Georges Hardier
email: hardier@onera.fr

Gilles Ferreres

Cédric Seren

Seren A Cédric

Recursive

A Recursive Estimation Algorithm to Track Aircraft Model Parameters

On-line parameter tracking is often an attractive option for advanced FDD/FTC systems that require a global monitoring in contrast with a local (component) one. This is the case for impaired aircraft, but also for under-equipped vehicles (UAVs, small airplanes), since an updated model is a prerequisite for many indirect adaptive or reconfigurable control algorithms. Many schemes have been proposed since the 80s to deal with the tricky problem of on-line estimation. However for aircraft, their computational burden is often incompatible with the constraints resulting from onboard implementation and certification issues. As a result, this paper relies on the well-known pros of frequency domain techniques, but proposes a fully recursive algorithm simple enough to comply with those requirements. The approach is applied to the parameter tracking of a civil aircraft on view of designing an adaptive gain-scheduled flight control law.

I. INTRODUCTION

This paper deals with the monitoring aspects related to FDD/FTC, and more precisely with the possibility to update the system modeling onboard, via a continuous estimation of its parameters. Actually, online tracking in near-real time is often the only alternative for some advanced FDD/FTC systems [START_REF] Hardier | On-line parameter identification for inflight aircraft monitoring[END_REF][START_REF] Melody | Parameter identification for in-flight detection and characterization of aircraft icing[END_REF][START_REF] Tang | Methodologies for adaptive flight envelope estimation and protection[END_REF] that require a global (system-wide) monitoring of the system in contrast with a local (component) level monitoring. An example for aircraft is the detection of icing (de-icing off or faulty), in contrast with actuator failures that can be sensed by servo mechanisms simply by comparing control orders with surface deflections when measured. But a loss of actuator efficiency resulting from airframe damages might be also undetected by electrical/mechanical subsystems, and a global monitoring can also be beneficial for underequipped vehicles (UAVs, small airplanes).

More generally, parameter estimation, also called Parameter IDentification (PID), refers to the task of updating the parameters of a mathematical model from i/o data. For aircraft, a batch or off-line version of PID is generally useful to adjust the prior initial modeling issued from CFD computations or wind tunnel testing. However, in abnormal conditions resulting from any impairment (fault, damage, etc.), that modeling can become unsuited again, and thus requires on-line adaptation, especially for FTC purposes since an updated model is a prerequisite for many indirect adaptive or reconfigurable control algorithms [START_REF] Chandler | System identification for adaptive and reconfigurable control[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]. When a post-fault model is available with an updated set of parameters, the Fault Accommodation process can be completed by scheduling or redesigning the control laws [START_REF] Buffington | Integration of on-line system identification and optimization-based control allocation[END_REF][START_REF] Ferreres | Adaptive control using robustness and gain-scheduling[END_REF][START_REF] Gustafsson | Adaptive Filtering and Change Detection[END_REF].

The approach presented in this paper is part of a FTC scheme based on indirect adaptive control whose principle is to estimate some of the model parameters and to adjust a controller on-line in terms of the updated estimates. Actually, gain-scheduling is classically used to control an airplane along a trajectory, the controller gains depending on measured or estimated flight parameters, such as airspeed, Mach number, mass and so on. However, in abnormal situations (actuator faults, icing), or if a scheduling parameter is unavailable due to a sensor fault (e.g., airspeed), adaptive control is an attractive solution to recover closed loop performance by scheduling the gains w.r.t. the aerodynamic model coefficients instead. This requires to obtain a priori guaranteed stability and performance properties of the adaptive closed loop, and to decrease the computational time and complexity to make onboard implementation possible. A detailed description of the associated FTC scheme is given in the companion paper [START_REF] Ferreres | Adaptive control of a civil aircraft through on-line parameter estimation[END_REF].

Regarding PID, the main issue is to develop a method for estimating the model parameters on-line, with a low computational burden and the capability to track time-varying parameters despite measurement noises, poor data information content (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 back-up controller ensuring minimal performance properties. Due to possible ill-conditioning, the method should also include some form of regularization [START_REF] Ljung | Adaptation and tracking in system identification: a survey[END_REF][START_REF] Smith | Regularization techniques for real-time identification of aircraft parameters[END_REF], although most on-line techniques are based on Equation Error (EE) minimizations [START_REF] Morelli | Practical aspects of the EE method for aircraft parameter estimation[END_REF] and involve only a few parameters in practical applications (e.g., via a tuneddown modeling). A contrario, Output Error methods [START_REF] Hardier | On-line parameter identification for inflight aircraft monitoring[END_REF] usually involve an extended set of parameters that are likely to reinforce the implementation issues. They are also less robust to the effect of unmodeled disturbances (turbulence).

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] Jategaonkar | Flight vehicle system identification: a time domain methodology[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 PID 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/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][START_REF] Ljung | Asymptotic behaviour of the extended KF as a parameter estimator for linear systems[END_REF]. More recently, this challenge has also been addressed by recursive subspace identification of linear or LPV 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. To give an example, these limitations preclude the use of any iterative algorithm which would result in some unbounded

A Recursive Estimation Algorithm to Track Aircraft Model Parameters

G. Hardier, G. Ferreres, and C. Seren computation loop through the data. Hence complex operations like non analytical matrix inversions should be banned. 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/FTC schemes.

Considering the implementation issues, TD algorithms could seem preferable at first sight, as the basic ones only involve very simple operations (e.g., by replacing matrix inversion with a scalar division). However, advanced strategies are usually required for regularizing the estimation process, and some of their 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, Frequency Domain (FD) techniques have many desirable features for on-line applications that other (TD) estimation methods do not have. The computation time is greatly 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., LF mismatch and HF noise) improves the estimation accuracy [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 inflight aircraft monitoring[END_REF][START_REF] Klein | Aircraft system identification: theory and practice[END_REF]. 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 PID process. Moreover, many issues specific to on-line PID 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).

noise

Aircraft

Frequency Model

Criterion J(θ θ θ θ) Otherwise, TD and FD estimation schemes can rely on a fully recursive algorithm (making use of measurements as soon as they are available), or alternatively on a kind of sequential procedure, 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 inflight aircraft monitoring[END_REF], a sequential estimation process was proposed for monitoring a civil aircraft, that permits pre and postprocessing stages to be included in the procedure to prevent from and to filter out inaccurate estimations. Hence, those 2 categories of methods have their own pros and cons [START_REF] Jiang | A revisit to block and recursive least squares for parameter estimation[END_REF], even if recursive schemes should be favored for on-line implementation. However, FD approaches are again more suitable to define hybrid recursive/sequential procedures because the information naturally accumulates in the signals' FT, which behave like storing memories and can be managed with more flexibility (truncation, forgetting, resetting).

