L1 ADAPTIVE CONTROLLER DESIGN FOR GUSTLOAD ALLEVIATION

Elisa Capello, Giorgio Guglieri, Fulvia QuagliottiPolitecnico di Torino, Department of Mechanical and Aerospace Engineering,

Corso Duca degli Abruzzi 24, 10129 Torino, ItalyKeywords: Adaptive Controller, Gust Load Alleviation, Flexible Aircraft

AbstractThe objective of this paper is the implementa-

tion and validation of an adaptive controller forgust load alleviation. The main contribution ofthis research is the design of a robust controllerthat guarantees the reduction of the gust loads,even when the nominal conditions change. Somepreliminary results are presented, considering thesymmetric aileron deflection as control device.The proposed approach is validated on a subsonictransport aircraft for different mass and flight con-ditions: even when the gust frequency changes, noparameter retuning is required.

1. IntroductionGust load alleviation systems are usually used to

reduce the airframe loads and to improve passengercomfort. Active control techniques for gust loadalleviation have been investigated extensively inthe last years to control the aeroelastic responseand to improve the handling qualities of the air-craft. Different approaches have been developedfor gust alleviation and load control, includingdesign of classical robust controllers, such as thelinear quadratic regulator [1] and gaussian theories[2], model predictive control algorithms [3], Hrobust control [4], [5].

Because of time varying characteristics of theaircraft dynamics, it is difficult to synthesize aunique control law for the whole flight envelopeand a gain scheduling should be required to ac-count for the time varying dynamics. For thisreason, adaptive feedback and/or feedforward con-trollers have been considered for adverse situations

due to their ability of modify a pre-existing controldesign.

The objective of the present paper is to derive acontroller that is robust in the presence of modeluncertainties, due to weight and flight conditionvariations. Moreover, the present paper aims atproposing an application of L1 adaptive controltechniques in the framework of flexible fixedwing subsonic aircraft, stabilizing the system underdifferent operating conditions and when differentgusts occur.

Some classical controllers, as H application,have limitations in terms of frequency band (onlylow frequencies are analyzed) and no uncertaintiesor variations on the nominal parameters are consid-ered,as in the works of [4], [6]. Recently, Jannsonand Eller [5] included uncertainties in the modelparameters but the gust is included as a Gaussianwhite noise of unit intensity and zero mean, that isnot a realistic case. In [7] two uncertainty modelsto decide what flexible modes to be truncatedfrom the original flexible system model whilepreserving closed-loop performance are presented.Robust control framework is used with the robustperformance criterion to show that the new inverseuncertainty representation of flexible modes givesgood closed-loop performance, but this controlleris not validated with different mass and flightconditions.

To overcome these limitations adaptive feedback[8], [9] and feedforward controllers [10], [11]are imnplemented. Wildschek [12] proposed anadaptive Multi Input - Multi Output (MIMO) feed-forward controller and a feedback H2 controller.No uncertainties are considered and the adaptivecontroller alleviated only the wing bending accel-

1

eration.L1 adaptive control theory reduces the tun-

ing effort required to achieve desired closed-loopperformance, particularly while operating in thepresence of uncertainties and failures. The adaptivelaw is a piecewise constant law, that guaranteesfast estimation, and the adaptation rate can beassociated with the sampling rate of the onboardCPU. Moreover, this adaptive algorithm guaranteesbounded inputs and outputs, uniform transient re-sponse and steady-state tracking.

The paper is organized as follows. In Section 2the aircraft model description and formulation ispresented. In the same Section the gust and loadmodels are described. In Section 3 the actuatormodels are analyzed. In Section 4 the controlarchitecture of the L1 controller is introduced. Thecontroller design and the simulation results are de-scribed in Section 5. Conclusions are summarizedin Section 6.

2. Aircraft and Gust Mathematical ModelFor the implemenatation of the proposed feed-

back controller only the longitudinal plane of asubsonic transport aircraft is considered. For themathematical formulation of the dynamic systema standard continous time-invariant state spaceformulation

x(t) = Ax(t) +Bu(t) +Bgwg(t),

y(t) = Cx(t) +Dgwg(t), x(0) = x0,(1)

