Active gust alleviation for a regional aircraft through static output feedback

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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

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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 systems by virtue of its simplicity compared with that of dynamical controllers. 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.

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L’interesse verso i sistemi attivi per l’attenuazione dei carichi da raffica è aumentato negli ultimi anni a seguito dello sviluppo di nuovi aerei ad alta efficienza caratterizzati da un’elevata flessibilità strutturale. Un controllore a retroazione diretta delle misure può rappresentare una scelta appropriata per questi sistemi data la maggiore semplicità implementativa rispetto ai controllori dinamici. Purtroppo non esistono metodi analitici per il calcolo della matrice dei guadagni di questi controllori, e un calcolo numerico può 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.

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CHAPTER

1

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.

Among all loads to which an aircraft is subjected during flight an important 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

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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.

The ability to sustain gust loads without introducing a decrease of efficiency 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

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composite layers in such a way to meet aeroelastic and structural performance requirements. This procedure is called aeroelastic tailoring [44] and can offer great possibilities of load reduction by exploiting the interaction between structural deformations and aerodynamic forces.

In the seventies some research work was performed in order to study the possible benefits of the adoption of a free wing [36]. A free wing is an aerodynamic surface which is free to rotate about a spanwise axis. Since the maximum achievable lift is decreased, this solution can be applied only to small aircrafts with low wing loading. For this reason this idea has been reproposed only recently by Welstead and Crouse [45], in view of a possible application to HALE vehicles.

A great number of active methods for gust load alleviation have been proposed, and a first attempt of implementing such devices in real aircrafts dates back to 1949 when the Bristol Brabazon was equipped with a gust load suppression system. This system operated in open loop, where control surfaces were moved basing on measurements of gust speed obtained through a gust vane detector located on the aircraft nose. This system was never applied since the aircraft never come into production [30].

In the following years many tests were performed on gust alleviation systems, and they were applied first on military aircrafts then on civil aircrafts [47], showing the effectiveness of active gust load alleviation systems.

In recent years the refinements of structural and aerodynamic models and the development of more powerful tools for control system design have increased the interest in active systems for gust load alleviation. One attractive feature of such systems is the possibility to deal simultaneously with flight stability augmentation systems, gust load alleviation and flutter suppression [38].

In this work an active gust alleviation system is designed by the use of a static output feedback controller. It uses the aircraft control surfaces for actuation and accelerometric sensors, thus special control surfaces or gust detectors are not needed.

A static output feedback controller represents the most simple control law which can be applied to a system. According to this configuration the control input is given directly by a linear combination of the measures taken from the system. This contrasts with the dynamic control approach, where the control input is affected also by internal states of the compensator and thus it is needed to simulate the dynamic of these states.

The great simplicity of static controllers make them advisable in many applications where simple control laws are desired, for example they are particularly well suited for distributed systems, where there is not a central controller but each actuator receives measurements from only a group of

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sensors. It is also easy to complement the static controller with a small dynamic controller in order to increase the overall performance, and the design of the mixed static–dynamic controller can be performed using the same analytical tools used for a pure static controller. Owing to their simplicity, static output feedback controllers can also be used as backup controllers, which are activated in case of failure of the main dynamic controller.

Despite the simplicity of the static output feedback controller, the design of such a controller is still an hard problem. This is due to the fact that the stabilizability of a system through static output feedback cannot be tested, and to the fact that numerical algorithms for the computation of the gain matrix cannot be shown to be convergent in general [43]. Nonetheless a lot of methods for the solution of this problem have been proposed, and a good surveys is given by Syrmos et al. [42].

Among various methods used for the computation of static output feedback controllers those based on the linear quadratic formulation of the control problem have seen a widespread use, since the presentation of the first algorithm by Levine and Athans [27]. In these methods the controller is computed by minimizing a cost function expressing the performances of the closed loop system. An example of aeronautical application of such algorithms is provided by the work of Patil and Hodges [34], where they used static output feedback for flutter suppression and gust alleviation of an HALE system. They also showed that a proper choice of sensor location could give results similar to those reachable by the use of a full state feedback controller. Miyazawa and Dowell [31] used a multimodel approach for flutter suppression, i.e. they used several different models in the formulation of the quadratic cost function, thus leading to a controller robust under changes of the system.

Also the algorithm presented in this work is based on the linear quadratic formulation of the cost function. It takes advantage of the possibility of the computation of the hessian of the cost function in order to speed up the convergence of the minimization procedure, which can be very slow if the function is quite flat near the minima and only gradients are available.

Recently, another approach to static output feedback control design has been proposed. It is based on the formulation of the control problem as a system of linear matrix inequalities [8]. Though the algorithm presented in this work is based on the linear quadratic formulation, these methods are discussed since they offer a good comparison for assessing the properties of the algorithm.

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CHAPTER

2

Aircraft model

The aircraft model considered in the present work is the Green Regional Aircraft, one of the six Integrated Technology Demonstrators developed within the Clean Sky research program. A view of the aircraft is presented in fig. (2.1). The Clean Sky Joint Technology Initiative is a european research program started in 2008, whose objective is to develop new technologies able to reduce the environmental impact of aircrafts. These technological improvements are required in order to follow the guidelines set by the Advisory Council for Aeronautics Research in Europe (ACARE), which dictates a reduction of fuel consumption, of emission of pollutants and of noise. The research activity is divided in the development of six Integrated Technology Demonstrators, each of them focused on the introduction of new technologies in a particular area of air transportation. The Green Regional Aircraft is one of these Demonstrators, within this project a regional transport aircraft is designed, and benefits coming from the introduction of new technologies are evaluated. These improvements include, for example, a widespread use of composite materials and of lower-density aluminium alloys in the design of the airframe. Another important feature of this aircraft is the use of electric compressors for cabin pressurization, flight controls, gears extension etc, thus reducing the need for hydraulic and pneumatic power.

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Figure 2.1 View of the Green Regional Aircraft. Retrieved November 18, 2013, from Clean Sky website: http://www.cleansky.eu/content/interview/focus-associate-member-cira.

2.1 Structural model

A Finite Element beam stick structural model with lumped masses has been chosen tho represent the inertial and stiffness properties of the structure. Beam stick models are very simple, but nonetheless they are able to describe properly the behaviour of the structure. They are particularly well suited for the conceptual design phase of the development of an aircraft when, on one hand, they provide simple models with which a lot of configurations can be tested efficiently, on the other hand in this design phase a complete description of the structure is not available, and thus the creation of a more detailed Finite Element Model would not be possible. The structural and the aerodynamic model for the GRA aircraft was created by the use of the NeoCASS software [12], [11]. NeoCASS is a suite of Matlab modules that combines state of the art computational, analytical and semi-empirical methods to tackle all the aspects of the aero-structural analysis of a design layout at conceptual design stage. In particular there are two modules, Weights and balance, and GUESS (Generic Unknowns Estimator in Structural Sizing) which permit to obtain a first guess analytical sizing of the airframe and mass distribution, starting from geometrical data of the aircraft.

