G Ferreres
email: gilles.ferreres@onera.fr

G Hardier

this paper describes the application of an indirect LFT-based state-space adaptive control scheme to a transport aircraft, within the context of the European project RECONFIGURE. The principle of the scheme is to off-line design and validate a gain-scheduled controller, depending on the plant parameters to be estimated, and to on-line combine it with a model estimator, so as to minimize the on-board computational time and complexity. A modal approach, very classical for the design of a flight control law, is used to directly synthesize the static output feedback LFT controller depending on the control and stability derivatives, i.e. the parameters of the linearized aerodynamic state-space model to be estimated. Since the gain-scheduled LFT controller online depends on the parameter estimates instead of the true values, its robustness to transient and asymptotic estimation errors needs to be assessed using µ and IQC analysis techniques. A primary concern being on-line implementation, a fully recursive frequency domain estimation technique is proposed, with a low on-line computational burden and the capability to track timevarying parameters. Full non-linear simulations along a trajectory validate the good performance properties of the combined estimator and gain-scheduled flight controller. To some extent, minimal guaranteed stability and performance properties of the adaptive scheme can be ensured, by switching to a robust controller when the parameter estimates are not reliable enough, thus bypassing the Certainty Equivalence Principle.

INTRODUCTION

Flight control has long been a privileged applicative field for robust and gain-scheduling control, see e.g. [START_REF] Aouf | Scheduling schemes for an integrated flight and propulsion control system[END_REF][START_REF] Biannic | Design and robustness analysis of fighter aircraft control laws[END_REF][START_REF] Ferreres | A practical approach to robustness analysis with aeronautical applications[END_REF][START_REF] Nichols | Gain scheduling for Hinfinity controllers: a flight control example[END_REF][START_REF] Rugh | Research on gain scheduling[END_REF][START_REF] Siwakosit | Robust flight control design with handling qualities constraints using scheduled linear dynamic inversion and loop-shaping[END_REF], and more recently for fault-tolerant and adaptive control [START_REF] Barkana | Classical and simple adaptive control for non-minimum phase autopilot design[END_REF][START_REF] Beard | An L1 adaptive pitch controller for miniature air vehicles[END_REF][START_REF] Ducard | A reconfigurable flight control system based on the EMMAE method[END_REF][START_REF] Ferreres | Adaptive control of a civil aircraft through on-line parameter estimation[END_REF][START_REF] Kim | A robust adaptive nonlinear control approach to missile autopilot design[END_REF][START_REF] Lombaerts | Nonlinear reconfiguring flight control based on online physical model identification[END_REF][START_REF] Oudin | Adaptive LFT control of a transport aircraft on the lateral axis[END_REF][START_REF] Oudin | Design and worst-case validation of a longitudinal adaptive flight control law: a practical approach[END_REF][START_REF] Shin | Adaptive Linear Parameter Varying control synthesis for actuator failure[END_REF][START_REF] Song | Online parameter estimation techniques comparison within a fault tolerant flight control system[END_REF][START_REF] Wahi | A survey of recent work in adaptive flight control[END_REF][START_REF] Wise | Robust and adaptive control of X-45A J-UCAS: a design trade study[END_REF][START_REF] Wise | Adaptive control of flight: theory, applications, and open problems[END_REF][START_REF] Ye | Adaptive fault-tolerant tracking control against actuator faults with application to flight control[END_REF][START_REF] Yu | Fault-Tolerant Flight Control Design with Explicit Consideration of Reconfiguration Transients[END_REF][START_REF] Yu | Fault-Tolerant Flight Control with Finite-Time Adaptation under Actuator Stuck Failures[END_REF]. Noting that strong links exist between all these fields. Indeed, robust and adaptive control can be seen as competing/complementary techniques for solving the same problem of controlling an uncertain plant: if a robust controller aims to stabilize the whole set of possible plant instances, an adaptive controller uses online measurements of the plant Inputs/Outputs (I/O) to stabilize the plant itself. On the other hand, gainscheduling can be seen as a special case of indirect adaptive control, whose principle is to estimate the plant parameters and to adjust the controller as a function of these on-line estimates: in the same way, a linear gain-scheduled controller adapts to the nonlinear plant using measured scheduling parameters.

Gain-scheduling is classically used to control an airplane along a trajectory, where the controller gains depend on measured or estimated flight parameters, such as airspeed, Mach, mass or Center of Gravity (CoG) position. However, situations may occur where the performance of the gain-scheduled controller is no more satisfactory, e.g. in the presence of icing, actuator faults, or when a scheduling parameter becomes unavailable due to a sensor fault, especially airspeed [START_REF] Goupil | AIRBUS efforts towards advanced real-time fault diagnosis and fault tolerant control[END_REF]. Adaptive control is an attractive solution in this context to recover closed loop performance.

Adaptive LFT control of a civil aircraft with on-line frequency domain parameter estimation

Most of the adaptive control literature focuses on the use of Input/Output (i.e. black box) models to control the system [START_REF] Landau | Adaptive control: algorithms, analysis and applications[END_REF]. Nevertheless, a physical aerodynamic state-space model is available for an aircraft (A/C), and state-space methods are typically used to design robust or gain-scheduled flight control laws, so that adaptive flight controllers are usually designed using state-space methods, see e.g. [START_REF] Beard | An L1 adaptive pitch controller for miniature air vehicles[END_REF][START_REF] Ferreres | Adaptive control of a civil aircraft through on-line parameter estimation[END_REF][START_REF] Kim | A robust adaptive nonlinear control approach to missile autopilot design[END_REF][START_REF] Lombaerts | Nonlinear reconfiguring flight control based on online physical model identification[END_REF][START_REF] Oudin | Adaptive LFT control of a transport aircraft on the lateral axis[END_REF][START_REF] Oudin | Design and worst-case validation of a longitudinal adaptive flight control law: a practical approach[END_REF][START_REF] Shin | Adaptive Linear Parameter Varying control synthesis for actuator failure[END_REF][START_REF] Song | Online parameter estimation techniques comparison within a fault tolerant flight control system[END_REF][START_REF] Wise | Robust and adaptive control of X-45A J-UCAS: a design trade study[END_REF][START_REF] Wise | Adaptive control of flight: theory, applications, and open problems[END_REF][START_REF] Ye | Adaptive fault-tolerant tracking control against actuator faults with application to flight control[END_REF][START_REF] Yu | Fault-Tolerant Flight Control Design with Explicit Consideration of Reconfiguration Transients[END_REF][START_REF] Yu | Fault-Tolerant Flight Control with Finite-Time Adaptation under Actuator Stuck Failures[END_REF].

Generally speaking, two main problems need to be solved in adaptive control, namely to obtain a priori guaranteed stability and performance properties of the adaptive closed loop, and to decrease the on-line computational time and complexity. In the context of indirect adaptive control, following [START_REF] Ferreres | Robust gain scheduling techniques for adaptive control[END_REF][START_REF] Ferreres | Adaptive control using robustness and gain-scheduling[END_REF], a solution based on robust and gain-scheduled control tools is to off-line design a gain-scheduled controller depending on the plant parameters to be estimated, to avoid the complexity of implementing a control design algorithm on-line.

On the other hand, LFT (Linear Fractional Transformation) representations of the plant (depending on the parameters to be estimated) and controller (depending on the parameter estimates) are connected to obtain an LFT closed loop depending on the true values of the plant parameters and on the estimation error. Using µ analysis with constant and frequency-dependent D, G scalings [START_REF] Fan | Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics[END_REF][START_REF] Ferreres | Robust feedforward design in the presence of LTI/LTV uncertainties[END_REF], and more generally Integral Quadratic Constraints (IQCs) [START_REF] Kao | A Matlab toolbox for robustness analysis[END_REF][START_REF] Megretski | System analysis via Integral Quadratic Constraints[END_REF], it is possible to study the stability and H ∞ /L 2 performance properties of the adaptive closed loop, considered as a gain-scheduled closed loop with an error on the scheduling parameters. As a specificity of this validation technique, the estimator is not included in the analysis which just provides specifications, i.e. a maximal amount of the allowable estimation error, and possibly a maximal rate of variation of the estimates. This estimation error can be considered as time-invariant or timevarying, thus studying the robustness of the adaptive scheme in the face of an asymptotic or transient estimation error.

Several gain-scheduling state-space control design techniques can be used, which assume time-varying or time-invariant scheduling parameters, see [START_REF] Rugh | Research on gain scheduling[END_REF] for a survey. In the sequel, we will focus on the modal technique of [START_REF] Magni | An LFT approach to robust gain scheduling[END_REF] because this one appears especially suitable for aeronautical applications: a static output feedback controller placing the main closed loop poles is synthesized under an LFT form. It only depends on the main A/C parameters to minimize on-line complexity, and to avoid an unsatisfactory transient response of the adaptive closed loop. Indeed, the quality of the estimated secondary A/C parameters is generally poor, so that a bad transient estimation error would give a bad transient closed loop response. This static LFT controller can be implemented either using the efficient recursive technique of [START_REF] Magni | Practical implementation of LFT-scheduled feedback gains[END_REF], which avoids any matrix inversion, or using a surrogate model, see section 6.