Parameter optimization

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 TD case. This paper improves the state-ofthe-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]. To cover these aspects, §II outlines the characteristics of the proposed recursive FD method. §III depicts the performances achieved in realistic flight conditions using the industrial non linear simulator of the RECONFIGURE project. §IV draws conclusions and gives some prospects.

II. PRINCIPLES OF THE FREQUENCY DOMAIN PID

A. From time domain to frequency domain

The transition to the FD is classically realized by means of the standard FT of the TD signals. As these ones are only available over a limited period of time ] , 0 [ T , the finite FT is used instead, which leads to the following relations for a signal ) (t x , its time derivative ) (t x & , and a constant value b (associated to a bias for instance):

[ ] [ ] [ ]        - + ω ω = ⇔ ω - = ⇔ = ω = ⇔ ω - ω - ω - ∫ ) 0 ( ) ( ) , ( ) ( ) ( ) 1 ( 1 ( ) ( ) , ( ) ( ) ( T ) T 0 T x e T x T X j t x t x j e b b b dt e t x T X t x t x T j T j t j T & & F F F
Practically, from the sampled values of ) (t x , the finite FT can be computed via a rectangular numerical integration:

) ( ) ( ) , ( 1 0 ω ∆ = ∆ ∆ ≈ ω ∑ - = ∆ ω - X t e t n x t T X N n t n j (2)
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 z-transform (CZT). The latter permits a desired frequency resolution to be chosen independently of the length of the time interval used, but is less effective as far as the computation time is considered. If we assume that the DFT is computed via standard FFT, from N data samples equally spaced over the time interval ] , 0 [ T , this algorithm calculates N values of the DFT over the frequency interval

] / 2 , 0 [ T N π , equally spaced too with a step T / 2π = ω ∆ .
More elaborated algorithms are available to reduce the inaccuracy resulting from (2), but these refinements are generally not useful when the sampling period is higher enough [START_REF] Morelli | Real-time parameter estimation in the frequency domain[END_REF]. Otherwise, we will see in the sequel that the recursive algorithm can provide FT at any selected frequency without the extra computational cost of CZT.

B. Standard formulation of a sequential algorithm

From now, we will consider a general formulation of the problem, leading to other types of parameters being considered in addition to the stability and control derivatives. When applying PID techniques, it is generally advisable to estimate initial conditions and state/output biases in order to cope with i/o measurement offsets or model structure uncertainties. Consequently, the dynamics will be expressed in state-space form by the following linearized equation:

   + Θ + Θ = + Θ + Θ = y x b t u D t x C t y b t u B t x A t x ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( & (3)
with the initial condition given by 0

) 0 ( x x = . Matrices D C B A , , ,
include the stability and control derivatives Θ to be estimated, that are assumed to be constant or at least to vary slowly during the flight with respect to the PID process and updating rate. In the case of aircraft for instance, the states of vector x are the airplane speeds and angular velocities in body axis, as well as Euler angles. The vector u collects the surface deflections, whereas the measurement vector y includes state components, air data angles and load factors. x b and y b represent the state and output biases.

Thanks to (1), if ω is chosen to be a multiple of the frequency step ω ∆ . (which happens for instance when using the FFT), by omitting the dependencies on Θ and by denoting ) (

≡ ω ≡ Y Y U U X X , ), ( ), ( ω ≡ ω
becomes [START_REF] Hardier | On-line parameter identification for inflight aircraft monitoring[END_REF]:

   ω + + = - + ω + + = ω ) ( ) ( ) 0 ( ) ( δ T b U D CX Y T x x δ T b U B AX X j y x (4) where ) (ω δ denotes a Dirac function in the FD, such that 1 ) ( = ω δ for 0 = ω , and 0 ) ( = ω δ for 0 / 2 ≠ π = ω T k (since 0 for 0 and , 0 for / ) 1 ( ≠ = = ω - ω - k k T j e T j
).

As a result, the initial and final conditions ) ( ), 0 ( T x x are 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 but just modify the zero frequency. To get the most out of these specificities, it is generally worthwhile to discard the zero frequency during the identification stage, which avoids the state and output biases having to be estimated. Thus, (4) is further simplified and reduces to:

   + = - + + = ω DU CX Y T x x BU AX X j ) ( ) 0 ( (5)
Hence, the discrepancies between the initial and final conditions of the states 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 inflight 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. Otherwise, the vector of parameters should be conveniently extended to include the ) dim(x corresponding values. If we also assume that all the state components are measured, in addition to the i/o ones, an EE 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]:

∑ = - ω - ω - ω ω = Θ M k i k i k i k i k i b U B X A x j J 1 2 ) ( ) ( ) ( 2 1 ) ( ( 6
)
where the subscript i corresponds to the i th state equation, and k ω to one of the M frequencies available from the TD to FD transformation ( N M ≤ ).

i i B A ,
are the i th rows of matrices A and B , whereas

) ( k i x ω is the i th component of the vector ) ( k X ω and ) ( ) 0 ( T x x b i i i - =
. The same kind of expression can also be established for each output equation, permitting the parameters included in matrices D C, to be estimated.

By collecting the different frequencies in vector and matrix forms, to make the summation of (6) disappear and by denoting Ξ the extended vector of unknown parameters in i A , i B and i b , the cost (6) turns to the LS criterion [START_REF] Morelli | Real-time dynamic modeling -Data information requirements and flight test results[END_REF]:

[ ]

) 7 ( ) ( ... ) ( 1 ... 1 ) ( ... ) ( ) ( ... ) ( : with ) ( ) ( 2 1 ) ( ) ( 2
1 ) ( 1 1 1 1 0 1 0 0        ω ω ω ω =         ω ω ω ω = Ξ - Ξ Σ Ξ - Ξ + Ξ - Ξ - = Ξ - M M M M i i T T T i x j x j U U X X † J Y X X Y X Y
where † is the complex conjugate transpose operator. In [START_REF] Ferreres | Adaptive control using robustness and gain-scheduling[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 Ξ increments w.r.t. prior values 0 Ξ through a corresponding covariance 0 Σ (usually a diagonal matrix). ( 7) is nothing but a standard LS regression with complex data instead of real ones. The well-known solution to this problem and the estimated error covariance are given by [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]:

     Σ + σ = Ξ Ξ Σ + Σ + = Ξ - - - - - 1 1 0 2 0 1 0 1 1 0 ] ) ( [ ) ( Cov ) ( ) ( ˆ] [ ] [ X X Y X X X R R R † † † (8)
where () R is the real part of a complex value. Practically, to compute the covariance matrix from the available data, especially the diagonal terms giving access to the parameter errors, the EE variance 2 σ is usually evaluated a posteriori from the residual errors, resulting in [START_REF] Morelli | Real-time dynamic modeling -Data information requirements and flight test results[END_REF]:

)] ( dim [ ) ( ) ( ˆ2 Ξ - Ξ - Ξ - = σ M † / / / / X Y X Y (9)

C. Towards a fully recursive algorithm

As regards practical aspects for on-board implementation, it should be noticed that the summation of (2) makes very simple operations possible, as mentioned for instance in [START_REF] Morelli | Real-time parameter estimation in the frequency domain[END_REF]:

     = ∆ + 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 ) ( ( ( ~ (10)
to evaluate the DFT at time t n∆ from its previous value at time

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 is also likely to involve state derivatives, the measurements of which are usually not available, a recursive formulation similar to [START_REF] Gustafsson | Adaptive Filtering and Change Detection[END_REF] . From ( 1)-( 2), we easily get:

) 0 ( ) ( ) ( ) ( x e t n x X j X t n j n n - ∆ + ω ω = ω ∆ ω - & (11) with ) ( ) ( ω ∆ ≈ ω n n X t X
. Hence, recursive expressions can be implemented (for any signal ) (t x even if it is not a state component consistent with the general formulation of §II.B), by denoting

) ( ), ( ), ( t n x x X X X X n n n n n ∆ ≡ ω ≡ ω ≡ & & :      ∆ = - ∆ ≈ - - + - ω = - ∆ ω - - - ∆ ω - - ∆ ω - - - ) 12 ( ) ( ) ( ) ( 1 1 1 1 1 t n j n n n n n t 1) - n ( j n t j n n n n n e x t X X t X X e x e x X X j X X & &
When using these coupled recursive updates for on-line purposes, older information can be overweighted regarding to recent ones, which can result in much delay in the adaptation process. That's why data forgetting can be used in conjunction with [START_REF] Gustafsson | Adaptive Filtering and Change Detection[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 of the detection method 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 inflight aircraft monitoring[END_REF], resulting in a succession of piecewise constant values, updated only after some seconds and hence delayed w.r.t. the varying parameters. Whatever approach followed, a time history of the past data could be required to enable those calculations. To avoid such a storage, and to develop a recursive algorithm, it is still possible to benefit from the linearity of the FT 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], and that 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]. Applying this trick to (12) yields (denoting

t j n n e x x ∆ ω - * = ):      ∆ + λ ≈ - + ∆ ω + λ ≈ ∆ ω - * - ∆ ω - - * - t 1) - n ( j n n n t 1) - n ( j n n n n e x t X X e x x t j X X 1 1 1 ] ) 1 [( & & (13)
We are now going to address a key feature of the proposed 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 contributes to modify all the M rows of the matrix X through the FT updates of [START_REF] Hanlon | Multiple-model adaptive estimation using a residual correlation KF bank[END_REF]. That is why the usual algorithms often 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, a recursive version can still be derived, almost similar to the TD ones, by noticing that: [START_REF] Ferreres | Adaptive control using robustness and gain-scheduling[END_REF]. In [START_REF] Hardier | An extended U-D algorithm with multiple forgetting factors for RLS estimation of model parameters[END_REF], the vector

t n j n n e t n t ∆ ω - - ∆ ∆ + ω λ = ω ) ( ) ( ) ( 1 x X X (14) for 1 m R x × ∈ defined as T n t n u t n x t n ] ) ( ) ( [ ) ( ∆ ∆ = ≡ ∆ x x , cf.
) (ω n X is issued from: [ ] ) ( ) ( ) ( ) ( ) ( ) ( 1 1 1 1 1 M M M m m T ω ω =         ω ω ω ω = X X X X X X X L L M M M L ( 15
)
As a result, denoting ] [

1 t n j t n j n M e e † ∆ ω ∆ ω = L E
, we can set the basis of a time-recursion as:

† t † † n n n n E x X X ∆ + λ = -1
, that yields (with

n n n † M X X = ): ∑ = - - - - + λ = λ + λ + ∆ ∆ + λ = 3 1 1 2 1 1 1 2 ] ) 16 ( [ 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 x E E x x x X X with:        = ∆ = = = λ = λ = = = ∆ = = = ∆ = - - T T n n n n n n T n B t D C † D † B † D C B M t B B D C t M B 2 3 3 2 1 3 1 2 2 1 2 1 1 1 1 1 1 1 x E E x x X X (17) As 1 m R × ∈ 1 B
, we also benefit from:

T D B D B D B M R n n n ] [ ) ( ) ( ) ( 2 2 2 2 1 1 1 2 R R I R + + + λ = - = (18) Hence 1 - n M (or 1 - n R
) can be computed recursively by applying the matrix inversion lemma 3 times to the 3 additions of ( 16):

1 1 1 1 1 1 1 ] [ ) ( - - - - - - - + - = + A D C B A D B A A BCD A ( 19
)
with:

     + = + + λ = + = + λ = λ = - - - 2 2 2 2 2 2 2 1 1 1 1 2 3 1 1 1 1 1 1 1 1 2 2 1 2 1 ) 20 ( D C B A D C B D C B M A D C B A D C B M A M A n n n
resulting in the following sequence of computation steps:

) ( / 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 2 ] [ - - - - - - - - λ = + = n n M s D B I M D C B A A ) ( / 2 1 2 2 2 2 1 2 1 2 2 2 2 1 3 ] [ - - - - - = + = A s D B I A D C B A A ( 21)
) ( / 3 1 3 3 3 3 1 3 1 3 3 3 3 1 ] [ - - - - - = + = A s D B I A D C B A M n with 1 s = 1 1 1 1 B M D n - - + 2 λ , 2 s = 2 1 2 2 B A D - +1 , 3 s = 3 1 3 3 B A D - +1 .
Finally, ( 17)-( 21) permit the direct updating of 1 n R to be computed without requiring onboard a tricky matrix inversion, by using only a series of matrix additions and products. For the sake of simplicity, Given that standard algorithms do not comply with certification and implementation constraints, another solution for computing 1

- n R when 3 ) dim( > n R
would consist in resorting to a blockwise inversion by iteratively partitioning n R into a series of submatrices of lower dimensions till analytical inverses can be got whenever dimensions 2 or 3 are reached. However, it is noteworthy that this approach requires the R matrix to be computed in addition to the matrix products of the blockwise inverse. Fig. 2 compares (for different values of m and M ) the computational costs, expressed in terms of the number of floating-point operations, associated to the recursive algorithm ( 17)-( 21) (including a priori knowledge) and to a blockwise inversion. For the usual values we are concerned with for on-line estimation (

6 3 ≤ ≤ m , 50 ≈ M
), the recursive algorithm proposed in this paper is clearly attractive: in the application case presented in §III, the cost is decreased more than twofold. Moreover, the blockwise algorithm would require the submatrices (and their Schur complement) to be invertible, a property that cannot be a priori guaranteed.