Geometrical properties of the considered aircraft are summarized in tab. (2.1)

The structural model is shown in fig. (2.2) and is composed by 85 beams; 10 in the fuselage, 18 for each semi-wing, 11 for the vetrical tail and 14 for

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2.1. Structural model

overall length m 40.69

overall height m 7.884

wing span m 34.96

wing root chord m 5.453

wing tip chord m 1.498

horizontal tail span m 12.30 horizontal tail root chord m 3.762 horizontal tail tip chord m 1.349

Table 2.1 GRA geometric properties.

each half of the horizontal tail.

Since only the symmetric response of the aircrat will be studied, only half of the structural model will be considered, reducing thus the number of beam elements to 53.

A dynamic analysis was performed using NASTRAN [40]. The computed structural modes are summarized in 2.2

Mode number frequency [Hz] Description

5 3.763 First wing bending

6 4.784 Wing and fuselage bending

7 9.339 Horizontal tail and fuselage bending

8 9.662 First wing in-plane bending

9 12.205 Second wing bending

10 14.671 Wing, horizontal tail and fuselage bending 11 17.9622 Wing, horizontal tail and fuselage bending

Table 2.2 Structural modes for the symmetric model.

In tab. (2.2) any wing torsional mode is present, since the frequency associated to this mode is very high, namely 31.74 Hz. This comes from both the high torsional stiffness and the small torsional inertia of the wing. Since the first torsional mode has high frequency, it is unlikely that a flutter instability can occur. This conjecture will be confirmed by the results of flutter computations presented in the section 2.5.

The modal analysis gives a transformation which relates the degrees of freedom of the structural finite element model to the modal coordinates

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0 5 10 15 20 25 30 35 40 −15 −10 −5 0 5 10 15 −5 0 5 10 y [m] x [m] z [m]

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2.2. Aerodynamic model

Since the modal coordinates represent an efficient basis, the equation of motion are expressed in this basis and take the form

Mjj¨qj + Kjjqj = fj (2.2)

where Mjj is the modal mass matrix, Kjj is the modal stiffness matrix and fj is the vector of external modal forces.

2.2 Aerodynamic model

Aerodynamic computation is performed using Doublet Lattice Method (DLM). This is a panel method based on the potential formulation of aerodynamics. Despite its simplicity compared to more complete computational fluid dy-namics methods, it can give fairly good results when applied to aerodynamic bodies and if the flow has a high Reynolds number. A doublet lattice is placed on the aerodynamic surfaces, and the intensity of each doublet is computed by imposing the no penetration boundary condition on some selected points called control points. Then aerodynamic pressures are computed and are transformed to forces acting on structural points.

The aerodynamic forces are computed by imposing boundary conditions whose shape is given by structural modal shapes, and that have an harmonic time variation characterized by the adimensional reduced frequency

k = ωla 2V∞

= ωta (2.3)

where ω is the pulsation, expressed in rad/s, la is a reference length, V∞ is the flow speed and ta = _2Vla_∞ is an aerodynamic reference time defined by la and V∞.

The modal transformation is applied to the nodal forces thus obtained in order to obtain modal forces. The computations are repeated for several values of reduced frequency k and of Mach number M∞. The result of this operation is the matrix of generalized aerodynamic forces (GAF) Qjj(k, M∞) whose generic element (m, n) expresses the aerodynamic force acting on the modal coordinate m, when the boundary condition has the modal shape associated to the mode n.

Modal Generalized aerodynamic forces were computed using NASTRAN [40] with the aerodynamic mesh shown in fig. (2.3). The computation was performed on a set of reduced frequencies spanning from 0 to 3, and was repeated for Mach numbers 0.5, 0.6, 0.71, 0.8.

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25 30 35 40 0 5 10 15 0 2 4 6 y [m] x [m] z [m]

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2.2. Aerodynamic model

2.2.1 Convergence

In order to check the convergence of aerodynamic coefficients with the panel number three analysis were performed: in the first case 258 aerodynamic panels were used with a maximum streamwise dimension on wing ∆x = 0.54 m and three chordwise panels were present on each control surface; in the second case the number of aerodynamic panels was 601, the maximum streamwise dimension on wing was ∆x = 0.32 m and six chordwise panels were present on each control surface; while in the third case 846 panels were used with ∆x = 0.26 m and eight chordwise panels on each control surface.

In tab. (2.3) the rigid stability derivatives computed in the three cases are presented , along with hinge moment coefficients for aileron and elevator. From these results it can be seen that the relative difference between coefficients computed in cases 2 and 3 is less than 10%, and thus a smaller variation is expected for more refined meshes.

Cl/α Cmαc.g. CHa/δa CHe/δe Case 1 3.5458 −2.7085 −1.4826 −3.9295 Case 2 3.5553 −2.6936 −1.3965 −3.7011 Case 3 3.5594 −2.6900 −1.3879 −3.6487

Table 2.3 Convergence of rigid stability derivatives.

All quantities in tab. (2.3) are related to static aerodynamic, the effect of the increase in panels number for unsteady aerodynamic is presented in fig. (2.4) where some coefficients of the Qjj matrix are presented for the three

cases. All coefficients in fig. (2.4) refers to diagonal entries of the Qjj matrix and thus represent aerodynamic forces acting on the same mode used for the imposition of the boundary condition. Element 5 refers to the first bending mode of the wing, element 17 refers to the first torsional mode of the wing, while modes 3 and 4 are associated to the deflection of the elevator and of the aileron, respectively.

2.2.2 Gust boundary conditions

The evaluation of the aerodynamic forces coming from a gust requires the introduction of a boundary condition which takes into account the gust penetration. This leads to delay terms on the transfer function from the gust velocity to the generalized aerodynamic forces acting on modal coordinates, Qjg(k, M∞). If a rational approximation of the aerodynamic transfer function is desired, then a high order model may be necessary in order to represent

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−0.05 0 0.05 0.1 0.15 −0.25 −0.2 −0.15 −0.1 −0.05 0 Q hh(5,5)

Real part [rad]

Imaginary part [rad]

0.0001 0.01 0.05 0.1 1 1.5 2 2.5 k = 3 Case 1 Case 2 Case 3 0 0.05 0.1 0.15 0.2 −0.2 −0.15 −0.1 −0.05 0 Q hh(17,17)

Real part [rad]

0.0001 0.01 0.05 0.1 1 1.5 2 2.5 k = 3 −0.045 −0.04 −0.035 −0.03 −0.1 −0.08 −0.06 −0.04 −0.02 0 Q hh(3,3)

Real part [rad]

0.0001 0.01 0.05_0.1 1 1.5 2 2.5 k = 3 0.0001 0.01 0.05_0.1 1 1.5 2 2.5 k = 3 −0.016 −0.015 −0.014 −0.013 −0.03 −0.025 −0.02 −0.015 −0.01 −0.005 0 Q hh(4,4)

Real part [rad]

0.0001 0.01_0.05_0.1 1 1.5 2 2.5 k = 3 0.0001 0.01_0.05_0.1 1 1.5 2 2.5 k = 3

Figure 2.4 Convergence of elements of Qjj matrix with increasing panel

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2.3. Control surfaces

these delay terms. Since the time delay is proportional to the distance between the gust entry point and each aerodynamic panel, a proper choice of this point can reduce this effect, allowing a simpler approximation of aerodynamic forces. Another approach is presented in [25], where a different gust entry point is defined for each group of panels. There are thus several gust input, which in turns are traced back to the true gust velocity by the introduction of time delays, either directly or by means of a state space approximation. In the present work this approach was not used, since good results were obtained by placing the gust entry point on the wing root leading edge.