Another main issue is to develop a method for estimating the A/C model parameters on-line, with a low computational burden and the capability to track time-varying parameters despite measurement noises, poor data information contents (e.g. in cruise condition when variables are likely to be almost constant during long periods), and external disturbances (turbulence). Whatever method used, residual errors can sometimes be large and a measure of the parameter accuracy should also be provided, so that some logic can be introduced: if the estimated scheduling parameters are not reliable enough, one can switch to a robust backup controller ensuring minimal performance properties.

Algorithms for recursive parameter estimation have been developed for real-time applications in various fields such as industrial processes, robotics and aerospace [START_REF] Hardier | An extended U-D algorithm with multiple forgetting factors for RLS estimation of model parameters[END_REF][START_REF] Ljung | Adaptation and tracking in system identification: a survey[END_REF]. Most theoretical aspects were covered by early reference publications [START_REF] Astrom | Adaptive Control[END_REF][START_REF] Ljung | Theory and practice of recursive identification[END_REF]. In aeronautics, the early attempts to benefit from on-line parameter identification date back to the 80s, but they were limited by the available computational power of onboard processors. The common Time Domain (TD) methods are usually based on recursive or sequential Least-Squares (LS), or Extended Kalman Filters to cope with nonlinearities [START_REF] Ducard | A reconfigurable flight control system based on the EMMAE method[END_REF][START_REF] Hanlon | Multiple-model adaptive estimation using a residual correlation KF bank[END_REF]. More recently, this challenge has also been addressed by recursive subspace identification of linear or LPV (Linear Parameter-Varying) models. However, the relevance of such advanced multi-model or nonlinear schemes for on-board implementations is dubious due to the present limitations of on-board computers, especially for A/C because of certification issues and code verification. That is why we look in this paper for a simple enough approach, more likely to satisfy those real time constraints, and that could constitute a viable alternative for updating the models required by the most advanced FDD (Fault Detection and Diagnosis) and FTC (Fault Tolerant Control) schemes.

The paper is organized as follows. §2 presents the aircraft problem. The general principles of the estimation process are described in §3, see [START_REF] Hardier | A recursive estimation algorithm to track aircraft model parameters[END_REF] for a more complete presentation. An LFT flight controller is synthesized in §5 using the tools briefly described in §4, and some issues related to implementation aspects are covered by §6. Adaptive control results using the realistic non linear simulator of the RECONFIGURE project are presented in §7. Concluding remarks end the paper.

THE AIRCRAFT PROBLEM

The indirect adaptive control strategy presented in this paper has been applied in the framework of the EU-FP7 funded project RECONFIGURE (REconfiguration of CONtrol in Flight for Integral Global Upset REcovery). The goal of this project is to investigate and to evaluate aircraft guidance and control technologies that facilitate the automated handling of off-nominal/abnormal events, alleviate the pilot workload, and that optimize the aircraft status by automatically reconfiguring the aircraft to an optimal flight condition (see http://reconfigure.deimos-space.com/). A detailed description of the benchmark model and the fault/failure scenarios is given in [START_REF] Goupil | AIRBUS efforts towards advanced real-time fault diagnosis and fault tolerant control[END_REF][START_REF] Kerr | RECONFIGURE FP7 project: preliminary results and contributions[END_REF]. The V&V process involves a nonlinear highly representative model of a generic Airbus civil aircraft. The benchmark contains a baseline gain scheduled PI controller, actuator/sensor models, measurement filters, as well as control law protections. The different control designs developed during the project had to demonstrate their performance and robustness thanks to the industrial benchmarking proposed by Airbus, by using an extensive set of simulations including a wide range of operational conditions (realistic flight scenarios, external disturbances, pilot maneuvers). The computational burden and complexity of the schemes had also to be compliant with stringent computational constraints in order to be possibly implemented on embedded computers. Robustness issues were tackled by using tools like a Functional Engineering Simulator, implemented during the project to achieve a traditional Monte Carlo analysis as a preliminary step of the industrial V&V process [START_REF] Kerr | RECONFIGURE FP7 project: preliminary results and contributions[END_REF]. It is also noteworthy that avenues do exist for improving the V&V of more complex controllers like the proposed adaptive scheme; for instance, a practical approach for the worst-case analysis of a similar adaptive controller was already investigated by Airbus in [START_REF] Oudin | Design and worst-case validation of a longitudinal adaptive flight control law: a practical approach[END_REF]. The highly realistic nonlinear simulator of the RECONFIGURE project, which is very close to a real civil aircraft, includes a nonlinear aerodynamic model with 5 longitudinal states: the angle of attack α , the pitch rate q, the pitch angle θ , the true airspeed V and the altitude z. The acceleration z N is also to be accounted for as an additional output2 . Two control loops are considered: the inner loop is used to control the fast longitudinal states α and q, as well as the output z N , using the flight controller. The outer loop is used to control the other longitudinal states using an autopilot and an auto-throttle. In the context of the RECONFIGURE project the autopilot and auto-throttle are given, only the flight controller is allowed to be modified to recover the performance of the inner loop, e.g. in the presence of an actuator/sensor failure. An adaptive flight controller is synthesized in this paper.

In the context of our indirect adaptive control scheme, a linearized longitudinal aerodynamic model with 4 states α , q, θ , V and the output z N should be a priori estimated:

         + + + = + + + = = + + + = + + + = V N N q N N N V X X q X X V q V M M q M M q V Z Z q Z Z V m q z V m q V m q V m q δ α δ α θ δ α δ α α δ α δ α δ α δ α (1)
m δ is the elevator input. The effect of altitude, which is the slowest longitudinal state, is neglected. The aircraft is supposed to move about a given flight condition and a constant engine thrust is assumed. Nevertheless, as explained in the following, only the first two lines and the last line of model (1) need to be estimated in practice, i.e. those corresponding to α & , q & and z N . More precisely, the issue is to estimate the control and stability derivatives, i.e. the parameters α Z , q Z … of the linearized aerodynamic model. A linear in the parameters model is obtained, i.e.:

[ ]

T m T V q Nz q ] [ δ α α Θ = & & (2)
The Θ matrix includes the 12 parameters to be estimated. In the context of the adaptive scheme, the main goal of the on-line estimator is to update the values of the 5 main control and stability derivatives

α δ α α N M M M Z q , , , ,
used to schedule the controller ( §5.1), although all the 12 coefficients have to be estimated. Only the first two lines and the last line of model (1) need to be estimated because the linearized model used to design the flight controller only contains the fast longitudinal states α and q, as well as the output z N :

       + + = + + = + + = m q z m q m q N q N N N M q M M q Z q Z Z δ α δ α δ α α δ α δ α δ α & & (3)
The flight controller, designed using the 2 states linearized model above, is a posteriori validated on the 4 states linearized model [START_REF] Aouf | Scheduling schemes for an integrated flight and propulsion control system[END_REF]. Its architecture is:

c c I q Nz m Nz L dt Nz Nz K q K Nz K + - + + = ∫ ) ( δ ( 4
)
where c Nz is the reference acceleration input.

A FREQUENCY DOMAIN APPROACH FOR ON-LINE PARAMETER ESTIMATION

Adaptive control strategies are rather uncommon in the aircraft field, even if they begin to develop for UAVs, and this has to do with stringent certification issues. Otherwise, practical implementation of indirect adaptive schemes mostly rely on TD algorithms for estimating time-varying or uncertain model parameters [START_REF] Tao | Adaptive control design and analysis[END_REF]. Actually, TD algorithms could seem preferable at first sight, as the basic ones only involve very simple operations (e.g., by replacing a matrix inversion with a scalar division). However, advanced strategies are usually required for regularizing the estimation process, and those advantages can thus be wasted (see [START_REF] Hardier | An extended U-D algorithm with multiple forgetting factors for RLS estimation of model parameters[END_REF] for an overview of the pros and cons of TD methods). By contrast, FD techniques have many desirable features for on-line applications [START_REF] Han | Frequency and time domain online parameter estimation for reconfigurable flight control system[END_REF][START_REF] Hardier | On-line parameter identification for in-flight aircraft monitoring[END_REF][START_REF] Klein | Aircraft system identification: theory and practice[END_REF]. The computation time can be reduced by processing only a limited amount of frequencies within the bandwidth of interest, and the resulting indirect filtering of the wide-band disturbances (e.g., Low Frequency mismatch and High Frequency noise) improves the estimation accuracy. The standard deviation of the estimation errors can also be evaluated without any additional cost.

Furthermore, the availability of efficient tools making the transition from TD to FD possible, such as the recursive Fourier Transform (FT), greatly facilitates their use, and this was stressed in many publications during the last 20 years [START_REF] Basappa | Evaluation of recursive methods for aircraft parameter estimation[END_REF][START_REF] Han | Frequency and time domain online parameter estimation for reconfigurable flight control system[END_REF][START_REF] Klein | Aircraft system identification: theory and practice[END_REF][START_REF] Morelli | Real-time parameter estimation in the frequency domain[END_REF][START_REF] Morelli | Practical aspects of the EE method for aircraft parameter estimation[END_REF][START_REF] Morelli | Real-time dynamic modeling -Data information requirements and flight test results[END_REF][START_REF] Smith | Real-Time Stability and Control Derivative Extraction from F-15 Flight Data[END_REF]. Fig. 1 gives a schematic representation of the way the FD information is linked to the estimation process. Moreover, many issues specific to on-line estimation are mitigated when using FD approaches instead of TD ones: estimates are nearly unbiased even in the presence of noisy I/O data in case of collinearity in the regressors, the estimation errors do not require to be improved afterwards due to possible colored residuals, and so on (see [START_REF] Morelli | Practical aspects of the EE method for aircraft parameter estimation[END_REF] for all these practical aspects).

