 Open Access
 Total Downloads : 32
 Authors : Rabearivelo Apotheken Gericha , Andrianaharison Yvon , Randriamitantsoa Paul August , Randriamitantsoa Andry August
 Paper ID : IJERTV7IS030064
 Volume & Issue : Volume 07, Issue 03 (March 2018)
 Published (First Online): 23032018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Contribution to the Modeling of the Robust Control, of the Longitudinal and Lateral Helicopter Motion Analysis
Rabearivelo A. G.,
Ph D Student
University of Antananarivo in the STII,
Andrianaharison Y.
Full Professor,
Ecole SupÃ©rieure Polytechnique dAntananarivo (ESPA), University of Antananarivo,
Randriamitantsoa P. A., Randriamitantsoa A. A., Reziky Zafimarina S. H. Z. T Full Professor,
Ecole SupÃ©rieure Polytechnique dAntananarivo (ESPA), University of Antananarivo.
Abstract The purpose of this article is to present the state of the art which concerns the application of robust synthesis techniques of linear systems and muanalysis of longitudinal and lateral flight of the helicopter.
Index Terms– Helicopter, state feedback control, state representation, robust control, muanalysis.

INTRODUCTION
The robust control of the helicopter is a type of control that aims to guarantee the performance and stability of a system during flight in the face of environmental disturbances and uncertainties of the model and a difficult task since the dynamics of the system is no linear, unstable on certain flight ranges and has a strongly coupled dynamic. Indeed, the mathematical model that models a real system is a representation that aims to best approximate, with simplifying assumptions, the system we want to control.
2.2 Robust control in longitudinal and lateral flight of the helicopter
The parameters of the system are marred by parametric uncertainty. Each uncertain coefficient is modeled as:
= (1 + ) ,  < 1 (1) Or :
:is the nominal value of the parameter considered;
: the corresponding weighting coefficient. We have chosen = 0,2 for all the parameters.
By higher fractional linear transformation, we have:
= [22 + 21( 11)112] (2)
Is :

METHODOLOGY
In practice, the modeling is not precise enough to faithfully
= [ 0 ] (3)
reflect the behavior of a system. Often designers use models with unstructured uncertainties that generally affect poorly known or intentionally neglected dynamics.
A direct additive form model error is adopted for the synthesis of the corrector for this system (Figure 1). In our case, we will take as scaling function the scalar function:
2.1 Simulation of the synthesis and analysis model Robustness analysis is performed by incorporating all model uncertainties into a single transfer matrix without imposing any particular structure on it. In this study, assuming that
( + 10)
= 107
10( + 500)
( + 5)
(4)
several sources of uncertainties of different natures coexist, we adopt a system with model errors of direct additive form. Two other input signals, applied in two different places of the
= 10
10(5 + 1000)
servocontrol, are then taken into account, and the evolution of the performance of the looped system is monitored. The problem therefore arises as the search for a compromise between the objective sought and the means necessary to correct the system by using different methods of synthesis.
To achieve the desired performance, it is necessary to satisfy
the inequality ( + )1 < 1.
Here, the weighting function is a scalar function, so the
singular values of the sensitivity function
( + )1 over all frequency ranges must be obtained by
1
the curve
. It means that
( + )1
< 1 if and only if all frequencies:
1298 s2 + 1.77 1014 s
+ 1.838 10^14
1.216 1016s2 + 8.485 1016s
+ 7.188 1016
1 s3 + 1.255 1011s2
s3 + 1.255 1011s2
[( + )1()] < 
.
()
+ 4.526 1014s + 6.02 1014
+ 4.526 1014s + 6.02 1014
27.2 s2 + 3.709 1012s
2.155 1012
s3 + 1.255 1011s2
2.535 1014s2 3.096 1015s
7.245 1015
s3 + 1.255 1011s2
+ 4.526 1014s + 6.02 1014
=
+ 4.526 1014s + 6.02 1014
218.1 s2 + 2.974 1013s
+ 2.111 1013
s3 + 1.255 1011s2
2.042 1015s2 + 7.905 1015s
7.002 1013
s3 + 1.255 1011s2
+ 4.526 1014s + 6.02 1014
361.2 s2 4.926 1013s
+ 4.526 1014s + 6.02 1014
3.383 1015s2 1.862 1016s
4.502 1013
s3 + 1.255 1011s2
1.104 1016
s3 + 1.255 1011s2
[+ 4.526 1014s + 6.02 1014 + 4.526 1014s + 6.02 1014 ]Block diagram of a slave system with direct additive shape model errors for robust synthesis and analysis.
By isolating the corrector K to determine the matrix of the openloop interconnection P and the closedloop transfer matrix M, FIG. 2 below:
Block diagram for the distinction between matrix M and matrix P. Considering all these hypotheses, we obtain as corrector in longitudinal and lateral flight by synthesis :
:

SIMULATION DIAGRAM
The helicopter is defined as consisting of several subsystems including fuselage, main rotor, tail rotor, empennage, engine,… The figure below illustrates the helicopter flight simulation model diagram and these different subsystems.
Simulation diagram of the longitudinal movement and these different
subsystems.
1,094 104 s3 1.163 105 s2
+ 3.637 107 s + 6.012 107
0.353 s3 3.906 109s2
7.569 1011s 7.819 109
s3 + 8.804 108s2 + 3.47 1011s + 4.22 1010
s3 + 8.804 108s2 +
3.47 1011s + 4.22 1010
4.888 105 s3 3.64 104s2
0.1577 s3 + 2.509 108s2
8.122 106s 1.998 107
+ 1.586 1011s 9.413 109
=
s3 + 8.804 108s2 + 3.47 1011s + 4.22 1010
s3 + 8.804 108s2 +
3.47 1011s + 4.22 1010
1.15 105s3 + 7810 s2
0.03708 s3 7.172 107s2
+ 1.498 106s + 1.206 107 4.357 1010s + 1.265 1010
s3 + 8.804 108s2 + 3.47 1011s + 4.22 1010
s3 + 8.804 108s2 +
3.47 1011s + 4.22 1010
1.051 105s3 8552 s2
0.0339 s3 + 4.168 107 s2 +
Simulation diagram of the lateral movement and these different
1.636 106s + 3.319 107 3.229 1010 s + 5.44 1010 s3 + 8.804 108s2 + s3 + 8.804 108s2 +
subsystems.
[ 3.47 1011s + 4.22 10103.47 1011s + 4.22 1010 ]
3.3 State equation of the flight subsystem of the helicopter
The equation of state of the longitudinal subsystem of the helicopter can be written in matrix form :
[] = [
] [ ] + [
0
0
1
1
] [ 0() ]
0
0
0
1 0
0 0
1 0
1()
With :
= [
] =
0
0
1
1
Robustesse en StabilitÃ© du SystÃ¨me BouclÃ© de M22 avec incertitudes structurÃ©es et non structurÃ©es en Vol Longitudinal de l'hÃ©licoptÃ¨re 0.8
inf(mu) M22H
sup(mu) M22H
0.7
0.6
M22 de V.S.S
0.5
0
0
1
0 0 1
1 0 0 0
0 [ 0 0 ]
0 0
0.4
0.3
0.2
= [0 1 0
0 ] = [0 0]
0.1
0 0 1 0 0 0
0 0 0 1 0 0
0
2 1 0
10 10 10
1 2 3
10 10 10
Frequence (rad/s)
4 5 6
10 10 10
One can write in matrix form the state equation of the lateral subsystem of th helicopter :
Robustness in Stability of the Curly System
= [
0
]
1
= 1
0
Looping
(rad/s)
max[(11)]
Stability guarantee
0,2341
0,7711
1
< 0,7711
0
0 1 0
0 0
1
0
0 0 0
0 0
[10 ]
Table.2 : Analysis of robustness in stability.
= [0 1 0 0 ] = [0 0]
0 0 1 0 0 0
0 0 0 1 0 0
The system is considered stable in robustness because
max[(11)] < 1 for this loopback.