Since gust boundary condition transfer matrices are not easily recovered from NASTRAN, the computation of Qjg(k, M∞) was performed using Neo-CASS, whose aeroelastic module, SMARTCAD, contains procedures for an aerodynamic computation using the doublet lattice method.

2.3 Control surfaces

The symmetric model has four control surfaces: aileron, elevator and two flaps. For control purposes only elevator and aileron are used since flaps have an actuation system with a too limited bandwidth, and their kinematic does not allow enough displacement, these two control surfaces are shown in fig. (2.5). The use of the aileron in a symmetric way requires that its actuation

system must allow both symmetric and antisymmetric displacements. One of the features of the model considered is the presence of electric, fly-by-wire actuation systems for the movement of control surfaces, this ensures that control surfaces can be controlled independently in both sides.

Since the model of the actuation system is unknown, it is assumed that it can be identified by a second order transfer function on the form

δ = ω 2 0 s2_{+ 2ξω} 0s + ω02 δc+ Mδ Kδ (2.4) where δ is the control surface deflection, δc is the commanded deflection, Mδ is the hinge moment and Kδ is the stiffness associated to the actuation system.

The frequency response of the actuator is determined by the two parame-ters ξ and ω0 = 2πf0. In the present model a value of f0 = 10 Hz was chosen, and the system had a nearly critically damping, with ξ = 0.99. This ensures a bandwidth of about 7 Hz.

The chosen value for the actuation stiffness is Kδ = 4 × 106 N which leads to a natural frequency of about 30 Hz.

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20 25 30 35 40 0 5 10 15 −5 0 5 10

Figure 2.5 Aircraft control surfaces.

The degree of freedom associated to the dynamic of the actuation systems are added to the modal degree of freedom to form the variable vector q_h, the motion equations of the aeroservoelastic system thus becomes:

Mhh¨qh+ Chhq˙h+ Khhqh− q∞Qhh(p, M ) qh = fh+ q∞Qhg(p, M ) vg V∞

(2.5) where Qhh and Qhg are directly obtained by padding with zeroes Qjj and Qjg, since the actuation system is not directly related to the aerodynamic.

Aerodynamic forces coming from the deflection of control surfaces are computed using the DLM method, this means that their amplitude can be quite inaccurate, in the design of the controller this must be taken in account, since it must be able to withstand a variation in the effectiveness of control surfaces.

2.4 Static analysis

A static analysis has been performed at different Mach numbers in order to assess longitudinal static stability, which is related to coefficients CL/α and Cm/α evaluated at the aircraft center of gravity [20]. The aerodynamic

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2.4. Static analysis

coefficients computed for the various Mach numbers are shown in tab. (2.4) along with the hinge moment coefficients for the aileron, CHa/δa and for the elevator, CHe/δe. The indicator of static stability is the Static Margin SM, also present in tab. (2.4) and computed as

SM = −Cmc.g./α Cl/α

la (2.6)

and it has to be positive if the aircraft is statically stable.

Mach number Cl/α Cmc.g./α SM [m] CHa/δa CHe/δe 0.5 3.099 −2.448 2.372 −2.308 × 10−3 _{−5.986 × 10}−3 0.6 3.272 −2.542 2.319 −2.443 × 10−3 _{−6.368 × 10}−3 0.71 3.559 −2.690 2.350 −2.669 × 10−3 _{−7.017 × 10}−3 0.8 3.933 −2.870 2.131 −2.969 × 10−3 _{−7.890 × 10}−3

Table 2.4 Rigid stability derivatives.

The variation of stability derivatives with dynamic pressure is shown in fig. ??; these coefficients take in account for aeroelastic effects which deform the structure. A quasi-steady aerodynamic was used and deformable modes were residualized.

From fig. ?? it can be seen that the variation of coefficients with dynamic pressure is smooth, and it does not affect the static stability.

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0.5 1 1.5 2 x 104 3

3.5 4

Dinamic pressure [Pa]

C l/ α 0.5 1 1.5 2 x 104 −3 −2.8 −2.6 −2.4

C m c.g. /α 0.5 1 1.5 2 x 104 2 2.2 2.4

SM [m] Mach = 0.5 Mach = 0.6 Mach = 0.71 Mach = 0.8 Mach = 0.5 Mach = 0.6 Mach = 0.71 Mach = 0.8 Mach = 0.5 Mach = 0.6 Mach = 0.71 Mach = 0.8

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2.5. Flutter computation

2.5 Flutter computation

The equations of the aeroservoelastic system are:

Mhhq¨h+ Chhq˙h+ Khhqh− q∞Qhh(p, M ) qh = fh+ q∞Qhg(p, M ) vg V∞

(2.7) Where qh are a union of modal coordinates of the structural model and of degree of freedom associated to actuators transfer functions. Mhh, Chh and Khh are, respectively, generalized mass, damping and stiffness matrices associated to qh.

To assess the stability of the aeroservoelastic system, the eigenvalues of the homogeneous part are computed in all the flight envelope. Since the matrix of aerodynamic generalized forces Qhh depends on the frequency through the adimensional Laplace variable p, a direct computation of eigenvalues is not possible and some iterative procedure is needed. The homogeneous system whose eigensolutions are sought is:

Mhhs2 + Chhs + Khh− q∞(v)Qhh(p(s), M (v)) q = 0 (2.8)

in short:

F(s, v)q = 0 (2.9)

Since eigenvectors q are defined up to a multiplicative factor a normaliza-tion equanormaliza-tion for eigenvectors must be added in order to obtain a well-posed problem. One possibility for the normalization consists of imposing 1

2q

H_{q = 1} A possible choice of a method for flutter tracking is a continuation method [29], which is based on the derivation of the flutter equation with respect to the flight speed v:

∂F(s,v) ∂s F(s, v) 0 qH ds dv dq dv =− ∂F(s,v) ∂v 0 (2.10) The problem is now a standard Ordinary Differential Equation (ODE) problem in the unknowns s and q, which can be easily solve by integrating on the desired speed range. Two issues arise in the application of this method: the first one is the necessity to find a starting solution at a given speed. The second one is the fact that, since the constraint equation (2.9) is never applied directly there is no assurance that it will be satisfied exactly in all the flight speed range.

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In order to avoid problems related to the second issue, the constrain equation 2.9 is enforced at each integration step. This is done by applying a Newton-Raphson method on the linearized equation:

_∂F(s,v) ∂s F(s, v) 0 qH (sk_,v) ∆s ∆q =−F(s k_{, v)q}k 1 2(q k₎H_qk (2.11)

The initial solution for the Newton-Raphson method is the result of the integration step, which generally are small enough to guarantee a good starting guess for the iterative method. The result is that only few Newton iterations are required at each step to enforce the condition.

