POLITECNICO DI MILANO Active Gust Alleviation for a Regional Aircraft through Static Output ...

131 Pages·2013·2.66 MB·English

by  Federico Fonte

POLITECNICO DI MILANO
Facolt`a di Ingegneria Industriale
Corso di Laurea Magistrale in Ingegneria Aeronautica
Active Gust Alleviation for a Regional
Aircraft through Static Output Feedback
Relatore: Prof. Paolo MANTEGAZZA
Tesi di laurea di:
Federico FONTE
Matr. 780844
Anno Accademico 2012 - 2013

Abstract
The interest in active systems for gust load alleviation is increased in recent
years following the development of new, highly efficient and flexible aircrafts.
A static output feedback (SOF) controller can be a suitable choice for such
systemsbyvirtueofitssimplicitycomparedwiththatofdynamicalcontrollers.
Nonetheless there are no analitycal methods for the computation of the
static output feedback gain matrices, and a numerical computation can be
problematic.
In the present work a numerical method for the computation of the gain
matrix of a SOF controller is presented. The method is based on the linear
quadratic formulation of the control problem and the minimization of the
related cost function uses both its gradient and hessian. This allows the use
of second order optimization algorithm, thus accelerating the convergence.
The algorithm is then applied to the design of a gust alleviation system for a
modern regional transport aircraft.
Sommario
L’interesse verso i sistemi attivi per l’attenuazione dei carichi da raffica `e
aumentato negli ultimi anni a seguito dello sviluppo di nuovi aerei ad alta
efficienza caratterizzati da un’elevata flessibilit`a strutturale. Un controllore
a retroazione diretta delle misure pu`o rappresentare una scelta appropriata
per questi sistemi data la maggiore semplicit`a implementativa rispetto ai
controllori dinamici. Purtroppo non esistono metodi analitici per il calcolo
della matrice dei guadagni di questi controllori, e un calcolo numerico pu`o
essere di difficile implementazione.
In questo lavoro di tesi `e stato sviluppato un metodo numerico per il calcolo
della matrice dei guadagni, basato sulla formulazione quadratica del problema
del controllo. La cifra di merito associata a questa formulazione viene mini-
mizzata sfruttando la conoscenza sia del gradiente che dell’hessiano, in questo
modo `e possibile utilizzare algoritmi di secondo ordine per l’ottimizzazione e
di conseguenza accelerare la convergenza.
L’algoritmo sviluppato viene quindi applicato al progetto di un sistema per
l’alleviazione dei carichi da raffica per un moderno velivolo da trasporto
regionale.
Contents
1 Introduction 1
2 Aircraft model 5
2.1 Structural model . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Aerodynamic model . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Gust boundary conditions . . . . . . . . . . . . . . . . 11
2.3 Control surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Static analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Flutter computation . . . . . . . . . . . . . . . . . . . . . . . 17
2.5.1 Interpolation of aerodynamic matrices . . . . . . . . . 18
2.6 Approximations for the aerodynamic model . . . . . . . . . . 22
2.6.1 Quasi-steady approximation . . . . . . . . . . . . . . . 22
2.6.2 Quasi-steady approximation with high order . . . . . . 23
2.6.3 Matrix fraction approximation . . . . . . . . . . . . . . 24
2.7 Stress recovery . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.7.1 Quasi-steady approximation of aerodynamic forces . . . 28
2.7.2 State-space representation of aerodynamic forces . . . . 29
2.7.3 Gust response . . . . . . . . . . . . . . . . . . . . . . . 29
3 Static Output feedback control 35
3.1 Generic system . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.1 Static Output Feedback control . . . . . . . . . . . . . 36
3.1.2 State Feedback control . . . . . . . . . . . . . . . . . . 38
3.1.3 State Estimation . . . . . . . . . . . . . . . . . . . . . 39
3.2 Quadratic control . . . . . . . . . . . . . . . . . . . . . . . . . 39
v
3.2.1 Deterministic setting . . . . . . . . . . . . . . . . . . . 40
3.2.2 Stochastic setting . . . . . . . . . . . . . . . . . . . . . 43
3.2.3 Interpretation as H control . . . . . . . . . . . . . . . 45
2
3.3 Computation of gain matrix . . . . . . . . . . . . . . . . . . . 46
3.3.1 Numerical methods . . . . . . . . . . . . . . . . . . . . 47
3.3.2 Issues on numerical methods . . . . . . . . . . . . . . . 48
3.4 Algorithm for gain computation . . . . . . . . . . . . . . . . . 48
3.4.1 Optimization Algorithms . . . . . . . . . . . . . . . . . 49
3.4.2 Continuation methods . . . . . . . . . . . . . . . . . . 51
3.5 Compuation of derivatives of cost function . . . . . . . . . . . 53
3.5.1 Lyapunov equation . . . . . . . . . . . . . . . . . . . . 55
3.5.2 Cost function derivatives for static output feedback . . 60
3.5.3 Algorithm description . . . . . . . . . . . . . . . . . . 63
3.5.4 Implementation of the algorithm . . . . . . . . . . . . 64
3.6 Linear Matrix Inequalities approach . . . . . . . . . . . . . . . 65
3.6.1 Linear matrix inequalities . . . . . . . . . . . . . . . . 65
3.6.2 Application to control system design . . . . . . . . . . 66
3.6.3 Algorithms for the solution of LMI problems . . . . . . 71
3.6.4 Numerical methods for static output feedback . . . . . 72
3.7 Example: helicopter model . . . . . . . . . . . . . . . . . . . . 73
4 Control setting 77
4.1 Control objectives . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3 Gust and turbulence model . . . . . . . . . . . . . . . . . . . 79
4.3.1 EASA regulation . . . . . . . . . . . . . . . . . . . . . 79
4.3.2 Gust shaping filter . . . . . . . . . . . . . . . . . . . . 82
4.4 Sensor transfer function . . . . . . . . . . . . . . . . . . . . . 83
5 Control Synthesis 87
5.1 Controller design . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1.1 Model for control synthesis . . . . . . . . . . . . . . . . 88
5.1.2 Optimal dynamic controller . . . . . . . . . . . . . . . 89
5.1.3 Selection of weights . . . . . . . . . . . . . . . . . . . . 89
5.1.4 Performances at nominal point . . . . . . . . . . . . . 92
5.1.5 Stochastic response . . . . . . . . . . . . . . . . . . . . 98
5.1.6 Use outside the nominal point . . . . . . . . . . . . . . 98
5.2 Analysis of optimization algorithms . . . . . . . . . . . . . . . 105
6 Concluding remarks 107
Appendix A Estratto in lingua italiana 109
A.1 Introduzione . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
A.2 Modello del velivolo . . . . . . . . . . . . . . . . . . . . . . . . 110
A.2.1 Modello strutturale . . . . . . . . . . . . . . . . . . . . 110
A.2.2 Modello aerodinamico . . . . . . . . . . . . . . . . . . 111
A.3 Controllo a retroazione diretta delle misure . . . . . . . . . . . 112
A.4 Progetto del controllore . . . . . . . . . . . . . . . . . . . . . . 114
Bibliography 119

