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 Authors : Ranarison Solofo Herizo , Randriamitantsoa Paul Auguste, Randriamitantsoa Andry Auguste, Reziky Zafimarina S.H.Z.T
 Paper ID : IJERTV7IS030093
 Volume & Issue : Volume 07, Issue 03 (March 2018)
 Published (First Online): 15032018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Contribution to Modeling a Robust Controller for a Flexible Aircraft
Ranarison Solofo Herizo, Randriamitantsoa Paul Auguste, Randriamitantsoa Andry Auguste,
Reziky Zafimarina S.H.Z.T
Abstract– This study uses the Hamilton method to derive the equation of motion of a flexible aircraft. The purpose is to design a robust controller using the and the looshaping methods. The method is used to analyze the system and the
gap, to compare the controllers.
Index Terms–Flexible aircraft, algorithm, loopshaping,
analysis, gap.
INTRODUCTION
Modeling the motion of a flexible aircraft is more complex than the case of a rigid aircraft. The flexibility of the structure increases the number of the parameters of the equations of motion. The components of the state vector are intensified, and it is complicating the design of an efficient controller for the system. In this article, only the longitudinal flight of a flexible airplane will be developed.

EQUATIONS OF MOTION
The function of Lagrange is defined by:
= (1)
And the Hamiltonian function is defined by:
= (2)
Fig. 1. Flexible aircraft model.
For the flexible aircraft showed in the Fig 1, the equations of energy are:
T 1 mu2 v2 w2
2
p
Where q represents the generalized coordinates, T is the kinetic
1 1 n 2

p q
2
rI q 2 M ii
energy, U is the potential energy and p is given by:
p L
(3)
r
1 n
i1
(5)
q
The equations of the motion of the system is given by the
U e
2 2 i M
i
i
2 i1
x
Hamilton canonic equations:

mg sin sin cos cos cos y
g
H
z
q p
p H Q
q
(4)
u v w : Velocities vector;
p q r : Angular velocities vector;
Where Q is the generalized force applied on the system.
m: Vehicle mass;
g: Gravity acceleration;
: The generalized displacement coordinates of the ith vibration mode;
: The undamped natural frequency of the ith vibration mode;
: The respective generalized mass of the ith vibration mode;
Ixx , I yy , Izz : The moments of inertia;
u
X u X


g X q
0 0
u

X

X E
Z Z
Zq Z
Z
Z
Z




, I
: The xy and yz inertial product;
0
u
0
E
xy yz
U 0 U 0
U 0 U 0
U 0 U 0
U 0
(9
, : The roll and the pitch angle;
0
0 0 1
0 0 0
0
0
q
E )
Ue: Elastic potential energy;
M u M
0 M q
M M
M
M
E
Ug: Elastic potential energy.
0
0 0 0 0
1 0 0
Then the function of Lagrange:
u
0 q
2
E
L 1
mu 2 v 2 w2
1 p
p
q rI q
1 M 2
x u q : The state vector;
n
2 2 2 i i
r
i 1
x
(6)
E : Thrust and elevator commands;
1 n 2 2
U : Equilibrium longitudinal velocity;
2
i i M i mg sin
i 1

sin cos

cos cos y 0
z
And the Hamiltonian function:
: Short notation for the derivative of
Qi ;
Mi
H mux mvy mwz 1 mu2 v2 w2
2
p
X ,Z ,M
: The derivative notation of X, Z and M
3. SYNTHESIS
1 1 n 2
3.1.
synthesis
p q
2
rI q 2 Mi i
r
i 1
x
(7)
The problem when designing a controller is to find a controller
K for a system P (Fig 2) that generates a signal u considering the

1 n 2 2i M mg sin sin cos cos cos y
information from y to mitigate the effect of w on z. In fact, the
2 i i
i 1
z
controller is synthesized while minimizing the closed loop norm
w to z.
With the Hamilton canonic equations, the longitudinal equation of
motionis given by:
Fl P,K
(10)
mu rv qw g sin X
mw qu pv g cos cos Z
qI yy ( p I xy rI yz ) (I xx I zz ) pr
(8)
Fl P,K : The lower linear fractional transformation of P and K, transfer function from w to z;
: A fixed real number.
yz
xy
xz
( pI rI
)q ( p2 r 2 )I M
2
Qi
i
i i
i
M
Where: X,Y,Z The longitudinal force, the normal force and the pitching moment;