The computation of the starting solution is performed using a PK iteration.

2.5.1 Interpolation of aerodynamic matrices

The matrix of generalized aerodynamic forces is known only at some values of reduced frequency and Mach number. It must be interpolated in order to obtain values in the entire imaginary axis, and it must be extrapolated in order to obtain values for Qhh also for complex p.

Interpolation on the imaginary axis is performed using a cubic spline. Extrapolation on complex plane is based on a second order series expansion of Qhh(p): Qhh(p) = Qhh(jk) + dQhh dp jk (p − jk) + 1 2 d2_Q hh dp2 jk (p − jk)2 (2.12)

It is assumed that Qhh is analytic on the imaginary axis, and then the computation of derivatives with respect of p reduces to the computation of derivatives with respect to k.

dQhh(p) dp jk = dQhh(p) djk jk = −j dQhh(k) dk k (2.13) d2Qhh(p) dp2 jk = d 2_Q hh(p) djk2 jk = − d 2_Q hh(k) dk2 k (2.14)

Derivatives with respect to k are computed using the same spline interpo-lation used for the computation of Qhh.

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2.5. Flutter computation

A linear interpolation over Mach number has been used in order to take into account compressibility effects.

For the PK computation a linear interpolation over reduced frequencies is used. Extrapolation is of zero order for the real part and of first order for the imaginary part. The accuracy of the method is preserved since this solution is a starting guess for a Newton method which enforces eq. (2.9).

In figg. (2.8) and (2.7) the computed aeroelastic eigenvalues with Mach M∞ = 0.71 are presented. Results obtained with the continuation method are compared to those computed by NASTRAN using a PK method, in order to have a validation for the continuation method.

−8 −6 −4 −2 0 0 50 100 150 200 250

Real part [rad/s]

Imaginary part [rad/s]

Mode 2 Mode 5 Mode 6 Mode 7 Mode 9 Mode 10 Mode 11 Mode 13 Mode 14 Mode 15 Mode 16 Mode 17 Mode 18

Figure 2.7 Flutter diagrams for symmetric aircraft: complex plane. Bold lines: NASTRAN results. Markers: continuation method. M∞ = 0.71, ρ =

0.53Kg/m3_.

In figg. (2.10) and (2.9) the computed aeroelastic eigenvalues obtained allowing the variation of Mach number are presented.

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1500 200 250 300 5 10 15 20 25 30 35 V [m/s] Frequency [Hz] 150 200 250 300 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 V [m/s] Damping [−]

Figure 2.8 Flutter diagrams for symmetric aircraft: frequency and damping. Bold lines: NASTRAN results. Markers: continuation method. M∞ = 0.71,

ρ = 0.53Kg/m3. −7 −6 −5 −4 −3 −2 −1 0 50 100 150 200

Real part [rad/s]

Imaginary part [rad/s]

Mode 2 Mode 5 Mode 6 Mode 7 Mode 9 Mode 10 Mode 11 Mode 13 Mode 14 Mode 15 Mode 16 Mode 17 Mode 18

Figure 2.9 Flutter diagrams for symmetric aircraft, with Mach dependence: complex plane, ρ = 0.53Kg/m3.

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2.5. Flutter computation 1400 160 180 200 220 240 260 280 5 10 15 20 25 30 35 V [m/s] Frequency [Hz] 140 160 180 200 220 240 260 280 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 V [m/s] Damping [−]

Figure 2.10 Flutter diagrams for symmetric aircraft, with Mach dependence: frequency and damping, ρ = 0.53Kg/m3.

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2.6 Approximations for the aerodynamic model

A steady state formulation for the aerodynamic generalized forces is desired since it would enable the use of the state space theory for control system synthesis. In order to obtain a state space representation of aerodynamic matrices some approximations must be introduced and there exist several possibilities to do that. A first approach approximates the aerodynamic transfer matrices as polynomials in k, in the case polynomial of order two this approach takes the name of quasi-steady approximation of aerodynamic matrices. Another approach consists of introducing a rational matrix fraction approximation (RMFA) of aerodynamic matrices [39]. Aerodynamic matrices are approximated using a model of the form

Q(p, M ) = D−1(p, M )N(p, M ) (2.15)

where D(p, M ) and N(p, M ) are matrix polynomials in the reduced Laplace variable p = tas.

2.6.1 Quasi-steady approximation

In order to obtain quasi-steady approximation for aerodynamic matrices a second order polynomial is fitted to the available values known for some dis-crete values of reduced frequency. It turns out that the result of interpolation is strongly influenced by the set of reduced frequencies used in the polynomial fitting. This is due to the fact that the interpolated function can approximate only locally the true function.

If a second order approximation is used, generalized aerodynamic forces depend only on modal coordinates up to their second derivative. Since these variables are already present in the structural model the aerodynamic model does not introduces new variables.

By allowing the use of different sets of reduced frequencies in the interpo-lation of each column of the aerodynamic transfer matrix, better results can be obtained. The natural frequency of modes associated to the column of qhh can be used as a criterion for the choice of the maximum reduced frequency used for the interpolation.

The damping of aeroelastic roots for the system with a quasi-steady representation of the aerodynamic is presented in fig. (2.11). From the figure it is possible to note that only low frequency modes are well approximated.

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2.6. Approximations for the aerodynamic model 150 200 250 300 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 V [m/s] Damping [−] Mode 7 Mode 8 Mode 9 Mode 10 Mode 11 Mode 12 Mode 13 Mode 14 Mode 15 Mode 16 Mode 17 Mode 18

Figure 2.11 Flutter diagrams for symmetric aircraft: damping. Bold lines: complete aerodynamic representation. Markers: Quasi-steady approximation, M∞= 0.71, ρ = 0.53.

2.6.2 Quasi-steady approximation with high order

If the order of interpolating polynomial is greater than two there is need to introduce new variables in order to represent aerodynamic forces.

Polynomial of order three If a third order polynomial is used, the aero-dynamic transfer function can be approximated as

Qhh(k) ≈ Kahh+ jkCahh− k2Mahh− jk3Dahh (2.16)

The damping of aeroelastic roots computed using this approximation is shown in fig. (2.12), where it can be seen that the quality of the approximation is better than those obtained by using a second order approximation.

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150 200 250 300 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 V [m/s] Damping [−] Mode 7 Mode 8 Mode 9 Mode 10 Mode 11 Mode 12 Mode 13 Mode 14 Mode 15 Mode 16 Mode 17 Mode 18

Figure 2.12 Flutter diagrams for symmetric aircraft: damping. Bold lines: complete aerodynamic representation. Markers: Third order approximation.

Polynomial of order four

If a fourth order polynomial is used, the aerodynamic transfer function can be approximated as

Qhh(k) ≈ Kahh+ jkCahh− k2Mahh− jk3Dahh+ k4Eahh (2.17)

Again, the damping of aeroelastic modes is presented, and it is shown in fig. (2.13), where a decrease in the quality of the approximation with respect to results obtained with a third order polynomial can be noticed.