where x(t) Rn is the state vector in whichboth rigid and flexible variables are considered,u(t) Rm is the control signal, y(t) Rl isthe controlled output, wg(t) Rm is the gustsignal. A Rnxn is the state matrix, B Rnxmis the control matrix, C Rlxn is the outputmatrix, Bg Rnxm is the input gust matrix andDg Rlxm is the output gust matrix. This com-plete aeroservoelastic model is obtained joiningtwo sub-models: (i) the flight dynamic model, thatdescribes the rigid body motion of the aircraftand (ii) the aeroelastic model, which is responsibleof the aircraft aeroelasticity. Hypothesis of smalldisturbances from a steady flight condition allowsto linearize the rigid body equations of motion [13]and to uncouple the longitudinal plane responsefrom the lateral one. The rigid state variables are

the longitudinal component of the total airspeedu, the angle of attack , the pitch angle and thepitch rate q. The control variables are the elevatordeflection e and the aileron deflection a (dividedin outboard deflection a,ou and inboard deflectiona,in). For both devices only symmetric deflectionare evaluated.

The mass and the elastic properties of the air-craft are given by a beam model made in Nastranand the flexible formulation can be obtained fromthe classical formulation of equations of motionfor multi degree of freedom systems, as in

Mx(t) + Ckx(t) +Kx(t) = F (t), (2)

where M is the mass matrix, Ck is the viscousdamping matrix and K is the stiffness matrix. Theflexible vector x(t) is the general time-varying dis-placement vector and F (t) can be divided into twoterms, one related to the aerodynamic componentsinduced by the structural normal modes and thesecond related to external forces that may not bedepending on aerodynamics.

The external aerodynamic force is modelledconsidering unsteady aerodynamics and is intro-duced in the statespace model (Eq. 1) using Padeinterpolation method [14]. The aerodynamic forcesFt and the gust forces Fg can be modeled as

Ft(s) =1

2V 2tasFGT (s)(s)

Fg(s) =1

2V 2tasFGM(s)w(s)

(3)

where Vtas is the aircraft true airspeed, i is thecontrol device deflection (i = e, aou, ain), w isthe gust speed, FGT (s) and FGM(s) are largescale improper transfer function matrices of theparticular forms

FGT (s) = Ft0 + Ft1s + Ft2ss2 +

Ni=1

Fi+2s

s+ i

FGM(s) = Fg0 + Fg1s + Fg2ss2 +

Ni=1

Fgi+2s

s+ gi(4)

with F and are interventing coefficient matricesand filter poles from FEM analysis [15].

Time domain aeroelastic analysis is performedto generate responses to active control and/or to ex-ternal force systems. Gust input causes a variation

2

E. Capello, G. Guglieri, F. Quagliotti

of the system aerodynamics that is simulated instate space formualtion using the Doublet-LatticeMethod ([16]). Panel incidence induced by thegust is computed for each control point of theaerodynamic mesh, that has to be introduced inthe model in terms of x, y and z coordinates. Thegeneration of the induced angle of attack due tothe gust profile (Fgustj(t)) for each aerodynamiccontrol point is expressed in Eq. 5 (discrete gust1 cosine model for a reference system with zupward and x backward),

Fgustj(t) =U

2Ucos(j)[

1 cos(

2fg

(t x0 xj

U

))] (5)where U is the vertical gust speed, U = Vtasis the aircraft airspeed, cos(j) is the dihedralcosine of each panel control point and fg is thegust frquency. The distance between the aircraftreference system center and the gust is defined asx0 = dgU, with dg = 0.1 s gust time delay.The variable xj is the x coordinate of the jth

aerodynamic control point.The gust dynamic loads have been calculated by

means of the mode displacement (MD) method,which recovers the loads directly from the modaldisplacements. The MD approach assumes that themodal superposition assumption, used to constructthe generalized aeroelastic equations of motion,can also be used to recover the load distributions.Since gust excitation cases are characterized byfairly well-distributed loads, the MD method canbe successfully used to calculate the actual loadswith a sufficient number of modes. The modalsuperposition assumption is:

(x, t) = (x)(t)

where (x) is the matrix of modal displacementsand (t) is the vector of natural modes in therange of 1 to 50 Hz of frequency. This assumptionimplies that the aerodynamic and inertial modalload (forces and moments), integrated for obtain-ing section loads, can be expressed as:

F (x, t) = CLOAD(t) (6)

where CLOAD is the integrated stiffness matrixexpressed in modal form.

3. Model of Actuator SystemsThe simulation model includes two different

actuators: (i) one related to the elevator surface(for the pitch rate control) and (ii) one related tothe aileron deflection (for the angle of attack vari-ation). For the elevator control surface, an Electro-Mechanical Actuator (EMA) is implemented andis composed of a variable speed bi-directionalelectric motor coupled with mechanical gears toprovide speed reduction and torque amplification.The implem