approximated by a second-order system
W..LslW(s) -
The wing model used for the experimental study consists of wing elements and an elastic beam which is made of wing elements and an aluminium with an uniform cross section as show in Fig. 2. Ten standard wing elements whose airfoil section is NACA DVLOOO18-5540 and a tip wing e lemen t which has an aileron driven by an installed electric motor were attached on the elastic beam. The frequency response of the actuator is nonlinear and depends on the ampli tude. Al though the maximum deflection angle of the aileron is 40 deg. and the maximum angular velocity is 1700 deg/s, permissible values of the root mean square (rms) of the deflection angle f3 and angular velocity ~ are assumed to be 15 deg. and 800 deg/s, respectively. Based on the experimental data of a command input u ± 10 deg, the actuator dynamics are mode led as a second -order system
=
(23)
Two accelerometors were placed on the beam near the wing tip as shown in Fig. 2, while strain gages for sensing bending and torsional strains were put on the beam near the wing root and on the center of the span, respectively. Gust Generator The flag and the generate a gust. was adjusted to mode of the wing spectra based on
grid shown in Fig. 3 were used to The peak frequency of the gust excite the first natural bending model. A study of the gust power Akaike's AIC cri terion was shown
961 4\0 STRA I N GAGE
STIIA I N GAGE
+ I ~9
J:i2 s
(24)
85.96898.6 .
where W" is the gust velocity and W is a white noise with an intensity of 17.5 m2 /s". Control System Design Analytical model. The primary natural modes of the wing are the first and second bending modes and the first torsional mode as shown in Table 1. The second bending and the first torsional modes were neglected in the synthesis of the optimal regulator, because the natural frequencies of these modes are far from that of the first bending mode, and because the bandwidth of the actuator is about 8 Hz. Therefore, these modes were considered unmode led elements in the control system design. The analytical model for GLA consists of the first bending mode of the wing, gust model, and actuator dynamics. The rational fraction approximations of the Theodorsen and the Kiissner functions are respectively included in the aerodynamics to obtain a state representation of the wing model. Accordingly, the analytical model for GLA can be expressed as Eq. 0), where x is the ten th-order state variable, and U is the first-order input. The bending acceleration and/or the bending strain are chosen as the measurement outputs for implementing controllers. Optimal regulator. To obtain an optimal regulator which has good control performance and is robust for unmodeled higher modes, a synthesis of optimal regulator using singular value is applied (Ohta and FUjimori, 1988). The controlled system G(s) has to be of low sensi tivity in the low frequeny region. This requirement can be represented in terms of the loop transfer function G (s) K c as (25) where l! represents the minimum sinular value except zero, and represents the maximum singular value. Jl(w) must satisfy O
a
A I LERON
1\0
7\0 1050
~
I
I
1015
Fig. 2. Elastic wing model.
The requirement of stability robustness is to make a[G(jw)Ko] in the high frequency region as small as possible. Let the upper bound of a mul tiple uncertainty of the model be denoted as 2 m(W). Then, the condition of stability robustness can be expressed as (Dolye and Stein, 1981) (26) Table 1 Oscillatory modes of wing model
Mode No.
Fig. 3. Gust genera tor.
Mode name
Frequency (Hz)
Bending
1.250
2
Bend i ng 2
8.375
3
Torsion
1 \. 625
156
A. Fujimori. H . O h la and P. :-I . Nikifo ruk
40
20
'0 0 2.0
-20~--~------+---~--~--~
2
1
10 20
5 W
50 100 -- - -- - -- ----~~~-~
(rod / s)
Fig. 4. Singular value plot.
..0" - - - - - 0 - - - -:'::.0-
'0
n,
Table 2 Performance of oI1timal regulator and LQG controllers Ca i n H.
Phase 11 .
h ( ols')
(d O)
( des)
( . ( u )
P (de s )
iJ
Cos t
.s
( degls)
1.9OG 411.4
Contr o l o f(
'00
Opliul regu I a tor
y=ii y
=
c
11
y= [~ .J
'00
, 180. +70. 1
1.14 3 42.92
10 . 98 159 .8
1. 41 96 4
.1 6.42
'85 . 9 ' 96.5
1. 460 190.1
1 .581 72.86
4 .88546
'6.681
' 159. . 52.2
1. 385 109. 1
12 .1 6 148.8
2.13268
tW .G2
'119. . U5.9
1.342 11 3.0
10 .59 11 2.9
2.55584
'0
n,
where w u is the crossover frequency of 1I m( w ). The crossover frequency of (J [ G (j w) K o ], W o , is related to the response speed of the system. In this paper, W c is determined f r om the settling time which is an al ternative measure of the response speed . The specification for the GLA control system of an elastic rectangular wing is given as follows. A value of JHw)=0.2
'0
Fig. 5. Control perfo r mance of the GHR method (simulation resul t). Table 3 Control I1erformance of the NFA method (simulation result) Out put
ii
(
.
ii
( o/s')
P ( deg)
Cost
Ga in 11 .