2.6.3 Matrix fraction approximation

The aerodynamic transfer matrix is the transfer function of a dynamical system. The inputs of the system are amplitudes of boundary condition with which aerodynamic forces have been computed, they can be modal displacements or the gust intensity. Outputs of the system are the generalized aerodynamic forces, scaled by the dynamic pressure q∞. In the reduced

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2.6. Approximations for the aerodynamic model 150 200 250 300 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 V [m/s] Damping [−] Mode 7 Mode 8 Mode 9 Mode 10 Mode 11 Mode 12 Mode 13 Mode 14 Mode 15 Mode 16 Mode 17 Mode 18

Figure 2.13 Flutter diagrams for symmetric aircraft: damping. Bold lines: complete aerodynamic representation. Markers: Fourth order approximation.

frequency domain this can be expressed by

fa(p) = Q(p)u(p) (2.18)

The matrix fraction approximation approach consists of approximating the aerodynamic transfer function by a rational transfer function, which leads to a representation of the aerodynamic system as linear, time invariant system.

The aerodynamic transfer function is approximated by fitting a rational matrix fraction on the form

Q(p, M ) = D−1(p, M )N(p, M ) (2.19)

where D(p, M ) is a matrix polynomial of order n and N(p, M ) is another matrix polynomial of order n + 2. This expression can be rewritten as

Q(p, M ) = D0+ pD1+ p2D2+ C(pI − A) −1

B0+ pB1+ p2B2

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The presence of terms D1, D2, B1 and B2 indicates that system input is not only the input vector u, but also its first and second time derivatives. The inclusion of input’s temporal derivatives represent a residualization of the high frequency part of the aerodynamic system.

The state space representation associated to eq. (2.20) is        dxa ds = Axa+ B0u + B1 du ds + B2 d2u ds2 fa = Cxa+ D0u + D1 du ds + D2 d2_u ds2 (2.21)

where s is the non-dimensional time s = _tt

a, in the physical time it becomes        dxa dt = 1 ta Axa+ 1 ta B0u + B1 du dt + taB2 d2_u dt2 fa = Cxa+ D0u + taD1 du dt + t 2 aD2 d2u dt2 (2.22)

If matrices of both modal and gust boundary conditions are interpolated the result is

Qhh Qhg = D0hh D0hg + p D1hh D1hg + p2D2hh D2hg

+ Cha(pI − Aaa) −1

B0ah B0ag + p B1ah B1ag + p2B2ah B2ag (2.23)

The aeroservoelastic system, in state–space form is   I 0 0 0 Mhh− q∞t2_aD2hh 0 0 −t2 aB2ah taI   d dt   qh ˙ qh xa  =   0 I 0 −(Khh− q∞D0hh) −(Chh− q∞taD1hh) q∞Cha B0ah taB1ah Aaa     qh ˙ qh xa   + 1 V∞   0 0 0 q∞D0hg q∞taD1hg q∞t2aD2hg B0ag taB1ag t2aB2ag

    vg ˙ vg ¨ vg   (2.24)

The computation of state space representation of aerodynamic matrices has been performed using the algorithm presented in [39], which is included in NeoCASS.

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2.7. Stress recovery 150 200 250 300 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 V [m/s] Damping [−] Mode 7 Mode 8 Mode 9 Mode 10 Mode 11 Mode 12 Mode 13 Mode 14 Mode 15 Mode 16 Mode 17 Mode 18

Figure 2.14 Flutter diagrams for symmetric aircraft: damping. Bold lines: complete aerodynamic representation. Markers: Quasi-steady approximation.

Results obtained by using 19 aerodynamic states are presented in fig. (2.14)

Results obtained by using a state space approximation are better than those obtained by the use of a third order polynomial approximation, while the number of extra degree of freedom added to the system is almost equal in the two cases. For this reason the state space approximation will be used to represent aerodynamic forces. In cases where only the low frequency behaviour of the system is needed, a quasi-steady approximation can be used, allowing a substantial reduction of the system dimension.

2.7 Stress recovery

In order to accelerate the convergence of recovered stresses with the number of modes included in the model the mode acceleration procedure was used.

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nodes by means of the following equation

ud= K−1_dd [fd− Mdhq¨h + q∞Qdh(k, M )qh] (2.25)

where ud represents the deformable motions of the structure, i.e. ud is created in such a way that Kdd is nonsingular.

Since the only external loads comes from the atmospheric turbulence the equation becomes ud= K−1dd −Mdh¨qh− Knsdhqh+ q∞Qdh(k, M )qh+ q∞Qdg(k, M ) vg V∞ (2.26) where Kns

dh contains the contribution of the actuation system to the structural stiffness.

From displacements ud internal stresses at some selected points can be computed by means of a properly defined stiffness matrix Kstress

zs = Kstressud (2.27)

2.7.1 Quasi-steady approximation of aerodynamic forces

If a quasi-steady approximation for aerodynamic forces is used, the output relation assumes the following form

zs = KstressK−1dd − ¯Mdh¨qh− ¯Cdhq˙h− ¯Kdhqh (2.28) + q∞ Ma_dg(M )vg V∞ Ca_dg(M ) ˙vg V∞ taKadg(M ) ¨ vg V∞ t2_a (2.29) where ¯ Mdh = Mdh− q∞Madh(M )t 2 a (2.30) ¯ Cdh = −q∞Cadh(M )ta (2.31) ¯ Kdh = Knsdh− q∞Kadh(M ) (2.32)

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2.7. Stress recovery

2.7.2 State-space representation of aerodynamic forces

Since matrices Qdhand Qdg depend on frequency, they must be approximated in order to obtain a state space representation, this can be done by identifying an augmented aerodynamic matrix on the form

Q =Qhh Qhg Qsh Qsg = I 0 0 Kstress Qhh Qhg Qdh Qdg (2.33) By doing this, the output of the aerodynamic system contains both modal forces and contribution of aerodynamic forces to recovered stresses.

zs = KstressK−1dd [−Mdhq¨h− Knsdhqh] + q∞Csaxa (2.34) + q∞ D0shqh+ D1shtaq˙h+ D2sht2a¨qh (2.35) + q∞ V∞ D0sgvg+ D1sgta˙vg+ D2sgt2av¨g (2.36)

2.7.3 Gust response

The response of the aircraft to atmospheric gusts with shape

vg = 1 2 1 − cos 2π Lg x (2.37) and unit amplitude vg = 1 m/s, is analyzed

Convergence on mode number

Simulations are performed using an increasing number of structural modes present in the model, and the wing root bending moment computed using mode acceleration is shown in fig. (2.15), where it can be seen that only a slight decrease of the peak value of the response results from the use of a reduced basis.