Otherwise, TD/FD estimation schemes rely either on recursive algorithms (making use of measurements as soon as they are available), or alternatively on sequential procedures, processing moving data windows with a lower rate to get a succession of piecewise constant values [START_REF] Chandler | System identification for adaptive and reconfigurable control[END_REF][START_REF] Hardier | An extended U-D algorithm with multiple forgetting factors for RLS estimation of model parameters[END_REF]. In [START_REF] Hardier | On-line parameter identification for in-flight aircraft monitoring[END_REF], a sequential process was proposed for monitoring a civil aircraft, that permits pre and post-processing stages to be included in the procedure to prevent from and to filter out inaccurate estimations. Even if both approaches have their own pros and cons [START_REF] Jiang | A revisit to block and recursive least squares for parameter estimation[END_REF], recursive schemes should be favored for on-line implementation, and FD is more suitable to define hybrid recursive/sequential procedures because the information naturally accumulates in the signals' FT, which behave like storing memories and are managed with more flexibility (truncation, forgetting, resetting).

Finally, one of the major reasons for which recursive TD algorithms are more popular than FD ones comes from some additional computation complexity, which has prevented from developing a fully recursive formulation of the FD approach, unlike the TD case. This paper improves the state-of-the-art existing approaches and proposes a recursive procedure from start to finish, permitting the practical implementation constraints to be satisfied, and the algorithm to be possibly embedded in aircraft computers. Other minor issues in the FT expressions of the signal derivatives are also properly settled, which are often disregarded for sequential processing thanks to the detrending achieved prior to FT computations, but which cannot as far as recursive algorithms are concerned. To sum up, the point is that FD approaches can also be simplified in order to reduce their computational complexity, similarly to what is done for TD formulations, and they can also include similar advanced mechanisms like data forgetting [START_REF] Hardier | An extended U-D algorithm with multiple forgetting factors for RLS estimation of model parameters[END_REF][START_REF] Ljung | Theory and practice of recursive identification[END_REF]. Before developing the proposed algorithm in §3.2, section 3.1 recalls the basics of the usual sequential approach in state-space form.

Standard formulation of a sequential algorithm

The transition to the FD is classically realized by means of the standard FT. As the signals are only available over a limited period of time ] , 0

[ T , the finite FT is used instead, which permits the FT of any signal

) (t x , time derivative ) (t
x & , or constant value to be easily computed in the FD (see [START_REF] Hardier | A recursive estimation algorithm to track aircraft model parameters[END_REF] for details). Practically, from the sampled values of ) (t

x , the finite FT can be computed via numerical integration:

) , ( ) ( ) ( 1 0 T X t e t n x t X N n t n j ω ω ω ∆ = ∆ ∆ ≈ ∑ - = ∆ - (5)
using N values equally spaced over ] , 0

[ T , with a sampling period t ∆ (see for instance [START_REF] Klein | Aircraft system identification: theory and practice[END_REF]). The term

) ( ~ω X represents the Discrete Fourier Transform (DFT) of the samples { 1 , , 0 ), ( - = ∆ N n t n x K
}. Efficient tools are available for computing this quantity, namely the Fast Fourier Transform (FFT) and the Chirp ztransform (CZT). As the aircraft model we are interested in is expressed in state-space form (see §2), the stability and control derivatives Θ to be estimated are included in the usual state matrices

D C B A , , ,
; they are assumed to be constant or at least to vary slowly during the flight with respect to (w.r.t.) the estimation process and updating rate. However, when applying estimation techniques, it is generally advisable to estimate also state/output biases in order to cope with I/O measurement offsets or model structure uncertainties [START_REF] Jategaonkar | Flight vehicle system identification: a time domain methodology[END_REF]. By applying the FT to these state-space equations, if ω is chosen to be a multiple of the frequency step ω ∆ (which happens for instance when using the FFT), the FD model can be written as [START_REF] Hardier | On-line parameter identification for in-flight aircraft monitoring[END_REF]:

   + + = - + + + = ) ( ) ( ) 0 ( ) ( ω ω ω d T b U D CX Y T x x d T b U B AX X j y x (6) by denoting ) ( ), ( ), ( ω ω ω Y Y U U X X ≡ ≡ ≡
, and where

) ( = ω d for 0 / 2 ≠ π = ω
T k [START_REF] Hardier | A recursive estimation algorithm to track aircraft model parameters[END_REF]. As a result, the discrepancy

) ( ) 0 ( T x x - = δ
between the state initial and final conditions is translated into a bias that impacts on all frequencies: localized effects in the TD are translated into broadband effects in the FD, and vice versa. A contrario, the biases act as broadband inputs in the TD and just modify the zero frequency. To get the most out of these specificities, it is generally worthwhile to discard the zero frequency during the estimation, which avoids the state and output biases having to be estimated. Thus, ( 6) is further simplified and reduces to:

   + = - + + = DU CX Y T x x BU AX X j ) ( ) 0 ( ω (7)
Hence, the discrepancies δ should be estimated in addition to the other parameters Θ , if not zero. However, this issue has not been paid much attention and is often ignored in practice [START_REF] Hardier | On-line parameter identification for in-flight aircraft monitoring[END_REF][START_REF] Larsson | Sequential aerodynamic model parameter identification[END_REF], which could be only justified in a pinch if a suitable preprocessing is inserted in a sequential algorithm to filter out these effects. If we assume that all the state components are measured, an Equation Error approach can be used, that consists in minimizing a set of cost functions [START_REF] Morelli | Real-time parameter estimation in the frequency domain[END_REF][START_REF] Morelli | Practical aspects of the EE method for aircraft parameter estimation[END_REF][START_REF] Morelli | Real-time dynamic modeling -Data information requirements and flight test results[END_REF] for all the state and output equations in which a subset i Θ of stability and control derivatives needs to be estimated (

) dim( ) dim( 1 Y X i + ≤ ≤
), the costs summing on M frequency samples available from the FT ( N M ≤ ). By collecting these M terms in vector and matrix forms, and by denoting i Θ the extended vector of unknown parameters in δ , , B A ( D C, resp.), the cost functions lead to classical LS criteria [START_REF] Morelli | Real-time dynamic modeling -Data information requirements and flight test results[END_REF]:

[ ] [ ] [ ]      = = Θ - Θ Σ Θ - Θ + Θ - Θ - = Θ - .) resp ) ( ) ( (or ) ( ) ( 1 1 ); ( ) ( ); ( ) ( : with )
( ) ( ) ( ) ( ) ( 1 1 1 1 1 0 1 0 0 M M M M M i i i i T T i i T i i i i i i y y x j x j U U X X † J ω ω ω ω ω ω ω ω ω ω K K K K K Y X X Y X Y (8)
where † is the complex conjugate transpose operator. In [START_REF] Ducard | A reconfigurable flight control system based on the EMMAE method[END_REF], the conditioning of the resulting optimization has been possibly improved by introducing some a priori knowledge about the expected value of the parameters. This form of regularization can be especially useful for weakly identifiable parameters, by softening their variations and improving the convergence. The principle of this Bayes-like estimation consists in adding a penalty term to the LS criterion to weight the i Θ increments with respect to prior values 0 i Θ through a corresponding covariance 0 Σ (usually a diagonal matrix). ( 8) is nothing but a standard LS regression with complex data instead of real ones. The well-known solution [START_REF] Klein | Aircraft system identification: theory and practice[END_REF][START_REF] Morelli | Real-time dynamic modeling -Data information requirements and flight test results[END_REF] to this problem is then computed as

] [ ] [ 0 1 0 1 0 ) ( ) ( ˆ1 i i † † Θ Σ + Σ + = Θ - -- Y X X X R R
, where () R is the real part of a complex value.