Phase 11 .
(d B)
(dex)
(. ( u )
NII I
+\ 5. 16
'19.1 '72.2
1. 395 140. 9
9.965 \04. 4
3.2619 1
NII2
' 13.36
'60.8 +74.9
1.399 14 1.2
10.23 108 . 4
3.35128
NI 13
'1 2.34
'50 .1 +76 . 1
1.4 18 146.2
10.49 11 2.4
3.6 1363
N11 4
'16.01
' 111 . ' 66. 7
1.4 37 156.8
9.633 99.33
3.7752 1
N2 11
+\ 4 .52
' 180. ' 39.2
1. 138 24.76
13. 44 131. 5
I. 42509
N212
'1 2 .83
' 180. ' 35.6
1. 134 19.30
13.53 129.6
1. 45224
N2 13
'1 2.58
, 180. ' 34.0
1.1 32 15.5 1
13.6 1 127. 0
1. 46044
N2 14
+1 1.64
'180. +31.2
1. 131 12.71
13.68 123.6
1.50280
N215
+11 .2 1
' 180. +28.9
1. 131 10.54
13. 76 11 8. 7
\. 53486
Nallle
y
P (de'gl,)
The control performance of the closed-loop system with full-order LQG compensators is also shown in Table 2. Figure 5 illustrates the control performance of the controllers designed .Py the GHR method. Comparing the rms values of hand £ b in this table, the controlle r s with the bending strain output show larger reduction of their rms values than those with the acceleration output, although the control efforts, that Is, the rms values of fJ and iJ, increase. These values are far below the prescribed bounds, fJ = 15 deg and iJ = 800 deg/s. This shows that the GLA systems with a bending strain output are superior to the ones with an acceleration output. The first or second order
Gust Load Alleviation System con tro llers are obtained in almost every simulation case. They show nearly as good control performance as the LQG compensa tors. Table 3 shows the performance of the Roes designed by the NFA method. The order of the controllers Is set to be n .=1 In this design. In the acceleration output cases, the Roes have large negative phase margin, and are superior to the ROCs designed by the GHR method. This is due to the fact that the frequency responses of these Roes are designed so as to approximate that of the optimal regulator. In the bending strain output, the Roes which have nearly the same control performance as the optimal regulator are obtained by the NFA method. The rms values of {J and iJ of these Roes are generally larger than those by the GHR, which leads to better con tro 1 performance.
157
achieved. For all experimental cases, a satisfactory reduction of bending strain was achieved, bu t the acceleration responses were no t as much reduced as expected. Unlike the analytical results, lower order controllers among the Roes showed better control performance. A reason for this discrepancy may be due to its digitalization of the control laws, because the computation time becomes longer as the dimension of the controller becomes higher. Thus, the time for activating the
~
'"VI 0·02
i
:-C
0·01 0
Wind-Tunnel Test The block diagram of the measurement and control systems used for the wind-tunnel test is shown in Fig. 6. The output signals from the accelerometors or strain gages were sampled using an A/D converter, and the control law was calculated using a microprocessor PC-9801m2. The pulse signals to the aileron deflection angle were then output through a % board. At the same time, a digital spectrum analyzer was used to observe the input and output signals, and the accelerometors, strain gages and aileron angle signals were recorded.
VI
""... w
~ OFF
V ON
400 0
...VI
" .,
The wind-tunnel test was carried out using the ROCs derived in the previous section under the condition that the wing is initially oscillating at the peak frequency of 1.25 Hz. The frequency of the first bending mode of the wing model is shown in Fig. 7. Under this condition, the rms bending moment is about 369 kgmm at the root of the wing. The power spectrum obtained using the second -order controller is also shown in this figure. It is seen that the first bending mode is reduced to less than half of its control-off value and that the second bending and the first torsional modes are no t excited. The time history is shown in Fig. 8, where the amplitudes of the acceleration and bending strain are effectively reduced due to the active control of the aileron and that the amplitude of the torsional strain remains almost unchanged.
800
w
30
t~FF ON
15 0
0·2
0·5
1
2
5
10
20
Frequency (Hz ) Fig. 7. Power spectra of the bending acceleration and bending and torsional strains (6 m/s).