1
CHAPTER
Introduction
Modern transport aircrafts are characterized by a slender aerodynamic shape
and a low mass structure in order to reduce the aerodynamic drag, to increase
the payload and to reduce fuel consumption. In recent years, the increase
in fuel cost and a growing attention to environmental issues determined
a further impulse toward the creation of aircrafts with light and slender
airframes. Such structures are highly deformable and thus they are subjected
to large deformations during flight. This effect, however, is undesired since
it can reduce the structural life of the airframe and they can lead to fatigue
problems.
Amongallloadstowhichanaircraftissubjectedduringflightanimportant
role is played by forces coming from atmospheric turbulence. These forces
are always present since the atmosphere is never really steady. It is also
possible that an aircraft, while flight, enters in a region where the air has a
nonzero mean vertical velocity. The rapid change in incidence resulting from
the variation in vertical velocity seen by the aircraft generates high loads on
the structure. This situation is commonly referred as the entrance in a gust
and, in general, the air velocity can be directed arbitrarily and not only in
the vertical direction as here described.
In addition to structural loads, atmospheric turbulence and gusts deter-
mine also high accelerations in the fuselage. These accelerations are undesired
since they reduce passenger comfort, reduce the reliability of the avionic
instrumentation and can affect the crew effectiveness.
For these reasons gust and turbulence loads are of great importance in
determining flight quality and safety, and this is witnessed by the great
1
Chapter 1. Introduction
interest they received since the earliest years of flying when Wright brothers
were forced to modify the structure of the wing of their airplane in order to
minimize the effect of lateral gusts [19]. Following from the importance of
such loads, regulations require that transport aircrafts must be certified to
be able to sustain gust and turbulence loads without affecting the security
and the effectiveness of the aircraft [1].
In most cases, however, specifications on the response to gusts and turbu-
lence loads were met passively, by a strengthening of the structure and by an
increase of wing loading. These solutions lead to an increase of the aircraft
weight and of aerodynamic drag, thus the fuel consumptions also increased.
For this reason, as efficiency requirements become more stringent, a great
effort is done in the search for alternative solutions.
Theabilitytosustaingustloadswithoutintroducingadecreaseofefficiency
is very appealing for the development of high efficient, flexible transport
aircraft, but it is mandatory for new classes of vehicles such as the High-
Altitude, Long-Endurance (HALE) aircrafts or sensorcraft. HALE aircraft
[24] are very light unmanned flying vehicles with an endurance of several days.
To achieve this they usually are provided with electric motors and solar cells.
While sensorcrafts are small unmanned vehicles which try to incorporate all
the sensing devices usually included in large transportation aircrafts [28]. The
design of an airframe able to sustain gust loads would lead to an excessive
weight increment with the consequence that the requirements on endurance
for HALE vehicles and on payload for sensorcrafts would be impossible to
met.
Over years a variety of passive and active methods for gust loads reduction
have been proposed and the number of actual implementations of such devices
is increased in recent years. Passive solutions are those that do not require an
actuation system to be effective, which, instead, is required by active systems.
Both solutions present advantages and disadvantages, and in some cases they
can be combined in order to obtain better performances. An advantage of
passive systems is their great reliability which comes from their simplicity. In
fact, there is always a possibility of a failure on the actuation system of an
active device. In this case the device can turn to be completely ineffective,
if not detrimental, in reducing gust loads. Another advantage of passive
systems is the fact that they do not require a power source and thus their use
is less ”expensive” than that of active devices. Active systems, on the other
hand, can potentially adapt themselves to the environmental conditions, thus
resulting effective in a wide range of flight conditions.
A first example of a passive solution for the reduction of gust loads is
offered by the introduction of composite materials for the realization of the
airframe. There is in fact the possibility to manufacture the layup of the
2