Towards a fully recursive algorithm

The previous LS formulation is well known, but was mostly implemented in sequential estimation processes, ignoring the issues of the extra discrepancies δ as well as the need to invert the information matrix. Here, we are in a different prospect and wish to consider the limitations of on-board computers, as well as the certification and code verification issues. A practical implementation on civil aircraft should preclude the use of any iterative algorithm which would result in some unbounded computation loop through the data. Hence complex operations like non analytical matrix inversions should be banned. That is why we look for a simple enough approach, likely to satisfy those real time constraints, and based on a fully recursive algorithm. At first, it should be noticed that the summation (5) makes very simple operations possible (see [START_REF] Morelli | Real-time parameter estimation in the frequency domain[END_REF] for instance): [START_REF] Fan | Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics[END_REF] to evaluate the DFT at time t n∆ from its previous value at time

     = ∆ + ω = ω ∆ ω - ∆ ω - ∆ ω - ∆ ω - - t ) - n ( j t j t n j t n j n n e e e e t n x ) X ) X 1 1 ) ( ( ( ~
t n ∆ -) 1 (
. For a given frequency ω , the quantity

) exp( t j ∆ -ω
is constant, so that the updating of the DFT just requires two multiplications and one addition, resulting in a very low computational effort. As the estimation process also involves state derivatives, the measurements of which are usually not available, a recursive formulation similar to (9) can be worthwhile for signals

) (t x &
to avoid a pseudo-derivation from the available signal

) (t
x , or the extra δ parameters to be estimated if we had to use [START_REF] Chandler | System identification for adaptive and reconfigurable control[END_REF]. The idea is to derive LS costs similar to (8) but involving

) ( k i x ω &
directly in the computation of the error terms as this is the case for a measurement ) ( k i y ω . This idea is new and differs from the two usual ways consisting either in computing an estimate of

) (t x & in the TD, or in replacing ) (ω X & by ) (ω ω X j in the FD, which is incorrect whenever 0 ≠ δ
. Hence, the proposed FD algorithm permits extra parameters, like δ , to be eliminated from the Θ vector.

Otherwise, when using recursive updates, older information can be overweighted regarding to recent ones, which can result in much delay in the adaptation process. That is why data forgetting can be used in conjunction with [START_REF] Fan | Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics[END_REF], to remove the effect of oldest data by working on a limited time window [START_REF] Morelli | Real-time dynamic modeling -Data information requirements and flight test results[END_REF]. In that case, the choice of the window width results from a trade-off between the amount of information available from the data and the sensitivity to A/C parameter variations. This remark also applies to a pure sequential approach, where a non recursive form of FT is applied to the data [START_REF] Hardier | On-line parameter identification for in-flight aircraft monitoring[END_REF], resulting in a succession of piecewise constant values, updated only after some seconds and hence delayed with respect to the varying parameters. To develop a recursive algorithm and to avoid the storage of a time history of the past data, it is still possible to benefit from the FT linearity w.r.t. the signals to implement a forgetting process similar to the one of TD exponential forgetting [START_REF] Hardier | An extended U-D algorithm with multiple forgetting factors for RLS estimation of model parameters[END_REF]; this simply involves multiplying the previous FT by a forgetting factor λ ) 1 ( < λ [START_REF] Morelli | Real-time dynamic modeling -Data information requirements and flight test results[END_REF]. Combining all these ideas with (9) yields the following recursive expressions for any signal [START_REF] Hardier | A recursive estimation algorithm to track aircraft model parameters[END_REF] for details):

) (t x , denoting ) ( ), ( ), ( t n x x X X X X n n n n n ∆ ≡ ≡ ≡ ω ω & & and t j n n e x x ∆ - * = ω (see
     - + ∆ + ≈ ∆ + ≈ ∆ - - * - ∆ - * - t 1) - n ( j n n n n t 1) - n ( j n n n e x x t j X X e x t X X ω ω ω λ λ ] ) 1 [( 1 1 1 & & (10)
We are now going to address a key feature of the proposed estimation algorithm. With TD methods, the arrival of a new sample only results in the addition of a new row in the regression matrix [START_REF] Hardier | An extended U-D algorithm with multiple forgetting factors for RLS estimation of model parameters[END_REF]. So, the information matrix (or its inverse) can be updated via simple (and light) computations thanks to the Woodbury formula. With FD methods, a new sample modifies all the M rows of the matrix X through the FT updates of [START_REF] Ferreres | A practical approach to robustness analysis with aeronautical applications[END_REF]. That is why the usual algorithms generally rely on a sequential estimation of the parameters, due to the lack of a recursive update of the matrix

) ( X X R † R =
or of its inverse. However, it was established in [START_REF] Hardier | A recursive estimation algorithm to track aircraft model parameters[END_REF] that a recursive version is achievable anyway, almost similar to the TD ones, thanks to the 3 following lemma. Lemma 3.1 In case of

) dim( i m Θ =
regressors, the basis of a time-recursion for the matrix

m M × ∈R X can be set as: † t † † n n n n E x X X ∆ + = -1 λ (11) for 1 m R x × ∈
defined as

T n t n u t n x t n ] ) ( ) ( [ ) ( ∆ ∆ = ≡ ∆ x x and ] [ 1 t n j t n j n M e e † ∆ ω ∆ ω = L E . Lemma 3.2 As a result, the information matrix n n n † M X X
= can be broken down into 3 parts as:

∑ = - - - - + = + + ∆ ∆ + = 3 1 1 2 1 1 1 2 ] [ i i i i n T n n n n n n T n n n n D C B M † † t M t M M λ λ λ λ x E E x x x X X (12)
where . 1 and , , , , , ,

2 3 2 3 1 2 1 2 1 1 1 = = = = = = ∆ = - i T n n T n C B D † D B † D B M B B D t M B X E x λ Lemma 3.3 Hence, 1 - n M
can be computed recursively by applying the matrix inversion lemma 3 times to [START_REF] Ferreres | Adaptive control using robustness and gain-scheduling[END_REF], as (

1 = ∀k to 3): ) ( 1 1 1 1 1 1 ] [ - - - - - + - = + = k k k k k k k k k k A D B s I A D C B A A with 1 2 1 - = n M A λ , n M A = 4 , 1 1 + = - k k k k B A D s (13)
Finally, (13) permits the direct updating of 1 n M to be computed without requiring a tricky matrix inversion, by using only a series of matrix additions and products. For the sake of simplicity, 1 0 -Σ was assumed to be 0 in the previous expression of n M , but dealing with a priori knowledge is however possible by achieving at first a series of m rank 1 corrections to 1 1 -A with the diagonal values of 1 0 -Σ . See [START_REF] Hardier | A recursive estimation algorithm to track aircraft model parameters[END_REF] for the proofs and for a comparison of the computational cost of this recursive algorithm w.r.t. an iterative blockwise inversion.

GAIN-SCHEDULING AND ROBUSTNESS TOOLS

A modal design technique of a static output feedback controller under an LFT form is presented in the first subsection. The second one presents a robustness analysis technique in the presence of time invariant or time varying parameters. In both cases, a preliminary step is to put the open loop plant model

) , ( θ s G
, where θ is a vector of parameters, under the standard LFT form

u s H F u s G y l ] ), ( [ ) , ( ∆ = = θ , i.e.:       =       w u s H z y ) ( (14)
) ( i q i I diag θ = ∆
gathers the parametric uncertainties, i.e. each parameter i θ is repeated i q times on the diagonal of ∆ . The fictitious feedback ( 15) is used to introduce the parametric uncertainties in the model. Generally speaking, using the LFR Toolbox [START_REF] Magni | Linear Fractional Representation Toolbox[END_REF] for instance, any state-space representation with a polynomial or rational dependence on uncertain or varying parameters can be put under this form. Parameters i θ are supposed to be normalized in this section, namely the vector θ is supposed to belong to the unit hypercube H (i.e.

1 1 + ≤ ≤ - i θ
for all i ). Associated parameters, before normalization, typically belong to an hyper-rectangle.

Modal design of an LFT gain

The first issue is to design a gain-scheduled LFT controller, i.e. a static LFT gain , using the technique of [START_REF] Magni | An LFT approach to robust gain scheduling[END_REF]. All θ -dependent quantities in the following Lemma are supposed to be under an LFT form. Lemma 4.1 [START_REF] Magni | An LFT approach to robust gain scheduling[END_REF] Let ( [ ]

) ( ) ( ) ( ) ( ) ( ) ( θ θ θ θ θ θ W W V D C K =       (16) where ] ) ( ) ( [ ) ( 1 θ θ θ m v v V K = and ] ) ( ) ( [ ) ( 1 θ θ θ m w w W K = . Each pair of vectors ) (θ i v and ) (θ i w must satisfy: [ ] 0 ) ( ) ( ) ( ) ( ) ( =       - θ θ θ θ λ θ i i i w v B I A (17)
Then, the interconnection of are computed so as to satisfy [START_REF] Hanlon | Multiple-model adaptive estimation using a residual correlation KF bank[END_REF]. The feedback gain ) (θ K is computed using [START_REF] Han | Frequency and time domain online parameter estimation for reconfigurable flight control system[END_REF].

Remark 1: Equalities (16,17) must be exactly satisfied, so that the number of poles to be placed must be less or equal to the number of plant outputs. Thus, only the dominant plant poles are placed in practice, and the position of the closed loop secondary poles is checked a posteriori.

Remark 2: this modal method not only places the closed loop eigenvalues (i.e. the closed loop poles), but also the associated eigenvectors, i.e. this is an eigenstructure assignment method. The robustness of the controller strongly depends on the choice of the placed eigenstructure. Robust controllers can be expected when respecting as much as possible the open loop dynamics, for instance by projecting the open loop eigenvectors, i.e. by choosing closed loop eigenvectors which are as close as possible to the open loop ones.

In the same way, in §5.2, the closed loop frequency of the short period mode, which is the main mode of the aerodynamic model, will be chosen to be the same as the open loop one, just the damping ratio will be improved. This enables to design flight controllers which reveal (very) robust to parametric uncertainties.