Figure 9 illustrates the reduction ratiO 1/ of the bending moment at the wing root W the deflection angle of the ai leron {J. It is shown in this figure that gust load alleviation of 10.3 - 60.8 % is
COMPUTER A C E G
POTENTIOMETER. B: STRAIN GAGE ACCELEROMETER. D: DC MOTOR AMPLIFIER. F : DATA RECORDER SPECTRUM ANALYZER, H: SERVOPACK
Fig. 6. Signal flow of the wind-tunnel test.
o
20
40 Time (sec)
Fig. 8. Time history (6 m/g).
60
80
A. Fujimori . H. Ohta and P. N. Nikiforuk
158
100
0
"
t
.,. 50
0
y.h
15
€,
Ch
6 ,1'
..
0.
Ou tpu t
~
Na.e
h (lis' )
I: ,
I:
(/./ )
( /./ )
2 . 194
481 . 6
18 .05
NIII HI12 NI13 NI14
1. 651 I. 551 1.69 1 I. 808
273. 3 284.9 302 . 7 33 1. 9
N211 N212 N21 3 N214 N215
1.336 I. 235 1.23 1 I. 259 I. 215
162. 9 167 . 9 167 . 1 171.1 174.7
t
fJ
~
(deg)
(96 )
17 .3 1 17 . 82 16. 68 16 .09
12.40 12.12 12.42 12 . 05
43 . 3 40.8 37 . 1 31.1
17 . 82 17.31 16 .09 17 . 88 14.17
14.85 14.81 14. 98 14.92 14.76
66 . 2 65 . 1 65.3 64. 5 63.7
"'-
~ n,
Table 4 Control 2erformance of the NFA method (ex2erimen tal resulO
Control off
10
ii
10
n,
10
1:,
Flg.9. Control performance of the GHR method (experimen tal resulO . .10. '
10 ~If)
;;-
E
:.c
designed wind speed. respectively. Controllers with the bending strain output can achieve GLA effect of 30.6 % in the best case.
5 0
CONCLUDING REMARKS
-;n 80
'3 oD
w
40 0 15
tI1
~~
.
tU
10 5
0·2
0·5
2
5
10
20
Frequency (Hz)
Fig. 10. Power spectra of the bending acceleration and bending and torsional strains (9 m/s). aileron becomes shorter in one sampling period. Furthermore. the aileron movement may not be appropriate when the computation time is longer. This is observed in some cases where the torsional strain is increased in spite of lesser amounts of aileron activation. Controllers obtained by the NFA method show a good control performance in the experiment like in the simulation. The best reduction ratio of the bending moment at the wing root is 66.2 % as shown in Table 4. A GLA experiment in a higher wind velocity. U=9 m/so was carried out using controllers which show good control performance in the designed wind velocity U=6 m/so This experiment is a verification of the robustness for the parameter perturbation of the controlled system. The peak frequency of the wing oscillation increases from 1.25 Hz to 1.75 Hz as shown in Fig. 10. The open-loop magnitudes of the acceleration and the bending strain spectra decrease to 49 96 and 17 96 of the values in the
This paper proposed two design methods of reduced-order controllers for the gust load alleviation systems of a cantilevered elastic rectangular wing. and examined the performance of the designed controllers by both simulation study and Wind-tunnel experiments. The Ist- or the 2nd-order controllers was designed by these methods. and they showed as good performance as the LQG compensator. The difference of the results between the simulation and the experiment is due to that the calculation time of the control law is not taken into account and approximate expressions of the unsteady aerodynamics lead to the uncertainty of the mode ling. Nevertheless. the designed reduced-order controller showed the gust load alleViation effect of 66.2 96 in the best case. Furthermore. the ROCs designed by the GHR and the NFA methods are robust for the parameter variation of the controlled system. because they show the control effect at the wind velocity of 50 96 increase of the design poin 1.
ACKNOWLEDGEMENT Part of this research was supported by the Natural Sciences and Engineering Research Council of Canada under Gran ts No. A-5625 and A-IOSO. and part by Ishida Founda tion .
REFERENCES Dolye. J.C •• and G. Stein (19S1). Mul tivariable Feedback Design : Concepts for a Classical /Modern SyntheSiS. IEEE Trans. AutomatiC Control. AC-26. 1. 4-16. Ohta. H• • and A. Fujimori (1988). A Synthesis of Robust Optimal Regulators Using Singular Value with Application to Gust Load Alleviation. AIAA Pa2er. 88-4114-CP. 519-52S. Tse. E.C.Y •• J.V. Medanic. and W. R. Perkins <197S). Generalized Hessenberg Transformat i on for Reduced-Order Modeling of Large-Scale Systems. Int. J . Control. 27. 493-512.