RESULTS

Muanalysis in longitudinal flight
The frequency response of the lower and upper bounds of the structured and unstructured singular value of the matrix 11, the nominal performance analysis matrix, for the corrector is shown in Fig 4.
The figure below shows the robustness in performance.
inf(mu) MH
sup(mu) MH
Robustesse en Performance du SystÃ¨me BouclÃ© de Mu avec incertitudes structurÃ©es et non structurÃ©es en Vol Longitudinal de l'hÃ©licoptÃ¨re 0.8
0.7
0.6
Mu de V.S.S
0.5
0.4
Looping
(rad/s)
max[(11)]
103
0,2000
Table.1 : Analysis of the nominal performance.
0.3
0.2
0.1
0
2 1 0
10 10 10
1 2 3
10 10 10
Frequence (rad/s)
4 5 6
10 10 10
inf(mu) M11H
sup(mu) M11H
Performance Nominale du SystÃ¨me BouclÃ© de M11 avec incertitudes structurÃ©es et non structurÃ©es en Vol Longitudinal de l'hÃ©licoptÃ¨re 0.25
0.2
M11 de V.S.S
0.15
0.1
0.05
0
2 1 0 1 2 3 4 5 6
10 10 10 10 10 10 10 10 10
Frequence (rad/s)
Nominal Performance of the Curly System.
The system is considered efficient because max[(11)] < 1 for this loopback (Table 1).
The frequency response of the lower and upper bounds of the structured and unstructured singular value of the matrix 22, stability stability analysis matrix, for the corrector is shown in FIG. 4.
Robustness in Curly System Performance
The frequency response of the lower and upper bounds of the structured and unstructured singular value of the matrix , the system for analyzing the robustness of the system, for each type of corrector is shown in Fig. 5.
Looping
(rad/s)
max[()]
Garantie de la
pÃ©rformance
0,2341
0,7725
1
< 0,7725
Table.3 : Analysis of the robustness in performance.
The system is judged to be robust in performance because max[()] < 1 for this loopback (Table 3). Yet there is a large guarantee of performance for looping with the corrector obtained by the synthesis .

Muanalysis of systems in lateral flight The system is considered stable in robustness because
The block diagram for analyzing the system with unstructured uncertainties is shown in Fig. 6.
max[(11
)] < 1 for this loopback.
inf(mu) MH
sup(mu) MH
Robustesse en performance du systÃ¨me bouclÃ©: V.S.S de Mu avec incertitudes structurÃ©s et non structurÃ©s du vol latÃ©ral 0.8
0.75
0.7
0.65
0.6
mu
0.55
0.5
0.45
0.4
0.35
4 3
10 10
2 1 0
10 10 10
1 2
10 10
3 4 5 6
10 10 10 10
Block diagram of the disturbed system with unstructured uncertainties in lateral flight
The frequency response of the lower and upper bounds of the unstructured singular value of matrix 11, the nominal performance analysis matrix, for the corrector is shown in the Fig. 8 below.
Looping
(rad/s)
max[(11)]
0.7053
0,7741
Table.4 : Analysis of the nominal performance.
inf(mu) M11
Performance nominale du systÃ¨me bouclÃ©: V.S.S de M11 avec incertitudes structurÃ©s et non structurÃ©s du vol latÃ©ral 0.75
H
sup(mu) M11
0.7
H
0.65
0.6
mu
0.55
0.5
0.45
0.4
0.35
4 3 2 1 0 1 2 3 4 5 6
10 10 10 10 10 10 10 10 10 10 10
Pulsation (rad/s)
Nominal Performance of the Curly System
The system is considered efficient because max[(11)] < 1 for this loopback.
The frequency response of the upper and lower bounds of the unstructured singular value of the matrix 22, stability stability analysis matrix, for the corrector is shown in Fig. 9.
inf(mu) M22H
sup(mu) M22H
Robustesse en stabilitÃ© du systÃ¨me bouclÃ©: V.S.S de M22 avec incertitudes structurÃ©s et non structurÃ©s du vol latÃ©ral 0.16
0.14
0.12
0.1
mu
0.08
0.06
0.04
0.02
Pulsation (rad/s)
Robustness in Curly System Performance
The frequency response of the lower and upper bounds of the unstructured singular value of the matrix M, the system of analysis of the robustness of the system, for each type of corrector is shown in the figure below.
Looping
(rad/s)
max[()]
Stability guarantee
0,5341
0,7741
1
< 0,7741
Table.6 : Analysis of the robustness in performance.
The system is considered robust in robustness because max[()] < 1 for this loopback. Yet there is a large guarantee of performance for looping with the corrector obtained by the synthesis .

CONCLUSION
In this article, we have been able to develop robustness synthesis analysis tools for a linear system. Structured singular value is one of the very powerful tools for analyzing the robustness of a linear system tainted by uncertainty. It has been explained how the analysis method can analyze the robustness of stability and performance of a system. The standard problem explains the synthesis of a controller by minimizing the Hinfinite standard of the LFT F(, ). The resolution of this optimization leads to the determination of

The use of analysis and the problem of Hinfinity has established another synthesis tool that is synthesis.


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4 3 2 1 0
10 10 10 10 10
1 2
10 10
3 4 5 6
10 10 10 10
model, NASA TMX3185, 1975.
Pulsation (rad/s)
Looping
(rad/s)
max[(11)]
Stability guarantee
105
6x 10(7)
1
< 6x 107
Robustness in Stability of the Curly System
Table.5 : Analysis of robustness in stability.

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