Using the routine fb_sched.m of the LFR Toolbox [START_REF] Magni | Linear Fractional Representation Toolbox[END_REF], the technique provides a static feedback gain:

21 1 22 12 11 ) ( ) , ( ) ( K K I K K F K l - ∆ - ∆ + = ∆ = K θ (18)
where

) diag( i k i I θ = ∆ and       = 22 21 12 11 K K K K K .
After the design, a key point is to check the well-posedness radius of the LFT gain, i.e. the maximal size k for which the matrix

∆ -22 K I is invertible for all H k ∈ θ (i.e. k k i + ≤ ≤ - θ
for all i ). Using standard Matlab routines, a guaranteed value of this radius, i.e. a lower bound, can easily be obtained as the inverse of a µ upper bound of 22 K , where µ is the structured singular value [START_REF] Fan | Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics[END_REF]. Since the vector θ of plant parameters is assumed to belong to the unit hypercube H , the well-posedness radius must be greater than 1. . Lemma 4.2 [START_REF] Ferreres | A practical approach to robustness analysis with aeronautical applications[END_REF] The issue is to compute a guaranteed robust performance level γ , for which: A sufficient condition for [START_REF] Hardier | On-line parameter identification for in-flight aircraft monitoring[END_REF] to hold is that there exist scaling matrices D and G satisfying ω ∀ : are handled, and inequality (20) can be independently solved at each frequency. This is a skew µ analysis problem [START_REF] Ferreres | A practical approach to robustness analysis with aeronautical applications[END_REF] for which efficient computational tools exist, and especially the routine mu_margin.m of the System Modeling, Analysis and Control (SMAC) Toolbox developed by Onera in Matlab-Simulink (available at http://w3.onera.fr/smac). Conversely, when parameters i θ are supposed to be time varying, without bound on the rate of variation, real frequency-independent scaling matrices D and G are handled, and inequality (20) must be simultaneously solved on the frequency continuum using IQC tools, see e.g. [START_REF] Ferreres | Robust feedforward design in the presence of LTI/LTV uncertainties[END_REF][START_REF] Kao | A Matlab toolbox for robustness analysis[END_REF] and the SMAC Toolbox. Last note that a bound on the rate of variation of the parameters can be introduced, using a generalization of the D,G scalings, in the framework of IQC analysis [START_REF] Megretski | System analysis via Integral Quadratic Constraints[END_REF].

Robustness analysis

H || s M F || 2 L i l ∈ ∀ ≤ ∆ θ γ ] ), ( [ (19)
D G j P j P G j j P D j P ≤ - + * * ] ) ( ) ( [ ) ( ) ( ω ω ω ω (

DESIGN AND VALIDATION OF A GAIN-SCHEDULED FLIGHT CONTROLLER

To reconfigure the inner control loop of the A/C in faulty or unexpected situations, a gain-scheduled controller depending on the control and stability derivatives is designed in §5.1 and §5.2, and validated on the nonlinear simulator in §5.3. Its robustness to an estimation error is checked in §5.4. In §5.5, a robust controller is deduced from the gain scheduled one, by introducing mean or worst-case values of the scheduling parameters.

State space design model

For the RECONFIGURE project, a set of 214 longitudinal linearized models has been provided by Airbus, covering the flight domain with respect to speed, altitude, mass and center of gravity position variations. This set is used to design and to validate the gain-scheduled controller. Owing to the range of variation of the stability derivatives (

K , , q Z Z α
) over the set of 214 models, the aerodynamic model (3) can be simplified by considering only the main stability derivatives:

     = + + = + = α δ α α α α δ α α N N M q M M q q Z z m q & & (21) i.e. 1 ≈ q Z while 0 , 0 ≈ ≈ q N Z δ and 0 ≈ δ N
. Indeed, in practice, when validating the flight controller, designed using model [START_REF] Jategaonkar | Flight vehicle system identification: a time domain methodology[END_REF], on the full 2 state aerodynamic model (3), closed loop performance appears essentially to be the same using the simplified model ( 21) and the full model [START_REF] Barkana | Classical and simple adaptive control for non-minimum phase autopilot design[END_REF]. By normalizing δ M and α N to unity, the design model can be further simplified to become:

     = + + = + = α α α α α α z u q M M q q Z q & & (22)
After designing the gain-scheduled controller:

T q q zdt z Z M M K u ] )[ , , ( ∫ = α α (23)
The true feedback controller will be:

          - -           = ∫ q dt N N N N N N M Z M M K c c z z z z q m ) ( 1 0 0 0 1 0 0 0 1 ) , , ( α α δ α α δ (24)
More precisely, when considering the architecture (4) of the flight controller, the feedback gains Nz K , q K , I K can be deduced from [START_REF] Kerr | RECONFIGURE FP7 project: preliminary results and contributions[END_REF]. The feedforward term L will be retuned in §5.3.

Controller design

Following [START_REF] Magni | An LFT approach to robust gain scheduling[END_REF] an open loop LFT model under the form ( 22) is built using the LFR Toolbox [START_REF] Magni | Linear Fractional Representation Toolbox[END_REF], with input u , outputs z and q , states α and q . The parameters of the LFT model are q M M Z , , α α , whose ranges of variation correspond to the set of 214 linearized models. Actuator and sensor models are added to this LFT model, as well as a 2 nd order Padé model (used to represent a time-delay at the plant input) and an integrator on the z output, so that the LFT design model has 3 outputs

∫ zdt q z , ,
. As a result, the 3 main closed loop poles can be placed by the LFT feedback controller ) , , (

α α Z M M K q , namely:
• The integrator pole at a specified location λ , the same for all models, i.e. for all values of α α Z M M q , , .

• The short period (SP) mode as a complex mode with damping ratio 0.8 and frequency ω , where ω is chosen as the frequency of the open loop SP mode, i.e. the poles of the 2 states linearized model [START_REF] Jiang | A revisit to block and recursive least squares for parameter estimation[END_REF]. These closed loop poles are chosen to satisfy classical design specifications on the step response to a reference acceleration c z N , namely maximal settling time on z N and maximal overshoots on z N (10%) and q (30%). The criteria chosen for evaluation rely on good tracking performance of the acceleration with homogeneous responses.

Once the design LFT model is computed, depending on the free parameters

α α Z M M q , ,
, and after choosing the closed loop poles with an additional free parameter ω , the LFT feedback controller q , , will be used, not only as scheduling parameters, but also to compute ω as the SP mode frequency of the estimated 2 states model [START_REF] Jiang | A revisit to block and recursive least squares for parameter estimation[END_REF]. A minimal value is introduced for ω , to speed up the closed loop SP mode when it is too slow with respect to the closed loop integrator pole. This LFT controller, modified following [START_REF] Kerr | RECONFIGURE FP7 project: preliminary results and contributions[END_REF], is validated over the 214 linearized models under the form (3). Although the value of the LFT controller at a flight point is computed using the true values of the stability derivatives, small changes of the closed loop integrator and SP modes are expected, since the secondary stability derivatives are no more neglected. Moreover, the closed loop actuator, sensor and Padé model poles need to be checked since they were not explicitly placed in the design procedure. Fig. 2 presents the closed loop poles for all 214 models, when zooming on the integrator pole and SP mode. The result is quite satisfactory, as well as the step responses in Fig. 3. Note that the q output is normalized by its steady state value at each flight point.

Nonlinear validation

The LFT controller is validated on the nonlinear simulator for the 214 flight points. The testing maneuver usually selected for industrial validation involves joystick doublet-type signals. Both the autopilot and the auto-throttle are switched off. Time domain responses differ from those obtained with the linearized models on Fig. 3, primarily because of the additional slow longitudinal states

z V , ,θ
in the nonlinear aerodynamic model, so that the feedforward term L of the flight controller (4) needs to be retuned.

Remember I K is the feedback gain corresponding to the integrator on the z N output. The linearized responses of Fig. 3 were obtained with the classical choice

I K L - =
. Nevertheless, to obtain a satisfactory performance on the nonlinear simulator,

I K CoG mass A L ) , ( - =
needs to be chosen, where the multiplicative factor A only depends on the mass and center of gravity position to ease the implementation. Choosing A results from a trade-off between the fastness of the closed loop and its overshoots: increasing A increases the fastness, at the price of higher overshoots on outputs z N and q . Nonlinear responses, which are satisfactory, are displayed in red color in Fig. 4 for a given value of mass and center of gravity: this "C1 case" includes 25 flight points corresponding to variations of airspeed and altitude. The airplane needs to be trimmed, before applying the first filtered step on the reference acceleration input (in blue color).