III. RESULTS AND DISCUSSION

In §III, the recursive FD estimator described in §II is used to develop an indirect adaptive control strategy 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 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 of 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, plus actuator and sensor models, measurement filters, as well as the control law protections.

During this project, an adaptive gain-scheduled flight control law has been designed off-line at first, by using a state-space modal technique to synthesize a LFT controller which would only depend on the major model aerodynamic parameters (see the companion paper [START_REF] Ferreres | Adaptive control of a civil aircraft through on-line parameter estimation[END_REF]). The FD PID is implemented to track those time-varying parameters on-line, and to schedule the controller w.r.t. the provided estimates. In this paper, we only consider the longitudinal motions, and we focus on the A/C lift, pitch and load factor equations that characterize the short-period dynamics. Hence, about a given flight condition, the stability derivatives to be tracked (physical parameters of the aerodynamic model) appear as pseudo-constants in a linearized state-space model (3)-type:

     + δ + + α = + δ + + α = + δ + + α = α δ α δ α δ α V N N q N N N V M M q M M q V Z Z q Z Z V m q z V m q V m q & & (22)
where the states α and q are the angle of attack and the pitch rate, V is the airspeed, m δ is the elevator input, and z N is the load factor output. The influence of the altitude variations ( z ) are neglected, as the aerodynamic coefficients only indirectly depend on z . Accordingly, [START_REF] Ljung | Adaptation and tracking in system identification: a survey[END_REF] reduces to:

[ ] T T lon T lon T u x Nz q ] [ Θ = α & & ( 23
)
where the Θ matrix includes the 12 parameters to be tracked and estimated, lon x denotes the longitudinal states of interest and lon u reduces to the elevator control m δ . In the context of the adaptive scheme, the main goal of the on-line estimator is to update the values of the five main stability derivatives

α δ α α N M M M Z q , , , ,
used to schedule the controller [START_REF] Ferreres | Adaptive control of a civil aircraft through on-line parameter estimation[END_REF], although the twelve coefficients of [START_REF] Ljung | Adaptation and tracking in system identification: a survey[END_REF] have to be estimated.

To evaluate the proposed approach, the simulated scenario corresponds to a 380 sec. realistic flight profile including 4 successive stages (Fig. 3 . The fault scenario which is considered is a loss of airspeed information; hence it is assumed that c V is no more available throughout the flight. The testing maneuvers selected by Airbus for the industrial validation of the controllers involve joystick doublet-type signals. So, to check the performances of the adaptive process, usual stick inputs are applied to the c z N set points (in blue) during the 1 st and 4 th parts of the flight with autopilot and auto-throttle switched off. The aircraft mass is 260t and the CoG is located at 28% (of the mean chord). Regarding the tuning of the FD Algorithm (FDA) described in §II, the frequency band chosen for the FT computations comprises 43 = M frequency samples equally spaced in the range 0.1-0.52 Hz with a frequency resolution of 0.01 Hz, to include the rigid-body modes of the airplane and to filter out low frequency phenomena. It is worth noting that the spacing does not need to be uniform for this FDA since no standard FFT is used. All the data are high-pass filtered prior to performing FT to comply with the assumption of linearization about a trim condition, and to filter out low frequencies. A 1 st order filtering is implemented with a time constant of 3 sec. After encoding the algorithm with operators compliant to the SCADE software (Safety Critical Application Development Environment), the resulting computational burden evaluated by Airbus is about 1.5ms, which is acceptable and could be further improved by updating the estimates at a lower rate. The results achieved by the FDA are displayed with solid blue lines in Fig. 4, that only plot the 5 (out of 12) coefficients required for the adaptive control. The parameter uncertainties (3σ) are plotted in blue dashed lines and are used to freeze the estimates whenever they are too high (straight lines in the plots). While the estimates are invalid, the last valid estimated value is used to schedule the controller as long as the estimates are not declared reliable again. The green curves give an idea of the reference values drawn from the closest LTI model (just one of 25 for this mass/CoG). Due to a loose mesh of the models in the flight domain, these values are only plotted as a rough guide, and this must be kept in mind when judging the estimates (the true linearized values being unavailable).

Owing to the simulated scenario, the variations that should be 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). Hence, 2 options are available to manage the FDA results, and to feed the adaptive controller in case of ill-conditioning:

to proceed with a continuous estimation while reinforcing the regularization temporarily using a priori knowledge; to lock the parameters to their last estimated values as long as they are unreliable.

In addition, the forgetting mechanism of ( 13) can be activated during level changes to speed up the convergence of the parameters to their new steady values, or the FDA can be automatically reset as soon as the new flight condition is reached. Both strategies have their own pros and cons. The former requires the value of the forgetting factors to be tuned (which could depend on the rate of changes), but permits the new estimates to be updated as soon as some excitation is available about the new flight condition. The latter avoids any tuning and relies on frozen values during the dead zone, but slows down the updating about the new flight condition due to a transient phase after resetting the FDA. It is noteworthy that all these possibilities (reinforced regularization/locked parameters, forgetting/resetting) have been evaluated but cannot be displayed by lack of space; they result only in very slight differences in the performances plotted in Fig. 3.

The point is that, thanks to a simple monitoring of the changes in flight condition and of the parameter uncertainty provided by the FDA itself, different strategies can be temporarily triggered resulting in different settings of the PID process: to activate the forgetting mechanism by switching to a 1 < λ , to strengthen the regularization by decreasing some components of 0 Σ , to freeze the values of the estimates delivered to the controller, and possibly to reset the procedure. On the other hand, owing to the properties of the recursive FT, it should be understood that the information provided by the data continuously accumulates and contributes to improve the parameter accuracy. Consequently, if excitation is lacking for a while, no parameter divergence is to be feared, except in case older information is progressively forgotten. That is why forgetting should not be permanently activated, and why the estimation accuracy is so valuable and should also be monitored to reconfigure the FDA whenever necessary.