In fig. (2.16) the wing root torsional moment computed with the mode acceleration and with the direct recovery are compared. It can be seen that, if the direct recovery is used, the torsional moment is recovered only if the mode associated to wing torsion (the 17th) is included in the model. If mode acceleration is used, instead, static response of modes not included in the model contribute to the response, and thus there is no need for the inclusion of the torsional mode in the model.

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0 1 2 3 4 5 −4 −3 −2 −1 0 1 2 3 4 5 6x 10 4 t [s] Bending moment [Nm] 18 modes 13 modes 14 modes 15 modes 16 modes 17 modes 19 modes

Figure 2.15 Gust response with increasing model order. Wing root bending moment. Gust lenght Lg = 18 m, M = 0.71, q∞= 1.282 × 104 P a.

Aerodynamic approximations

Responses to gust excitation were performed also by using the quasi-steady and the state-space approximations of aerodynamic transfer matrices. Results are shown for three different gust lengths: Lg = 18 m, which represent the shorter gust indicated by regulations, Lg = 70 m, which is the gust that gives the maximum wing root bending moment and Lg = 220 m, which is the longest gust indicated in regulations.

It can be seen that the state space representation of aerodynamic forces gives stresses which are close to those obtained by the use of the complete aerodynamic transfer function. This holds for all cases considered. The quasi-steady approximation, instead, gives fairly good results for longer gust but does not allow the recovery of the torsional moment for the shortest gust.

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2.7. Stress recovery 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −4000 −2000 0 2000 4000 6000 8000 10000 t [s] Torsional moment [Nm] 18 modes 13 modes 14 modes 15 modes 16 modes 17 modes 19 modes

(a) Mode Acceleration.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 −4000 −2000 0 2000 4000 6000 8000 10000 t [s] Torsional moment [Nm] 18 modes 13 modes 14 modes 15 modes 16 modes 17 modes 19 modes (b) Direct recovery.

Figure 2.16 Gust response with increasing model order. Wing root torsional moment. Gust lenght Lg = 18 m, M = 0.71, q∞= 1.282 × 104 P a.

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0 0.2 0.4 0.6 0.8 1 −6 −4 −2 0 2 4 6 8x 10 4 t [s] Bending moment [Nm] Complete aerodynamic State space representation Quasi−steady approximation

(a) Bending moment.

0 0.2 0.4 0.6 0.8 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2x 10 4 t [s] Torsional moment [Nm] Complete aerodynamic State space representation Quasi−steady approximation

(b) Torsional moment.

Figure 2.17 Gust response with different aerodynamic models. Wing root bending and torsional moment. Gust length Lg = 18 m. M = 0.71, q∞ =

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2.7. Stress recovery 0 0.5 1 1.5 2 2.5 3 −1 −0.5 0 0.5 1 1.5x 10 5 t [s] Bending moment [Nm] Complete aerodynamic State space representation Quasi−steady approximation

(a) Bending moment.

0 0.5 1 1.5 2 2.5 3 −5000 0 5000 10000 t [s] Torsional moment [Nm] Complete aerodynamic State space representation Quasi−steady approximation

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0 1 2 3 4 5 6 7 8 −1.5 −1 −0.5 0 0.5 1x 10 5 t [s] Bending moment [Nm] Complete aerodynamic State space representation Quasi−steady approximation

(a) Bending moment.

0 1 2 3 4 5 6 7 8 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1x 10 4 t [s] Torsional moment [Nm] Complete aerodynamic State space representation Quasi−steady approximation

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CHAPTER

3

Static Output feedback control

A classical approach to control system design is the so-called linear quadratic formulation. The basic idea consists of formulating an objective function J which measures how much variables of interest are different from desired values. Then the controller which minimizes the cost function J is sought. This approach can also be applied to the design of a static output feedback (SOF) controller, and indeed many methods have been proposed which are

based on the optimization of a quadratic cost function.

Another different approach has been developed only recently and is based on the formulation of the control problem in the form of a system of linear matrix inequalities (LMI). Linear matrix inequalities are used to express control objectives such as stability, decay rate and so on. Once the system of LMIs has been formulated, efficient algorithms exist which are able to find a controller which satisfies the system of inequalities, or to determine if such a controller does not exist.

In this chapter both approaches are described, and an algorithm for the computation of a static output feedback controller, based on the linear quadratic formulation, is presented.

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3.1 Generic system

The generic system considered throughout this chapter is a linear, time– invariant system, whose representation in state space form is:

     ˙ x = Ax + Buu + Bdd y = Cyx + Dyuu + Dydd + Dynn z = Czx + Dzuu + Dzdd (3.1)

where x ∈ Rn is the state vector, u ∈ Rmu _{is the control vector, d ∈ R}md is the vector of external disturbances, y ∈ Rly _{are the measures, n ∈ R}mn _are measurement noises, and z ∈ Rlz _{is the vector of response performances of} interest.

The classification of inputs and outputs with different terms has been introduced to reflect both the physical origin of each term and the way it will be used within the control law synthesis.

Control input u represents the only way the system can be modified by the user. In the case of an airplane, for example, the control input vector can contain the control surfaces displacement.

Disturbance inputs d are unmodifiable inputs which affect system dy-namics, a direct measure of these disturbances is rarely available, in general informations about disturbance input come from measurement output.

Measurement vector y represents the only quantity known for the all system. It is usually an output from sensors placed in the physical system. They can depend directly by control and disturbance inputs and they are corrupted by a third input, the measurements noises n. These outputs do not affect system dynamic but only available informations about system.

Finally, performance outputs z represent quantities of interest of the system, they can be not accessible through direct measurement, and usually it is desired that they stay as close as possible to the null value.

3.1.1 Static Output Feedback control

In a static output feedback controller the control input u is taken as a linear combination of measured outputs y

u = − ˆGy (3.2)

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3.1. Generic system as u = −I + ˆGDyu −1 ˆ G (Cyx + Dydd + Dynn) (3.3)

If Dyu 6= 0 the expression of u is nonlinear in the gain matrix ˆG, and this complicates the computation of the gain matrix. The variable change G = I + ˆGDyu

−1 ˆ

G, however, leads to an expression linear in G, and can be used to obtain ˆG once G has been computed. In some cases there is need to impose a structure to the matrix ˆG, for example some selected elements may be forced to be null. A typical situation when this is done is in the design of distributed controllers, where each actuator is connected to only one sensor, the resulting gain matrix is thus diagonal. In general constraints refers to elements of ˆG separately, but this does not happen in general for the transformed gain G, where each constraint can relate all elements at the same time. There is need to take account for these constraints in the gain computation, as it will be seen in section 3.5, where the algorithm for the computation of G will be discussed.