Linear validation via µ and IQC analysis

The issue is to check the robustness of the LFT controller in the presence of estimation errors on the 5 main derivatives

α δ α α N M M M Z q , , , ,
at each of the 214 available flight points. To this aim, an error is introduced in each stability derivative at a given flight point, e.g.:

) 1 . 0 1 ( 1 0 δ α α + = Z Z (25)
where 0 α Z is the value of α Z at this flight point. When the normalized uncertainty 1 δ varies between -1 and 1, the corresponding variation of α Z is % 10 ±

. When introducing an uncertainty on each main stability derivative in the LFT controller, a new LFT controller is deduced using the LFR Toolbox [START_REF] Magni | Linear Fractional Representation Toolbox[END_REF], which depends on

] 5 , 1 [ ) ( ∈ i i δ
. More precisely, in the structured model perturbation, the normalized estimation errors on q M M Z , , α α are repeated 3 times on the diagonal, 1 time for δ M , and 2 times for α N . Two approximations are made to get this rather reasonable complexity: the frequency ω in the initial LFT controller is fixed to its nominal value, i.e. it is computed using the true values of the stability derivatives at the flight point; in the feedforward term

I K A L - =
, I K is also fixed to its nominal value. A time invariant estimation error is supposed at first. µ analysis gives a range of variation between 34.2% and 49.7% over the set of 214 flight points for the LTI (Linear Time Invariant) robust stability margin in the Left Half Plane, with an average value of 38.8%. For instance, a margin of 38.8% means that closed loop stability is guaranteed despite a simultaneous uncertainty of ±38.8% on all five main stability derivatives, which appears very satisfactory. Note that the result does not depend on the weight of 0.1, which was just introduced to normalize the uncertainty in [START_REF] Kim | A robust adaptive nonlinear control approach to missile autopilot design[END_REF].

The results obtained with a time-varying estimation error reveal not so different with a range of variation between 23.0% and 35.8% for the LTV (Linear Time Varying) robustness margin (average value of 31.5%). These results are obtained with IQC analysis and constant scaling, i.e. without bounding the variation rate of the uncertainties. If the gap between the LTI and LTV robustness margins was larger, it would be worth introducing a bounded rate of variation for the estimation error, to explore the intermediate cases, but this appears useless here.

The robust performance margins are computed in the same way, leading to a range of variation between 5.8% and 21.0% for the LTI margin (average value of 14.2%), and a range between 4.9% and 18.2% for the LTV margin (average value of 12.3%). Again, the gap between LTI and LTV robustness margins is reasonable and the results are satisfactory, when considering that performance is much more constraining than stability.

Design of a robust controller

To apply the proposed adaptive strategy, a robust controller is needed which only depends on mass and center of gravity position, i.e. it should not depend on airspeed, altitude nor on stability derivatives. For the sake of simplicity, only the C1 case is considered here, corresponding to 25 flight points. This robust controller can be deduced from the gain-scheduled one of §5.1 and §5.2 by introducing mean or worst-case values of the 5 scheduling parameters.

More precisely, 3 possible values are considered for each of the 5 parameters, namely the minimum, mean or maximum values of the derivative over the 25 linearized aerodynamic models: 3 5 =243 combinations are involved. The criterion is the minimal value of the degree of stability over the 25 linearized closed loops. Remember that the degree of stability is defined as the opposite of the maximal value of the real part of the closed loop poles. The maximal value of the criterion over the 243 combinations is selected, leading to choose the mean value for α M , the minimum value for α N , and the maximum values for

α δ Z M M q , ,
. Fig. 6 presents the closed loop step responses for the gain-scheduled and robust controllers. As expected, the performance of the latter is degraded, see the overshoots on z N and q . Nevertheless, this controller ensures minimal performance properties.

Remark: in the same way, it would be necessary to design a robust fixed controller for each value of mass and center of gravity position, so that when considering all the 214 available flight points, the back up controller would be scheduled/interpolated as a function of mass and center of gravity position. The back up controller was chosen not to depend on airspeed, because the airspeed measurement, which may be missing due to a sensor fault, is less reliable than the measurement/estimation of mass and center of gravity position. Conversely, we have chosen not to design a robust fixed controller over all 214 available flight points because its robust performance properties would be too bad: when designing the robust fixed controller, it is worth reducing the range of variation of the model parameters as much as possible, to obtain robust performance properties which are as satisfactory as possible.

PRACTICAL IMPLEMENTATION ASPECTS

To be possibly implemented onboard, the frequency domain estimator and the gain-scheduled controller should comply with the stringent constraints resulting from the present industrial coding process applicable to Airbus aircraft. Accordingly, the algorithms have been firstly encoded in full Simulink ® , i.e. without any call to Mex nor Matlab ® functions. In a second step, The Simulink ® schemes were encoded by using the existing symbols of the Flight Control Computer (FCC) library provided by Airbus to be compliant with the SCADE process (Safety Critical Application Development Environment). After encoding the algorithm with FCC operators, the resulting computational burden evaluated by Airbus during the RECONFIGURE project is about 1.5ms, which is quite acceptable and could be further improved by updating the estimates at a lower rate. As this library only includes basic operators (logical and mathematical type), the second step has required higher level modules to be developed for coding more complex functionalities, e.g. required by matrix calculus. This was especially true for the FD estimation algorithm, which involves matrix operations with complex numbers.

As regards the gain-scheduled controller, it is noteworthy that the usual way of computing (off-line) the value of the LFT controller is not suitable for on board implementations, as the computational complexity of this procedure is generally high, and would especially require a matrix inversion of dimension 27 in the present case. To bypass this difficulty several alternatives do exist [START_REF] Magni | Practical implementation of LFT-scheduled feedback gains[END_REF], for instance by resorting to a recursive method which avoids this inversion by using a fixed point technique (possibly with a relaxed sampling rate). However, the complexity of the LFT controller designed in §5.2 is rather high: even if each control or stability derivative is repeated 3 times, which is reasonable, the frequency of the SP mode is repeated 16 times, so that the on-line computational burden of the technique proposed by [START_REF] Magni | Practical implementation of LFT-scheduled feedback gains[END_REF] would be too high.

That is why we have preferred another option, consisting in developing off-line a surrogate model of the LFT gains, which could be used afterwards for on-line computations, and which could also be easily coded in terms of the basic FCC library operators by using only basic scalar operations available for onboard computers. The characteristics of these surrogate models and the principles of the techniques permitting rational-type representations to be obtained are detailed in [START_REF] Seren | On-line Estimation of Longitudinal Flight Parameters[END_REF]. To build this surrogate model, the tool named koala was used, which is part of the APRICOT library (Approximation of Polynomial and Rationaltype for Indeterminate Coefficients via Optimization Tools), included into the SMAC Toolbox (available at http://w3.onera.fr/smac/). The tools included in APRICOT permit sparse rational models to be generated, which lead to simple yet accurate Linear Fractional Representations. The computational complexity of the (rational) modeling is thus consistent with the FCC library coding requirements, as it only involves a set of scalar additions and multiplications.

In the present case, the 3 control gains

I q Nz K K K , ,
need to be modeled in terms of the 3 explanatory variables

α α Z M M q , ,
, the SP frequency being directly inferred from the open loop eigenvalue of the simplified state-space model [START_REF] Jiang | A revisit to block and recursive least squares for parameter estimation[END_REF]. To get a set of reference data from which the approximated rational function can be derived, a 3D gridding is firstly achieved by computing a set of LFT exact values from regularly spaced samples in the input space

α α × × Z M M q . Overestimated bounds of the aerodynamic coefficients ]} , [ ] , [ ] , [ { max min max min max min α α α α α α Z Z Z M M M M M M q q q ∈ × ∈ × ∈
were selected to define the expected cubic domain. Fig. 7 illustrates the result of this process for the 3D coefficient ) , , (

α α Z M M K q q
. The surrogate model used to approximate the 3 control gains simultaneously is computationally efficient and achieves a very low acceptable error in the whole selected domain.

TIME-DOMAIN RESULTS OF THE RECONFIGURE BENCHMARK

Adaptive control strategy along a trajectory

The issue is to deal with the time-varying aspect of the parameters to be estimated, despite an unsatisfactory closed loop excitation. This concerns the estimator but also the gain-scheduled controller, since the point is also to decide when the values of the scheduling parameters should be updated, knowing that a measure of their reliability is available as well. At the start, when controlling the A/C at a trim point, "robust" values of the stability derivatives are chosen as scheduling parameters, i.e. those corresponding to the robust controller of §5.5, to ensure minimal stability and performance closed loop properties. As early as the estimator provides (reliable) estimates, these are injected in the gain-scheduled controller. When moving from a flight point to another, time-varying parameters can be tracked either by benefiting from the forgetting factors, whose values may be adjusted along the trajectory, or by resetting the estimator whenever it is necessary [START_REF] Hardier | A recursive estimation algorithm to track aircraft model parameters[END_REF]. Estimates may also be frozen if the I/O data content is too poor and hence the estimation errors too high.

Regarding the adaptive control strategy, a first solution is to apply the Certainty Equivalence Principle (CEP), i.e. the estimates are continuously introduced in the gain-scheduled controller whatever their reliability is. Another solution is to switch to the robust controller whenever the estimates are not reliable enough. Now, the CEP is bypassed and the adjustment of the scheduling parameters is discontinuous, using a low pass filter to avoid jumps in the scheduling values. This 2 nd solution permits minimal stability and performance properties to be guaranteed for the adaptive closed loop, a crucial point if the flight controller was to be certified.

Conversely, the transient response of the adaptive closed loop may be shorter with the 1 st solution, as it may be useless to wait for fully reliable estimates thanks to the robustness of the controller with respect to estimation errors: a proper time domain response may be obtained despite less reliable estimates. The two solutions are illustrated in the following, and essentially produce the same type of results. Indeed, estimates are typically unreliable when closed loop excitation is poor, so that the difference between the closed loop responses of the robust and gain-scheduled controllers is not significant.