For example, in Fig. 4 the forgetting mechanism is (only) triggered when significant V z / changes are detected providing a way to adapt more rapidly to varying conditions. The results corresponding to the 2 previous options are illustrated: in blue color when the parameters are locked in case of unreliable estimates, in red dashdot lines when the regularization is reinforced instead. It should be pointed out that the latter was not used during the closed-loop simulation of Fig. 3, but is still plotted in Fig. 4 to evaluate the behavior of the estimates, should this option be used. However, regardless of the selected option, the closed loop responses of Fig. 3 are almost the same: the estimates are typically unreliable when closed loop excitation is poor, so that the gap between the performances is insignificant.

It should also be pointed out that with such a poor excitation (see c z N in Fig. 3 during the 2 nd /3 rd stages), 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 ( 7) is introduced, w.r.t. the values of the damping coefficients q Z (trickier to estimate even with proper excitation), and mostly of the speed derivatives V C (the speed variations about the trim condition are very small). When the regularization needs to be reinforced, the variation of the pitching derivatives is also overweighted, especially to avoid data collinearity between q M and δ M .

The plots of Fig. 4 , the excitation level is very low (remember that the data of Fig. 3 are high-pass filtered before being processed by the FDA), and hence the parameter inaccuracy results either in freezing or in strongly regularizing the estimation process (blue/red curves). This process is also not exactly the same for the different parameters as the lift, pitching and load factor equations are processed separately. Hence, the trends indicated by the (green) reference values cannot be mimicked by the estimates, and instead they are drawn towards the prior valid estimates or towards average reference values used as a priori knowledge (e.g., mean values for a given altitude). As soon as sufficient excitation is available again, and the accuracy is good enough (time to store information through the recursive updates of the FT), the estimates are freed again and converge rapidly to suitable values.

In steady state conditions, the estimates provided by the FDA compare very favorably to their reference values. Thanks to the noise filtering outside the selected bandwidth, the FDA does not require much time to accumulate enough information and to recover proper estimates. In case of continuous estimation, the convergence delay is even shorter as the estimates may be reliable enough although their standard errors are not fully satisfying.

As regards the performance of the adaptive controller, we have to refer to the plots of the load factors in Figs. 3 and As soon as new reliable estimates are available for the scheduling, much better performances are recovered thanks to the adaptive process resulting in proper overshoot and settling time (please refer to the companion paper [START_REF] Ferreres | Adaptive control of a civil aircraft through on-line parameter estimation[END_REF] for a comprehensive description of the control aspects). Hence in faulty situations [START_REF] Goupil | AIRBUS efforts towards advanced real-time fault diagnosis and fault tolerant control[END_REF], the proposed strategy can avoid switching to a more robust but less efficient control law (e.g., when the airspeed data is missing for the usual scheduling process due to sensor faults).

IV. CONCLUSIONS AND PROSPECTS

A fully recursive estimation scheme was developed in the frequency domain to track time-varying parameters, more likely to comply with the requirements of embedded A/C implementation than the standard existing algorithms. Some issues related to the computation of state derivatives are also properly settled to get recursive expressions of the Fourier Transforms. This recursive estimation algorithm is part of an indirect adaptation process, permitting an LFT flight controller to be designed and to be scheduled w.r.t. the estimated model coefficients when the airspeed information is missing. Nonlinear simulations, achieved via the industrial benchmark of the RECONFIGURE project, 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. The computational burden and complexity of the global adaptive scheme are also compliant with an embedded implementation on existing computers.

These results are very promising as regards a possible application to civil aircraft, but of course further industrial validation would be needed using extensive sets of simulations including a wide range of operational conditions (realistic flight scenarios, external disturbances, pilot maneuvers, and so on). Tools like the Functional Engineering Simulator, implemented during the RECONFIGURE 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], should be extended to be suited to this need. Actually, to evaluate the proposed adaptive approach, longer operational scenarios are required to benefit from on-line parameter estimation, including changes in the flight conditions and unexpected faults occurring during the flight. A transient stage at the start of the simulation is also required to initialize the algorithm and to accumulate some information prior to delivering reliable estimates to the gainscheduling process. These constraints spring from the fact that the adaptive controller is not pre-computed once for all, and hence is not available at the initial starting time.

To contribute to the V&V process, avenues do exist for such complex controllers; for instance, a practical approach for the worst-case validation of a similar adaptive scheme 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]. On the other hand, this validation involves integrated estimation and control components, but some theoretical guarantees regarding the robustness to transient or asymptotic estimation errors can be proved at first using µ and IQC analysis techniques [START_REF] Ferreres | Adaptive control of a civil aircraft through on-line parameter estimation[END_REF]. Stability and performance properties of the adaptive closed loop can also be enforced thanks to more elaborated strategies, e.g. by switching to a robust controller in some transient situations. More details are available in the companion paper [START_REF] Ferreres | Adaptive control of a civil aircraft through on-line parameter estimation[END_REF].

  be 0 in the previous expression of n M . Dealing with a priori knowledge is however possible by achieving at first a series of m rank 1

Figure 2 :

 2 Figure 2: Computational costs of the recursive and blockwise inversions

Figure 3 :

 3 Figure 3: Time history of the longitudinal aircraft flight parameters

Figure 4 :

 4 Figure 4: Time history of the estimated parameters

  Fig. 5 focuses on the responses during the first and last 40s of the flight. The blue line corresponds to the c z N stick input settings, and the black one to the z N signals.

Figure 5 :

 5 Figure 5: Zooms in on the load factor responses

Fourier transform Fourier transform inputs U(t) U(f) measurements Z(t) Z(f) outputs Y(f) initial knowledge

	inputs U(t)	Aircraft	+ +	+ +	measurements Z(t)
	Fourier transform		Fourier transform
	U(f)		parameter parameter update θ θ θ θ ^noise update θ θ θ θ ^	Z(f)
	initial knowledge θ θ θ θ 0 θ θ θ θ 0	Frequency Model	-+ + -outputs Y(f)	Criterion J(θ θ θ θ)
			Parameter optimization
	Fig. 1. The FD identification process

This work was performed in the framework of the RECONFIGURE European Project: Grant agreement №: FP7-314544.