SYSTEM DESCRIPTIONS AND MODELING
With some assumption in the parameters and for one vibration mode, the resulting of statespace model for linear control synthesis is given by:
Fig. 2. Standard interconnection for the synthesis.
3.2. Loop shaping synthesis
Loop shaping procedure shaped the nominal plant using a pre
~
M 2
~ N
2 1
N
M1
compensator W1 and/or a postcompensator W2. The controller
is synthesized with minimizing the norm of the McFarlan and
2
N
if det M
N1
M
G ,G
2 1
(17)
Glover stability margin, such that:
1 2
I I P K
1 ~ 1
1
N2
N1
K
s Ms
(11)
and wnodet M
M 0
Where Ps W2 PW1 and an optimal stability margin;
1 otherwise
2
1
P ~ 1 ~
is the coprime factor of P .
G N M 1
~ 1 ~ : The normalized right (left) coprime
s Ms N1 s
i i i
Mi Ni
The final feedback controller is:
K W1KW2

ANALYSIS
M is defined as a transfer function form, w to z:
M F1P,K
(12)
(13)
factorizations of the plants no denoted the winding number.
6. SIMULATION
The large highspeed is adopted as the simulation object. The singular value plot of the system is shown in Fig 3. The peak value is 80 dB at 0.356 rad/s, it is the necessary to design a controller to stabilize the system.
Singular Values
80
60
The structured singular value of a matrix M is defined as: 40
Singular Values (dB)
1
20
m in : detI M
(14) 0
20

denoted the upper singular value and a set of uncertainty.
The closedloop system achieves the nominal performance if only if:
0
40
60
80
2 1
10 10
0 1
10 10
Frequency (rad/s)
2 3
10 10
sup M22 1
(15)
Fig. 3. Diagram of the singular value
The closedloop system achieves the robust stability if, only if:
Controllers are synthesis with the openloop bock shown in fig
sup M11 1
(16)

The system is perturbed by an additive uncertainty. The objectives of the design are to maintain stability and
The closedloop system achieves the robust performance if, only
if:
performance in presence of a bounded uncertainty.
The weights are selected to maximize disturbance rejection, and
sup M 1
(16)
minimize wind gust effect and a sensor noise.

GAP
The gap metric is defined by the quantification of the distance between any two processes in terms of similarity of behavior when connected to a closed loop.
W1 1,5
s 2
3s 10
, W2
12,5
s 10
1 2
+
+
+
With:
dl s3 +1178s2 + 3075s + 2968
l11 2,346.106 s3 +0,02839s2 +0,562s – 0,00102 l12 – 2,541.105 s3 – 0,3548s2 – 6,977s – 0,2226
l21 – 1,099.109 s3 – 8,814.106 s2 – 0,000182s – 2,992.105 l22 1,253.108 s3 +0,0001101s2 +0,002254s +0,0004336 l31 – 3,9.109 s3 – 0,0004008s2 – 0,007566s – 0,002176
Fig. 4. Closeloop system with additive uncertainty.
The two controllers are obtained:
l32 – 3,934.108 s3 +0,004978s2 +0,09351s+0,02911
l – 2,964.109 s3 – 6,189.105 s2 – 0,00121s – 0,001066

controller:
h h
41
l42 2,307.108 s3 +0,0007675s2 +0,01483s+0,01324
11
dh
p1
p1
H
K dh
dh
h41
12
dh
p2 dh p2
dh
h42
The gap of the two controller is:
KH ,KL 0,0024
It mean that the two controller are close.
The closeloop matrix for the analysis (Fig 5) is :
W KS W KS
d d M 2 2
With:
h
h
Where S 1 GK1 .
W1S
W1S
dh s3 +1,41s2 +0,4838s+0,04285
p1 1,947.107 s3 – 2,89.105 s2 + 9,614.107 s+1,91.107
p2 – 2,345.106 s3 +0,0003394s2 – 6,505.106 s – 1,925.106
p1 – 1,029.1010 s3 + 5,663.109 s2 – 7,473.1010 s 7,442.1011
p2 4,165.1010 s3 – 6,775.108 s2 +7,231.109 s+7,78.1010
p1 4,492.109 s3 +1,598.107 s2 + 4,066.108 s+1,987.109
p2 1,33.108 s3 – 1,773.106 s2 – 4,542.107 s – 2,234.108
h41 8,759.1010 s3 – 1,615.108 s2 + 4,456.109 s+ 3,662.1010
h42 – 4,752.109 s3 +1,983.107 s2 – 4,543.108 s – 3,889.109

Loop shaping controller:


Fig. 5. Closeloop matrix for analysis.
The frequency range of the analysis is [104; 104] rad/s.
Robust Stability: S.S.V of M11
inf[mu(M11H)]
sup[mu(M11H)]
inf[mu(M11LP)]
sup[mu(M11LP)]
0.03
0.025
l l
11
dl
l21
K dl
12
dl
l22
dl
0.02
mu
0.015
0.01
L l l
31
dl
32
dl
0.005
l l
0
4 3
2 1 0 1 2 3 4
41
42
10 10
10 10
10 10
Frequence (rad/s)
10 10 10
dl
dl
Fig. 6.plot for robust stabilityanalysis
TABLE 1 RESULTS OF ROBUST STABILITY ANALYSIS
Close
loop
(
/)
max[(11)]
Guaranteed of stability
0,1804
0,0165
1
< 0,0165
1.5167
0.0260
1
< 0.0260
The peak value of (11) is less than one for each case of close loop (Table 1, Fig 6). This implies that for all perturbations,
TABLE 2 RESULTS OF ROBUST STABILITY ANALYSIS
Close
loop
(
/)
max[()]
Guaranteed of
stability
0, 1804
0, 0262
1
< 0, 0262
1,5167
0,0307
1
< 0,0307
The peak value of () is less than one for each case of close
< 1 the stability is guaranteed. The guaranteed
(11)
loop (Table 3, Fig 8). This implies that for all
stability is large for thealgorithm.
Nominal performance: S.S.V of M22
inf[mu(M22H)]
sup[mu(M22H)]
inf[mu(M22LP)]
sup[mu(M22LP)]
0.01
0.009
perturbations,
< 1
()
the performance is guaranteed. The
0.008
0.007
0.006
mu
0.005
0.004
0.003
0.002
0.001
0
4 3 2 1 0 1 2 3 4
10 10 10 10 10 10 10 10 10
Frequence (rad/s)
Fig. 7.plot for nominal performance analysis
TABLE 2 RESULTS OF NOMINAL PERFORMANCE ANALYSIS
Closeloop
(/)
max[(22)]
0,1804
0,0097
0,0001
0,0070
The peak value of (22) is less than one for each case of close loop (Table 2, Fig 7). This implies that for all perturbations, nominal performance was achieved. However the performance specification is better for the closeloop with loop shaping algorithm.
Robust performance: S.S.V of M
inf[mu(MH)]
sup[mu(MH)]
inf[mu(MLP)]
sup[mu(MLP)]
0.035
0.03
0.025
mu
0.02
0.015
0.01
0.005
guaranteed performance is large for the algorithm.

CONCLUSION
The large high speed is instable. However, the two controllers designs, and loop shaping guarantee a robust stability, and a nominal and robust performance. The two controllers are close in reference of the gap.

REFERENCES


L. Meirovitch, Methods of Analytical Dynamics, Hill, 1970.W.K. Chen, Linear Networks and Systems. Belmont, Calif.: Wadsworth, pp. 123135, 1993. (Book style)

M.R. Waszak, D.K. Schmidt, Flight Dynamics of Aeroelastic Vehicles, Journal of Aircraft, 1988.

D.K. Schmidt, Modern Flight Dynamics, Hill, 2012.

C. Zhu, "Robustness analysis for power systems based on the structured singular value tools and the nu gap metric", PHD, Iowa State University, 2001.

G. Vinnicombe, Uncertainity and Feedback loopshaping and the
metric, Imperial College Press, 2001.

K. Zhou, J.C. Doyle, K. Glover, Robust and Optimal Control, Prentice Hall, New Jersey, 1996.

K. Zhou, J.Doyle,"Essentials of Robust Control", Prentice Hall, 1999.
RANARISON S. H., received his Master Diploma in Automatic from 2015 at Ecole SupÃ©rieure Polytechnique dAntananarivo (ESPA), University of Antananarivo. Currently he is a PhD student at University of Antananarivo in the STII,
RANDRIAMITANTSOA P.A., full Professor, Ecole SupÃ©rieure Polytechnique dAntananarivo (ESPA), University of Antananarivo.
RANDRIAMITANTSOA A.A., Doctor, Ecole SupÃ©rieure Polytechnique dAntananarivo (ESPA), University of Antananarivo.
REZIKY Z.S.H.Z.T, Doctor, Ecole SupÃ©rieure Polytechnique dAntananarivo (ESPA), University of Antananarivo.
0
4 3
10 10
2 1 0 1
10 10 10 10
2 3 4
10 10 10
Frequence (rad/s)
Figure 6 plot for robust stability analysis