Flight scenario

To evaluate the proposed adaptive scheme in the framework of RECONFIGURE, the simulated scenario is a 380 seconds realistic flight profile including 4 successive stages: The autopilot is used to regulate the altitude z and conventional airspeed c V at their reference values. See Fig. 8 for the time history of altitude and true airspeed (which differs from the conventional airspeed).

The testing maneuvers selected by Airbus for the industrial validation of the controllers involve joystick doublet-type signals. So, to check the performance of the adaptive flight controller, these usual stick input signals are applied to produce a reference input signal c z N during the 1 st and 4 th parts of the flight, with autopilot and auto-throttle switched off (which explains on Fig. 8 why the altitude is no more regulated at the beginning and at the end of the flight). See Fig. 9 for the time plot of the acceleration (more precisely the load factor in g) z N , the rotational rate q and the elevator (i.e. the A/C) input m δ . The mass and center of gravity correspond to the C1 case.

Adaptive control results

The performance of the adaptive controller can essentially be seen on the plots of the load factors in in both solutions, because the parameter estimates are unreliable or unavailable inside these time intervals). Regardless of the flight controller, the outer loop responses of the autopilot are satisfactory, see the plots of true airspeed and altitude in Fig. 8. Due to the poor excitation, it is difficult to judge the quality of the inner loop responses on Fig. 9, but they seem suitable. Fig. 10 focuses on the steady flights at the beginning and at the end of the simulation. In the upper plot, the 2 solutions (black/red lines) match exactly because the robust controller is used at the beginning in both cases: the estimator is initialized so that some time is needed before an estimate is available. In the bottom plot, the transient response of the 1 st CEP solution (in black) appears better, noting that the overshoot and the settling time for the 2 nd solution (in red) are those of the robust controller for 343 < t : once reliable estimates are available, the overshoot/settling time become satisfactory. Hence, the 2 nd solution guarantees minimal stability and performance properties for the adaptive closed loop at the price of a potentially less proper transient response.

Results of the recursive estimation algorithm

The reader is referred to [START_REF] Hardier | A recursive estimation algorithm to track aircraft model parameters[END_REF] for a comprehensive description of the estimation results. Fig. 11 briefly illustrates the behavior of the recursive FD scheme (tuned with 43 = M frequency samples equally spaced in the range 0.1-0.52 Hz), through the δ M estimate got during the simulation. The parameter uncertainty (3σ) is plotted in blue dashed lines and results in freezing the estimate (solid blue line) whenever it is too high. The green curve gives a rough idea of the reference value drawn from the closest LTI model (just one of 25 for the C1 case). Due to a loose mesh of the linearized models in the flight domain, these values are only plotted to give a trend (the true linearized values being unavailable). Owing to the simulated scenario, the variations that are tracked result either from a change in the flight condition or, at the start of the run, from a misknowledge of the initial parameter values. Both of them demonstrate the interest of the adaptation to cope with modeling uncertainties. It is worth noting that the continuous variation of V z / is monitored by the estimation algorithm during level changes, and that the parameters accuracy can also be used to trigger a freeze of the estimates whenever the uncertainty becomes too high (not enough excitation or rapid changes in the flight condition). In addition, the forgetting mechanism of ( 10) is only triggered in Fig. 11 when significant changes are detected in V z / , providing a way to adapt more rapidly to varying conditions.

It should also be pointed out that with such a poor excitation (during the 2 nd /3 rd parts of the scenario), some parameters can't be estimated properly. Hence to prevent from correlation issues or ill-conditioning, a tuned-down modeling or some regularization is required. Here, a priori knowledge via the Bayesian formulation of (8) is introduced; hence, the trends indicated by the (green) reference values cannot be instantly mimicked by the estimates, and instead these ones are drawn towards the prior valid estimates or towards average reference values used as a priori knowledge by the estimation algorithm (e.g., mean values for a given altitude). As soon as sufficient excitation is available, and the accuracy is good enough (some time is needed to store information through the recursive updates of the FT), the estimates are freed again and converge rapidly to suitable values. Thanks to the noise filtering outside the selected bandwidth, the FD algorithm does not require much time to accumulate enough information and to recover proper estimates. In case of continuous estimation (CEP), the convergence delay is even shorter as the estimates may be reliable enough although the errors are not fully satisfying.

CONCLUSION

The contribution of this paper is twofold. First, an adaptive scheme is presented, which deeply differs from classical indirect adaptive control schemes: these use a time domain (Least Squares or gradient) estimator, as well as a (SISO or square MIMO) transfer function to describe the plant. The controller is designed on-line. On the contrary, in our scheme, a non-square MIMO plant model can be used under a state-space form. Moreover, to minimize the computational time, one uses a frequency domain plant model estimator and a gain-scheduled controller, designed off-line.

As a second contribution, the practical interest of our scheme is illustrated by the design of an adaptive flight controller, with a computational time and a complexity that comply with the requirements of embedded A/C implementation. The indirect adaptive control strategy combines an efficient recursive frequency domain estimator with an LFT controller scheduled with respect to the parameters to be estimated. Robustness to transient and asymptotic estimation errors was proved using µ and IQC analysis techniques, and demonstrated via the industrial benchmark. Nonlinear simulations also show the capability to control the airplane along a trajectory, a more difficult problem than controlling about a steady flight point since the time-varying parameters have to be estimated despite poor closed loop excitation.

These results are very promising as regards a possible application to civil aircraft. The next step would be to further validate the adaptive flight control law. Such a validation seems very difficult using analytical methods, due to the complexity of the frequency domain model estimator and gain-scheduled LFT controller. For example, finding a Lyapunov function for the adaptive closed loop, as classically done for simpler adaptive control schemes, appears very unlikely. In this context, different complementary solutions have been or can be proposed:

1/ Checking the robustness of the LFT controller to an estimation error on the scheduling parameters ( §5.4): the obtained robustness margins appear especially relevant, even if the model estimator is not accounted for in the analysis. Stability and performance properties of the adaptive closed loop can be guaranteed despite a maximal allowable amount of the estimation error, provided by the estimator.

2/ Using the available measure of the confidence on the parameter estimates, to switch to a robust controller whenever the estimates are not reliable enough ( §7): this enables to guarantee minimal stability and performance properties for the adaptive closed loop, even when excitation is poor.

3/ Using extensive sets of nonlinear simulations to check the robust stability and performance properties of the adaptive flight controller: it would be worth exploring this perspective of an industrial validation, noting that a practical approach for the worst-case validation of a similar indirect adaptive control scheme was already investigated by Airbus and Onera in [START_REF] Oudin | Design and worst-case validation of a longitudinal adaptive flight control law: a practical approach[END_REF].

  is a Dirac function in the FD, such that 1

H

  , with additional I/O w and z , is fixed while

  of ∆ ).

20 )

 20 When parameters i θ are supposed to be time invariant, complex frequency-dependent scaling matrices

  using Lemma 4.1. When implementing the controller on-line, the estimated values of α α Z M M

At a given flight

  point, the LFT closed loop is obtained by connecting the new LFT controller with the linearized aerodynamic model (3), augmented with actuator, sensor and Padé models, and with an integrator on is obtained either for robust stability analysis in the Left Half Plane, or for robust performance analysis of the closed loop transfer function between c z N and the tracking error c z z N N -. A first order frequency domain template is applied to c z z N N -to ensure a minimal steady-state error (1%), a maximal frequency domain overshoot (6 dB), and a minimal closed loop bandwidth (0.5 rad/s): see the plot in Fig.5, where the template is in green color and the nominal (without uncertainty) transfer functions between c

Figs. 9

 9 and 10 ( z N vs c z N ). Fig. 10 focuses on the responses at the beginning and at the end of the flight. The blue line corresponds to the reference input c z N , the black and red ones to the z N signals. As soon as reliable estimates are available for scheduling, adaptation enables to recover a much better performance, i.e. satisfactory overshoot and settling time.

Fig. 9

 9 Fig.9presents the results along the whole trajectory. As expected, closed loop excitation is globally poor during the 2 nd /3 rd stages corresponding to altitude/airspeed changes: see the reference input signal c z N in blue color on the z N plot. As a result, during these 2 stages, either the robust controller, or the gainscheduled one are used depending on the reliability of the estimates and on the adaptive control strategy chosen (CEP or switch). The black curves of Figs. 9 and 10 correspond to the 1 st solution (CEP, i.e. the parameter estimates are used as scheduling parameters whatever their reliability), whereas the red ones correspond to the 2 nd solution (switch for ] 343 , 194 [ ∈ t to the robust controller, also active for ] 18 , 0 [ ∈ t

Figure 1 :Figure 2 :

 12 Figure 1: The FD estimation process

Figure 3 :Figure 4 :

 34 Figure 3: linear step responses at the 214 flight points

Figure 5 :

 5 Figure 5: Nominal frequency-domain closed loop responses for the C1 case

Figure 6 :Figure 7 :Figure 8 :Figure 9 :Figure 10 :Figure 11 :

 67891011 Figure 6: Closed loop step responses of the controllers for the C1 case (left = gain-scheduled, right = robust)

  , where ∆ contains normalized uncertain parameters i θ . The aim of the following Lemma is to compute a guaranteed value of the worst-case ∞

	Consider a closed loop LFT model	F l	[	M	(	s	),	∆	]
	F l	[	M	(	s	),	∆	]	, i.e. of its maximal value over	θ	∈	H	H norm / 2 L gain of

In the following, the variables α, q and so on of the linearized models should be understood as departures from trim variables, whereas they correspond to global variables when using the non-linear simulator.