Ruth Meyers indicated that the Commission did not feel that the Gospel passage about Cana is really about Jesus adorning marriage by his presence, as the introduction to the 1979 rite puts it, and there was a desire in the Commission to move away trom this when drafting the 2012 liturgy. Presumably, this explains the deletion in the gender-neutral

EXPLORING THE NEW CREATION: ESCHATOLOGICAL IMAGINATION AND THE BLESSING OF SAME-SEX UNIONS Rémy Bethmont

TransCrit / Université Paris 8

The liturgies for same-sex marriages authorised by the General Convention of the Episcopal Church last summer were the result of a 6-year process whose first stage was the production of theological and liturgical resources which did not label same-sex relationships as marriages. The liturgy authorised in 2012 was tellingly entitled "The Witness and Blessing of a Lifelong Covenant." As I shall make clear, the definition of lifelong covenant was in no way inferior to that of marriage since marriage itself was defined as a lifelong covenant. Talking about covenanted relationships placed both same-sex and different-sex couples within a larger frame of reference in which commitments that did not center on a couple were equally honoured, such as vows binding a monastic community. The breadth of this frame of reference could accommodate the various ways in which monogamous gay and lesbian relationships had been thought of in queer religious discourse since the 1990s. Marriage had certainly been an important template, which had been given prominence in much of the Western World by the secular campaign for marriage equality, but another script had also been put forward, that of covenanted friendship, favoured by people who felt uncomfortable with the idea of marriage. Having said this, the fact remains that the use of the word marriage in the 2012 liturgy was avoided for political reasons. The Standing Committee on Liturgy and Music (SCLM), who had been commissioned by the 2009 General Convention to develop resources for the blessing of same-sex unions, had not been given any mandate to apply the word marriage to these unions. Given the apprehension and misgivings that this move gave rise to within sections of the Church, the Commission's chair, Ruth Meyers, felt it very important not to overstep the bounds of their mandate. But the positive consequence of the prohibition on the word marriage was an openness in the liturgical and theological resources to the way in which committed gay and lesbian relationships should be labelled and imagined.

At least this was how I felt the prohibition could be received in the European context of the Convocation of Episcopal Churches in Europe whose Bishop sent me as a delegate to the Consultation on same-sex blessings in June 2014 in Kansas City. This paper comes out of my initial, probably naïve surprise at how much of an issue the absence of the word marriage was for many American delegates. Although the American context provided an explanation for this, I could not help wondering whether the Episcopal Church was not becoming obsessed with the very family values that queer theology had been encouraging the Church to move away from: how were the queer resources authorised in 2012 being received by people who campaigned for the blessing of same-sex relationships in the Church? In Kansas City, indepth discussions took place in small groups. In my own small group, there seemed to be an agreement that the Commission had produced a very valuable theological document. At the same time the question of how much it related to the way in which ordinary church members actually viewed marriage or any of the covenanted relationships mentioned in the document was asked. Episcopalians still have some way to go before they conceive of the union of a couple as a covenant. In this context of Episcopal groping for a covenantal understanding of committed relationships in general, what, we ask, is the meaning of the inclusion of same-sex unions in Christian marriage that the last General Convention officially validated?

In fact the strong eschatological underpinning of the various queer ways of describing and labelling gay and lesbian relationships in SCLM's work means that choosing the word marriage to name same-sex unions does not necessarily exclude the theological insights behind the choice of other images. Using the examples of Michael Vasey and Eugene Rogers' writings, I will show how much same-sex unions viewed in terms of friendship or of marriage have in common when they are both viewed eschatologically. In that perspective, the significance of the choice of one label over another is very relative. The resources produced by SCLM are very much inscribed in this eschatogical perspective. Given the American legal, historical and political context, the choice of the word marriage in 2015 makes perfect sense without necessarily compromising the queer renewal of the Christian imagination that SCLM is inviting the Church to enter into. What seems clear, however, is that the Episcopal Church still has some way to go before this renewal actually happens. The revision of the marriage canons and the authorisation of inclusive marriage rites are hardly the end of the road but merely a station on the journey -albeit a significant one.

Naming gay and lesbian relationships eschatologically

The new marriage liturgy authorised by the Episcopal Church is heavily indebted to the recovery of an eschatological view of the Eucharist as a foretaste of the heavenly feast and as the Kingdom of God breaking into the here and now. In Anglicanism worship plays a central role in articulating doctrine (lex credendi, lex orandi). In doing so it also plays a central role in forming the Christian imagination. It provides images and words that form and reform the sensus fidelium. There is a continuum between eucharistic eschatology and a renewed way of thinking about Christian morals. 1 The natural paradigm which has dominated Western thinking since at least Thomas Aquinas has been increasingly challenged by an eschatological paradigm. What I call the natural paradigm is that way of thinking by which moral rules are derived from the natural order which, in spite of sin, still bears witness to the perfection of God's Creation in the beginning, to the Creator's intention for his Creation. Classically in the natural paradigm, salvation goes hand in hand with conformity to the order of Creation. 2 The natural paradigm has tended to look to what was in the beginning to find the moral truth of humankind: true morality is found in Adam and Eve's innocency before the Fall and the quest for holiness is about finding one's way back to that original truth and purity, with the assistance of grace.

A stronger eschatological awareness in the Western Church has meant that the quest for the natural order in the beginning has no longer been as obviously necessary to the moral quest and indeed to salvation. One emphasizes the new reality of the New Creation which surpasses rather than restores the perfection of the old. Therefore morality can be defined in terms that need not refer to the law of Nature and salvation is rediscovered, as James Alison puts it, as being led to inhabiting a space of "serenity about nothing human being simply 'natural', but everything being part of a human social construct, to the extent where we can begin to imagine God quite removed from any justification of the present order, and yet ever palpitating beneath the vertiginous possibilities of the bringing of a divine order into being."3

Queer theology in its fresh approach to homosexuality has made great use of what I call the eschatological paradigm, along the line of a great deal of patristic theology, by which the truth of humankind is defined by the endtimes, before being defined by what God created in the beginning. Negotiating the shift to the eschatological paradigm has been anything but straightforward, however. The eschatological paradigm enables one to recast one's discourse on homosexuality away from its naturalness or unnaturalness, towards the way in which gay and lesbian relationships relate to the realities of the New Creation. But traditionally, while eschatological thinking (in the patristic Church notably) allowed for a positive assessment of affectionate relationships that were not procreative, it tended to view the sexual dimension of any relationship in a negative light. 4 The ideal marriage for the Patristic Church is a spiritual marriage, devoid of sex. 5 Relating gay and lesbian relationships to eschatological imagination has meant rethinking theological tradition and experimenting with a variety of images to describe the truth of gay and lesbian relationships in the light of the New Creation.