If the expression for u is inserted in (3.1), the following closed-loop system results:                  ˙ x = (A − BuGCy) x +Bd− BuGDyd −BuGDyn d n z = (Cz− DzuGCy) x +Dzd− DzuGDyd −DzuGDyn d n u = (−GCy) x +−GDyd −GDyn d n (3.4)

By introducing the notation indicating closed loop matrices by an overbar, the equations for the closed loop system can be rewritten in a more compact form:                  ˙ x = ¯Ax +_¯ Bd B¯n d n z = ¯Czx + _¯ Dzd D¯zn d n u = −GCyx + _¯ Dud D¯un d n (3.5)

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with ¯ A = A − BuGCy ¯ Bd= Bd− BuGDyd ¯ Bn= −BuGDyn ¯ Cz = Cz − DzuGCy ¯ Dzd= Dzd− DzuGDyd ¯ Dzn = −DzuGDyn ¯ Dud= −GDyd ¯ Dun = −GDyn

Therefore the closed loop system is a closed loop system with inputs d and n and the performance outputs z as outputs. This form of the closed loop system will be used in the specification of control objectives.

3.1.2 State Feedback control

In a state feedback controller the control input depends linearly on the entire state vector x. It can be demonstrated (see for example [26]) that, if a quadratic cost function is used as a control specification, the optimal controller is on this form. The control input has expression

u = −Gx (3.6)

If the expression for u is inserted in (3.1), the following closed-loop system results:      ˙ x = (A − BuG) x + Bdd z = (Cz− DzuG) x + Dzdd u = −Gx (3.7)

The equation for the measurement output y has been neglected since it does not concurs in the controller synthesis. By introducing the notation of indicating closed loop matrices by an overbar, the equations for the closed loop system can be rewritten in a more compact form:

     ˙ x = ¯Ax + Bdd z = ¯Czx + Dzdd u = −Gx (3.8)

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3.2. Quadratic control

The state feedback problem can be viewed as a static output feedback problem where the whole state is available as measurement, without any noise corrupting it. Formulae for state feedback can be directly derived from those of static output feedback by imposing Cy = I, Dyu= 0, Dyd = 0, and Dyn = 0.

3.1.3 State Estimation

The objective of state estimation problem is to find an estimate of the state x by means of a reconstructed system. The reconstructed system has the same dynamic matrix as the true system and is forced by the same inputs. An additional forcing term is added to the reconstructed system, this term is proportional to the difference between actual measures y, and their reconstructed counterparts. The dynamics can be written as

(

˙o = Ao + Buu + L (y − yo) yo = Cyo + Dyuu

(3.9)

The objective of the state reconstruction is to nullify the estimation error e = x − o, thus the state estimation problem can be stated in a more natural way by considering the error dynamic, obtained by subtracting eq. (3.9) from the first equation in eq. (3.1)

             ˙e = (A − LCy) e +Bd− LDyd −LDyn d n z = e uo = (−LCy) e +−LDyd −LDyn d n (3.10)

Also the state estimation problem can be seen as a special case of the static output feedback, obtained by imposing Bu = I, Cz = I, Dzu= 0, and Dzd= 0.

3.2 Quadratic control

If the objective of the control system is to maintain the performance outputs as close as possible to zero, a performance index for the control system can be represented by a measure of z. For example a possible choice for a performance index can be the magnitude of the output of the closed loop system, when

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disturbances are impulses applied at time t = 0. Another possible choice is to consider as a performance index the variance of the output of the closed loop system when disturbances are white noises. These two possibilities are analyzed in a more detailed way in the following sections.

3.2.1 Deterministic setting

A typical approach in defining a quadratic objective function is to consider the free response of the closed loop system when it evolves from a generic initial condition [26]. If an infinite time interval is considered the cost function can be expressed as J = 1 2 Z ∞ 0 zT_W zzz + uTWuuu dt (3.11)

where Wzz = WTzz ≥ 0 and Wuu = WTuu> 0 are weighting matrices. The explicit presence of the control input as a separate term in the cost function deserves some comments; in any real system the amplitude of control input is limited by the presence of actuators saturation and by the presence of bounds on control energy available, in addition a low intensity of control action is desired since it reduces the control system energy requirements and can increase the robustness of the controller. For this reason u can be considered as a performance output for the system and included in z, in this case, however, the control input has been considered separately. The ratio between amplitude of Wzz and Wuu, in fact, strongly affects the closed loop behaviour of the closed loop system and thus it is convenient to consider these two terms separately.

The cost function in eq. (3.11) is associated to the constraint that z and u must result from the unforced, closed loop system

( ˙

x = ¯Ax x(0) = ¯x0

(3.12)

since the application of impulsive loadings at time zero is equivalent to a change of initial conditions, this formulation can take account for impulsive disturbances on form

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The initial condition of the state can thus be written as

¯

x0 = x0+ ¯Bww¯ (3.14)

Also generic disturbances with finite duration can be considered within this formulation by the introduction of shaping filters. A shape filter is a dynamical system whose impulse response represents the time evolution of the disturbance input. Then the control is designed on the augmented system where the disturbances affects the physical plant only through the shape filter dynamic.

Since both z and u are outputs of the closed loop unforced system they depend linearly on the state vector and on the disturbance input by a relation of the form

z = ¯Czx + ¯Dzww (3.15)

u = ¯Cux + ¯Duww (3.16)

If the closed loop system was not strictly proper, i.e. ¯Dzw 6= 0 or ¯Duw 6= 0, the expression of the cost function would contain the product of two impulses and thus it would not be defined. For this reason it is assumed that the closed loop transfer function is strictly proper and that disturbances can affects z and u only through system dynamics. With this assumption the cost function can be written as J = 1 2 Z ∞ 0 xTWd₁xdt = 1 2 Z ∞ 0 ¯ xT₀eA¯TtWd₁eAt¯ x¯0dt = 1 2T r Z ∞ 0 ¯ xT₀eA¯TtW₁deAt¯ dt ¯x0¯xT0 = 1 2T rΛW d 3 (3.17) where Wd₁ = ¯CT zWzzC¯z + ¯CTuWuuC¯u and Wd3 = ¯x0¯xT0. W d 3 represent the influence of initial conditions and disturbances on the state dynamics and the superscript d indicates the deterministic definition of the cost function. The matrix Λ satisfies the following Lyapunov equation

¯

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Application to the static output feedback If a static output feedback controller is used, the closed loop system is defined by eq. (3.5), the dis-turbance acting on closed loop system is composed by both d and n; the perturbed initial condition is

¯

x0 = x0+ ¯Bdd + ¯¯ Bnn¯ (3.19)

in this case Wd₁ and Wd₃ have expressions

Wd₁(G) = Qd_xx− Qd_xuGCy − CTyG T (Qd_xu)T + CT_yGTQuuGCy (3.20a) Wd₃(G) = Sd_xx − (Sd yx) T_GT_BT u − BuGSdyx+ BuGSdyyG T_BT u (3.20b) where Qd_xx = CT_zWzzCz Qd_xu = CT_zxWzzDzu Qd_uu = DT_zuWzzDzu+ Wuu (3.21) and Sd_xx = BdMddBTd + Mxx+ MxdBTd + BdMdx Sd_yx= DydMddBTd + DynMndBTd + DydMdx+ DynMnx Sd_yy = DydMddDTyd+ DynMndDTyd+ DydMdnDTyn+ DynMnnDTyn (3.22) All matrices on form Mpq in eq. (3.22) are defined as

Mpq = pqT (3.23)

and are an indication of the relative magnitude of influence of initial condition and disturbances on system dynamics.