& & &


Sandra Nabil

Raffaella Balzarini

Frederic Devernay

James Crowley

Designing objective quality metrics for panoramic videos based on human perception

Keywords: Panoramic Videos, Quality Assessment, Human Perception, Parallax Errors

The creation of high quality panoramic videos for immersive VR content is commonly done using a rig with multiple cameras covering the required scene. Unfortunately, this setup introduces both spatial and temporal artifacts due to the difference in optical centers as well as imperfect synchronization of cameras. Traditional image quality metrics are inadequate for describing geometric distortions in panoramic videos. In this paper, we propose an objective quality assessment approach for calculating these distortions based on optical flow coupled with an existing salience detection model. Our approach is validated with a humancentered study using error annotations and eye-tracking. Preliminary results indicate a good correlation between errors detected by the algorithm and human perception of errors.

Introduction

The richness of panoramic videos within immersive environments makes it possible to capture outstanding scenes where the user can experience presence. However, realism of such videos is highly affected by visual artifacts such as parallax errors that appear in the form of deformations or ghosting. Unfortunately, this is unavoidable in panoramic video capture due to the use of multiple overlapping cameras. Several attempts have striven to minimize these errors [START_REF] Perazzi | Panoramic video from unstructured camera arrays[END_REF], Lee et al., 2016], but neither could completely remove them. Quantifying those errors is important to video producers for quality monitoring and to researchers for algorithms optimization. Traditional image/video quality metrics are not well suited for capturing the geometric nature of panoramic video defects.

In this paper, we propose a quality assessment method based on a relative comparison of optical flow between the set of input videos overlapping at a pixel location and its corresponding pixel in the output panorama. We enhance our results with the calculation of a salience map proposed by [START_REF] Conze | Objective view synthesis quality assessment[END_REF]. Finally, we assess our method using a human-centered empirical experiment by means of an error annotation interface along with eye-tracking Tobii glasses. Results show a good correlation between the way human perceive errors and the proposed objective metric as well as put the light on the importance of salience calculation.

Optical flow-based quality metrics

Several techniques for panoramic video quality assessment focus on user experiments within a virtual environment where salience data is recorded and analyzed [START_REF] Xu | A subjective visual quality assessment method of panoramic videos[END_REF], Zhang et al., 2017]. Although interesting, subjective experiments are expensive and unpractical. As an alternative, we have investigated optical flowbased methods for this problem. A quality metric for stitched images was proposed by [START_REF] Cheung | A content-aware metric for stitched panoramic image quality assessment[END_REF] which calculates optical flow in the overlapping regions between views along with a salience map that guides the calculation. Unlike our approach, this method only works for images and is not applicable on videos. Another limitation is the constraint they put on camera setup where the central image in a 3-view scene can be considered the reference. Similar to our method, [K. [START_REF] Channappayya | [END_REF]] calculate a temporal feature from optical flow then use feature statistics to compare the reference and distorted videos on a frame-by-frame basis. Their approach is intended, however, to single-view videos hence does not have to deal with multiple inputs. The success of these methods in identifying geometrical defects has motivated us to work on an optical flow-based method for panoramic videos.

Proposed objective quality metric

In a panoramic video, a final output video frame is a composite novel view from a number of input views that go through a series of geometric transformations. Panoramic video stitching methods usually add further transformations for parallax compensation [START_REF] Perazzi | Panoramic video from unstructured camera arrays[END_REF], Lee et al., 2016]. Our method suggests using the original videos as a reference to the single output panorama by comparing the difference in motion between two given frames in the original videos and motion in the final panorama. However, this is not directly applicable since a given pixel x pano in the final panorama can have one to N sources corresponding to N input videos with overlapping regions. To overcome this, we have explored the calculation of the deviation of displacements of all source pixels x i , i ∈ N from the displacement of the panorama x pano between times t and t + 1. In this way, we calculate the optical flow between two frames at times t and t + 1 of each input video and the final panorama. We calculate the standard deviation of the end points of the vector field at each pixel x of the panoramic image to produce our distortion map M d as follows:

M d (x t ) = (x pano,t +1 -n i x i ,t +1 ) 2 n (1)
where n ∈ [1, N ] is the number of overlapping images at pixel x and x t +1 = x t + µ(x t ) and µ is the motion field of a pixel between times t and t + 1.

For more accurate results, we determine visual salience using visibility maps as suggested by [START_REF] Conze | Objective view synthesis quality assessment[END_REF]]. Visibility maps model three features that have been shown to mask errors, which are contrast, texture and variation of gradient orientations. Equations for these maps are detailed in [START_REF] Conze | Objective view synthesis quality assessment[END_REF]. We define our salience map M s for a panoramic frame I pano as:

M s = W c (I pano ) * W t (I pano ) * W o (I pano ) (2)
where W c , W t and W o correspond to contrast, texture and orientation weighting maps respectively.

Building on the assumption that a human would gaze a region if it is distorted or if it salient, we propose to produce a distortion-salience map M d s from M d and M s using a weighted sum. The parameter ω can be changed depending on the video content, i.e.: a highly salient content implies a lower weight for M s .

M d s = ωM d + (1 -ω)M s (3)
4 Human gaze and annotations: an empirical assessment Our objective visual quality assessment is meant to model human perception's sensitivity to distortions and salience. Thus, it was important to validate it with a human-centered study where we collected gaze data using eye tracking and user decision using annotation. In our experiment, we define an error as a region with high number of fixations and an annotation by at least one participant. The protocol for measuring such errors involved the observation of 20 participants wearing Tobii eye-tracking glasses while watching 4 short panoramic videos. Our experiment was designed to answer the following question: Do humans recognize errors that are not identified by an algorithm and vice-versa? The videos were displayed on a large screen for immersion and were played in a random order within an annotation interface that allows the observer to pause the video and mark a region they perceived as deformed. This setup allowed the collection of two types of data, participants' gaze and frames annotations. Our analysis was conducted on both types of data jointly. For each video, we chose one or more keyframes representing the central view within a sequence of frames. Depending on the scene being taken from a fixed view-point or a moving camera, we chose one keyframe or more.

To compare this with our objective metric, we calculated the temporal distortion on a sequence of frames whose center was a given keyframe. Temporal pooling of a given sequence was done by a simple OR then we overlayed our final composite map on the keyframe. Afterwards, data analysis was done using Tobii Pro Lab software. For each keyframe, we defined annotations collected by the 20 participants using Tobii Areas of Interest (AOI) corresponding to at least one human annotation. On the same keyframe, we defined the error regions identified by our distortion map as seen in figure 3. Qualitative results were obtained by generating heat-maps from gaze data recordings using an I-VT Tobii attention filter as suggested by [START_REF] Salvucci | Identifying fixations and saccades in eye-tracking protocols[END_REF]. Classical metrics such as total fixation count and total fixation duration [START_REF] Holmqvist | Eye tracking: A comprehensive guide to methods and measures[END_REF] were used to obtain descriptive statistics on the AOIs described earlier.

Preliminary results

In this paper, we present preliminary results that were obtained mainly on two videos Opera and Street taken in Zurich by Disney researchers [START_REF] Perazzi | Panoramic video from unstructured camera arrays[END_REF]. We subdivided Opera into 5 keyframes since the camera is constantly moving, and only one for Street which was taken using a fixed rig hence content was nearly stable. Distortion and salience maps are calculated on those keyframes and we used [Farnebäck, 2003] for optical flow calculation. Heatmaps were generated for the 20 observers and combined on a single keyframe. We assume that an area with high fixations on a heatmap correspond to perceiving a distortion and/or a salience.

In response to our experimental question, preliminary results shown in 1 indicate that a person's face deformation and a lamp ghosting were the most annotated errors by participants in Opera which correspond to error locations identified by the distortion maps (see figure ). Results on street (see figure 2) show a correlation between human gaze and the calculated salience map. Statistics on AOIs confirm our findings as in figure 3.

Conclusion

We presented a method for designing quality metrics for panoramic videos that was evaluated using a humancentered study. Our early results show that humans are more likely to see errors present in the foreground or moving objects, rather than those in the background. We were able to verify that the calculated distortion map was able to identify most error locations. However, the salience map is yet to be improved to reflect the new findings and be able to penalize distortions occurring in particular regions. Finally, we aim to complete our data analysis and provide a pooling method to obtain an index that can allow the comparison to other methods.

Figure 1 :Figure 3 :

 13 Figure 1: Results on Opera with corresonding heatmaps for gaze data and distortion map calculated by Eq. 1

Acknowledgements

This work was funded by the LIVE360TV FUI 19 project and supported by the French Agence Nationale de la Recherche (ANR), grant ANR-11-EQPX-0002 Amiqual4Home.