In the early years of queer theology6 frienship seemed to be as readily invoked as marriage as a template for gay and lesbian relationships. Michael Vasey, in Strangers and Friends (1995), saw gay relationships as a gift to Western Church and society, which had lost any sense of intimacy between males out of idolatrous conformity to the capitalist order. Capitalism turned men into competitors in the public sphere, leaving the married domestic sphere as the only space for emotional fulfilment and intimacy. 7 The 'family values' which the church identified with, said Vasey, would have been hardly comprehensible to the Primitive Church for which family was very much part of the order of this fallen world. Family values for early Christians would have embodied all those things belonging to natural man that one should die to: preference given to your own clan over universal fraternity with all those who had been baptised into the Church. Gay people, said Vasey, could help the Church reconnect with the breadth of covenanted relationships that Scripture and Tradition commended. While Vasey was by no means censorious of gay people adapting to themselves the marriage script to order their lives, his preference seemed to lie with the script of affectionate, covenanted friendships such as can be found in the monastic tradition represented by Aelred of Rievaulx. More than marriage, the script of intimate friendship would make more significant the gay contribution to the critique of an idolatrous capitalist order the Church had bought into.

For Vasey the classic Christian understanding of romantic love is not found in its relatively recent restriction to heterosexual marriage but in "an intuition of beauty, a moment of revelation both about God and about creation." (236) This revelationary quality of the erotic is linked to eschatology: it is a revelation of the Kingdom of God and a foretaste of heaven. It is not incidentally that Vasey ends his book with a section about heaven or "gay paradise" as he entitles it: "The classic Christian imagination saw the awakening of love, the sweetness of sexual pleasure, and the fruitfulness of the sexual act as real but partial anticipations of the true locus of human longings for joy and immortality. The true and lasting fulfilment of these hopes lay in heaven" (245). And Vasey laments the "loss of heaven from the modern imagination" (245), something he thinks gay people could help the Church recover: "The biblical and traditional images of heaven are so preoccupied with style and public celebration as to be almost camp. While relentlessly political, they have more in common with a Gay Pride event than with the sobriety of English political life or the leisurewear informality of evangelical Christian life. … The hope of heaven does not rest on fitting in with the way of the world but on the Lion and the Lamb -on the beauty of a king who strives for justice and the love of a gentle friend who takes to himself our pain and failure." (248)

The broadening of the Western imagination about affectionate relationships which gay people can contribute to, more easily perhaps than straight people, is closely linked to a renewed, enhanced eschatological imagination. Vasey strived to present gays and lesbians, indeed the gay culture, which was universally reviled by Evangelicals in his days, as a gift to the Church. His concern was to underline the differences between gay and straight people, rather than what they had in common, to show that the straight Church needed their gay brothers (and probably their lesbian sisters, too, although Vasey hardly ever mentions them). On one count at least, it seems that Vasey was right: the constancy with which eschatological reflections have underpinned queer theology has meant that gays and lesbians have helped the Church to recover the "hope of heaven" by relating moral thinking to the New Creation. An eschatological approach has characterised a number of queer theologians who have come to favour the template of marriage for gay and lesbian relationships.

Eugene Rogers is one such queer scholar. Steering away from the image of friendship, Rogers decidedly turned to marriage as the better image for same-sex unions. This made him a supporter of the civil campaign for marriage equality, but his queer theology of marriage is not formulated in terms of rights. Rather it moves away from considerations of individual rights to emphasize what a gay and lesbian partaking of marriage can offer the Church and the world, how it can further God's kingdom. In that sense, it is quite remote from the idolatrous attachment to bourgeois marriage and family values that Vasey writes against. Rogers' theology insists on relating gay and lesbian relationships to the hope of heaven, bringing it much closer to Vasey's than one might at first think.

Eugene Rogers defines marriage as having an "eschatological end in the grace and gratitude of the trinitarian life, apart from childbearing." 8 When desire is deepened in marriage, it may "trick lovers" into "acts of faith, hope and charity," it stretches forward "into things that are more desirable." 9 Gay and lesbian relationships can equally partake of a vision of marriage as "bodily means that God can use to catch human beings up into less and less conditioned acts of self-donation, finally into that unconditional response to God's self-donation that God's self gives in the Trinity." 10 Situating marriage eschatologically makes it the twin brother of celibacy 11 (as indeed 'spiritual marriages' did in patristic times) and makes its meaning independent from procreation. The fruitfulness of the marriage is situated in its straining forward into the life of mutual self-offering in the New Creation. In this sense, procreation links up with the Christian meaning of marriage only if the couple with their children form a community that seeks to partake of the life of the Trinity. Procreation in other words is merely one possible channel by which the eschatological fruitfulness of a marriage may bloom.

Vasey and Rogers use very different images for committed same-sex relationships but they both approach them eschatologically, thereby providing a critique, explicit or implicit, of the late 20 th -and early 21 st -century Church's obsession with bourgeois family values. The right ordering of relationships should certainly be of concern to the Church, but it should view this ordering in the light of the New Creation which God is bringing about. In this light, bourgeois family values do not mean enough.

This kind of eschatological thinking has been fundamental in the development of the recent liturgy for the blessing of same-sex unions in the Episcopal Church and for the even more recent revision of this liturgy, adapting it for marriage for both same-sex and different-sex couples.

The Episcopal theology of same-sex marriage

Last July, the General Convention of the Episcopal Church authorized for trial use a genderneutral version of the 1979 marriage rite, which parallelled the gender-neutral revision of the marriage canons. More interestingly, it seems to me, Convention also authorised The Witnessing and Blessing of a Marriage which has come out of the work of the Standing Commission on Liturgy and Music (SCLM) on the blessing of same-sex couples. This liturgy, based on the 2012 Witnessing and Blessing of a Lifelong Covenant, constitutes a thorough rewriting of the marriage rite, taking up the definition of marriage as covenant, found in several recent Anglican liturgies, and using the notion of covenant to shift the definition of marriage from taking another to giving oneself to another.12 This is powerfully conveyed in the vows the two spouses make, especially in the 2015 version, prior to the relatively minor revisions made by the competent legislative committee of the 2015 General Convention:

In the name of God, I, N., give myself to you, N. I will support and care for you [by the grace of God]13 : in times of sickness, in times of health. I will hold and cherish you [in the love of Christ]: in times of plenty, in times of want. I will honor and keep you [with the Spirit's help]: in times of anguish, in times of joy, forsaking all others, as long as we both shall live. This is my solemn vow.