The constraint equation can be taken in account by the use of the Lagrange multiplier method, by defining the augmented cost function ¯J

¯ J = 1

2T r h

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where P = PT is the matrix of Lagrange multipliers. If derivatives of the cost function with respect to P, Λ and G are imposed equal to zero, the following system of equations results

       ¯ ATΛ + Λ ¯A + Wd₁ = 0 ¯ AP + P ¯AT + Wd₃ = 0 BT_uΛBuGSdyy+ QduuGCyPCTy = BuTΛPCTy + BTuΛ(Syxd )T + QduxPCTy (3.25a)

3.2.2 Stochastic setting

An alternative way to define a quadratic cost function is to consider the response to disturbances on form of white noises, in this case the time integral of the response is not defined, but a measure of the output can be obtained by considering the variance of the response. The stochastic cost function for a linear, time-invariant system can thus be expressed by [26]

J = lim t→∞E h z(t)TWzzz(t) + u(t)TWuuu(t) i (3.26) as in the deterministic case the control input u has not been included to the performances in order to explicitly show the relative importance of u and z.

If disturbances are statistically stationary then also z and u are statistically stationary and then the expected value does not depend on time, the cost function becomes J = EzTWzzz + uTWuuu = T rWzzσzz2 + Wuuσuu2 (3.27) where σ2

zz = E[zzT] is the variance of performance outputs and σuu2 = E[uuT_{] is the variance of control input. If the closed loop system is strictly} proper output variances are related to the variance of the state P = σ2_xx through the relations

σ_zz2 = ¯CzP ¯CTz (3.28a)

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If disturbances are white noises with variance Www, then P must satisfy the following Lyapunov equation

¯

AP + P ¯AT + BwWwwBTw = (3.29)

¯

AP + P ¯AT + Ws₃ = 0 (3.30)

As in the deterministic case, a broader class of disturbances can be considered if a shape filter is included. In this case the shape filter must provide an output with the desired covariance when forced by a white noise.

If eqs. (3.28a) and (3.28b) are substituted in the cost function, it becomes

J = 1

2T r [PW s

1] (3.31)

where Ws₁ = ¯CT

zWzzC¯z+ ¯CTuWuuC¯u. Analogously as what done in the deterministic case, the superscript s has been used to indicate the stochastic definition of the cost function.

Application to the static output feedback If a static output feedback controller is used, the closed loop system is defined by eq. (3.5) and the disturbance acting on closed loop system is composed by both d and n; in general these two inputs can be correlated, with a variance matrix

Ed n dT _nT =Wdd Wdn Wnd Wnn (3.32) In this case Wd₁ and Wd₃ have expressions

Ws₁(G) = Qs_xx− Qs_xuGCy − CTyG T (Qs_xu)T + CT_yGTQuuGCy (3.33a) Ws₃(G) = Ss_xx − (Ss yx) T_GT_BT u − BuGSsyx+ BuGSsyyG T_BT u (3.33b) where Qs_xx = CT_zWzzCz Qs_xu = CT_zxWzzDzu Qs_uu = DT_zuWzzDzu+ Wuu (3.34)

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3.2. Quadratic control and Ss_xx = BdWddBTd (3.35) Ss_yx= DydWddBdT + DynWndBTd (3.36) Ss_yy = DydWddDTyd+ DynWndDTyd+ DydWdnDTyn+ DynWnnDTyn (3.37) Following the same procedure used in the deterministic case an augmented cost function ¯J is defined in order to account for the constraint equation (3.32) ¯ J = 1 2T r h PWs₁(G) + Λ ¯AP + P ¯AT + Ws₃(G) i (3.38) where Λ = ΛT is the matrix of Lagrange multipliers. If derivatives of the cost function with respect to P, Λ and G are imposed equal to zero, the following system of equations results

       ¯ AP + P ¯AT + Wd₃ = 0 ¯ ATΛ + Λ ¯A + Wd₁ = 0 BT_uΛBuGSdyy+ QduuGCyPCTy = BuTΛPCTy + BTuΛ(Syxd )T + QduxPCTy (3.39) The stationarity conditions of eq. (3.39) have the same expression as those of eq. (3.25), obtained in the deterministic case; also the augmented cost functions are the same in stochastic and deterministic case. The only difference between the two formulation is in the definition of weighting matrices and of shaping filters, if any. These differences do not affect the computation of the cost function, but they can strongly affect the cost function itself, leading to very different performances of the closed loop system.

3.2.3 Interpretation as H

2

control

For a strictly proper, asymptotically stable transfer function G(jω) the H2 norm is defined as [49] kG(jω)k2 2 = 1 2π Z ∞ −∞ T rG(jω)HG(jω) dω (3.40)

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By exploiting the Parseval’s theorem the H2 norm can be rewritten in the following form: kG(jω)k2₂ = Z ∞ 0 T rg(t)Tg(t) dt (3.41)

where g(t) is the inverse Fourier transform of G(jω), g(t) = F−1[G(jω)] and it is the impulse response associated with G(jω).

Another interpretation of the H2 norm can be given in term of the variance of the output of the system represented by G, as shown in [41]

kG(jω)k2

2 = T rEz(t)z(t) T

(3.42) where z(t) is the system response when the input is a white noise whose variance matrix is the identity matrix.

The same relationships hold if a weighting term is introduced in the definition of the norm:

kG(jω)k2_W = 1 2π

Z ∞ −∞

T rG(jω)HWG(jω) dω (3.43)

Both deterministic and stochastic cost functions can thus be viewed as a weighted H2 norm of the closed loop transfer function.

3.3 Computation of gain matrix

The problem of quadratic control applied to the static output feedback case can be formulated as the problem of finding a matrix G which satisfies

G = arg minJ (G) (3.44) where J (G) = 1 2T r [ΛW3] = 1 2T r [PW1] (3.45)

with the constraints

Λ ¯A + ¯ATΛ + W1 = 0 (3.46a)

¯

Active gust alleviation for a regional aircraft through static output feedback

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

Contents

CHAPTER

1

Introduction

CHAPTER

2

Aircraft model

2.1

Structural model

2.2

Aerodynamic model

2.2.1

Convergence

2.2.2

Gust boundary conditions

2.3

Control surfaces

2.4

Static analysis

2.5

Flutter computation

2.5.1

Interpolation of aerodynamic matrices

2.6

Approximations for the aerodynamic model

2.6.1

Quasi-steady approximation

2.6.2

Quasi-steady approximation with high order

2.6.3

Matrix fraction approximation

2.7

Stress recovery

2.7.1

Quasi-steady approximation of aerodynamic forces

2.7.2

State-space representation of aerodynamic forces

2.7.3

Gust response

CHAPTER

3

Static Output feedback control

3.1

Generic system

3.1.1

Static Output Feedback control

3.1.2

State Feedback control

3.1.3

State Estimation

3.2

Quadratic control

3.2.1

Deterministic setting

3.2.2

Stochastic setting

3.2.3

Interpretation as H

control

3.3

Computation of gain matrix