Legislative committee changed the first sentence of the vow to "In the Name of God, I, N., give myself to you, N, and take you to myself". This was done at the instigation of a priest who insisted that the taking importantly signified the total acceptance of one's partner. 14 Giving, he said, should be complemented by receiving. The rite was authorized by Convention with this revision. Although the authorized version of the vows objectively moves the rite closer to traditional liturgies where the two spouses "take" each other, it does not ultimately change the focus of the liturgy on the self-offering of the partners. In particular, the celebrant's first question to each partner still reads: "N., do you freely and unreservedly offer yourself to N.," a striking rewriting of the traditional question asking whether each partner will "have" or "take" the other. 15 Marriage as self-offering is reminiscent of Rogers' theology. And indeed, Rogers sat on the SCLM sub-committee responsible for drafting the theological document accompanying the 2012 liturgy. But the work of SCLM was thoroughly collective and the liturgy and theological resources that the Commission produced cannot be traced back to one scholarly influence.

The central notion of the theological document, "Faith, Hope and Love: Theological Resources for Blessing Same-Sex Relationships" is that of Covenanted households, something, as it appears, borrowed from Bishop Thomas Breidenthal. 16 Tellingly, although the document was produced at a time when SCLM had not been authorised to use the word marriage for same-sex relationships, it was the same document that was again submitted to the 2015 General Convention, together with the request that equal marriages rites be authorised. The Commission only added responses to the theological document by people representing various backgrounds and shades of opinion in the Episcopal Church. This sufficiently shows that the Commission saw their theological work of the 2009-2012 trienium as a sufficient theological rationale for same-sex marriage rites: whether one calls same-sex unions marriage or not does not change the meaning of the blessing imparted by the church on the couple. We must note that the SCLM material was complemented by the work of the task force on marriage which strived to show that same-sex covenanted relationships were the same thing as marriage. But the fact remains that the SCLM theological document was seen as equally valid to accompany marriage rites in 2015 as they had been to accompany a liturgy which eschewed the word marriage in 2012.

It seems to me that this was made possible by the focus on the notion of covenanted households. It enabled the Commission to write about same-sex unions in the wider context of covenanted relationships of which marriage is but one example, next to monastic communities. What these various households have in common is their eschatological vocation: they are all called to "contribute to the Church's witness to the new life God offers in Christ and through the Spirit, which the Church celebrates in the 'sacraments of the new creation.'" 17 The term "household," like "covenant," has a sacramental resonance for Episcopalians. In the 1979 baptismal liturgy, celebrant and people together declare to the newly baptized: "We receive you into the household of God. Confess the faith of Christ crucified, proclaim his resurrection, and share with us in his eternal priesthood". And of course the 1979 Prayer Book puts the baptismal covenant at the heart of the baptismal rite and, one should say, at the heart of contemporary Episcopalian theology. A covenanted household is therefore an expression of what it is to be Church, caught up in the dynamic of sacramental life. "In these covenants, two people vow to give themselves bodily and wholeheartedly to each other. They do this, in part, to live out the promises of baptism while also living into the self-offering of Christ, as expressed in the eucharistic table: 'This is my body, given for you'". 18 Strikingly, next to the Prayer Book marriage service, the 1979 baptismal liturgy was an important source for the 2012 liturgy of blessing which became the 2015 inclusive marriage liturgy. The possibility for the couple to be presented to the celebrant is a fascinating rewriting of the question found in many traditional Anglican liturgies "Who giveth this woman to be married to this man?" It has followed the 1979 baptismal practice of sponsors presenting the person to be baptized, 19 the understanding being that the sponsor is someone who will support the newly baptized in their life of faith. The patriarchal giving away of the bride has been rewritten as a presentation by members of the Christian community who promise to "love, respect, and pray for N. and N., and do all in [their] power to stand with them in the life they will share." 20 The covenantal vows here are subsumed in the baptismal vows in which the whole Church share and the covenantal relationship becomes a commitment to ministry and mission.

Following Rowan Williams in his 1989 paper "The Body's Grace" and Thomas Breidenthal's reflections about sanctifying nearness, the theological document affirms the vocation of sex as fruitfulness quite independently from procreation. Rather, the vocation of sex is about radical availability and vulnerability to another, leading to a giving of oneself that mirrors Christ's own self-offering.

The movement of sexual desire toward intimacy and into commitment begins as we give ourselves over to another in faithful relation and continues toward the final moment of committal, surrendering our lives to God. This movement describes a lifelong, deliberate process that, with obedience and faithfulness, produces visible holiness and the fruits of the Spirit. Both for the good of the couple and for the good of the Church, God blesses this loving, intimate commitment. This blessing, in turn, empowers the couple for their ministry in the world and energizes the Church for mission. 21 Covenantal language enables the Commission to relate all committed relationships to the eschatological meaning of the sacramental life of the Church. While the focus was initially on the blessing of same-sex unions, many in the Commission, right from the start, felt that they were engaged in the sort of work that may lay the foundation for a renewed way of looking at different-sex marriage:

Some will find this kind of theological reflection on same-sex relationships unfamiliar and perhaps unwarranted. Many opposite-sex couples would likewise find this to be a new way of thinking about their own marital vows. Thus, General Convention Resolution 2009-C056, which called for these theological resources, becomes an opportunity for reflecting more broadly on the role of covenantal relationship in the life of the Church. 22 As the chair of SCLM indicated to me, for many Commission members, the production of the resources for same-sex blessings was seen as preparatory work in view of recasting the marriage liturgy of the 1979 Book of Common Prayer at a time when a comprehensive revision of the BCP was looming. 23 The revision of the marriage canons and the authorisation of marriage rites for both same-sex and different-sex couples in 2015 can therefore be understood as a declaration about both kinds of couples sharing in the same eschatological hope and renewal.

An incomplete process complicated by the superposition of a political and of a theological logic

In a sense, Michael Vasey's hope that gays and lesbians might help the Church recover its sense of heaven may be becoming reality in the Episcopal Church. However, the process of rethinking marriage in the light of the New Creation is far from complete. Some eschatological images around marriage still sit uncomfortably with the inclusion of same-sex relationships in holy matrimony and take us back to the lexical instability of queer religious discourse since the early 1990s to name same-sex unions. It is striking, for example, that the image of the wedding feast, a central eschatological motive which Eugene Rogers uses extensively in his writings, was completely left out of the liturgy. When I asked Ruth Meyers, the chair of SCLM, for a reason, she answered that the Commission felt the gendered language around Christ and his bride was "complicated" to deal with. A church in which women priests and bishops represent Christ at the altar can only agree with Rogers that "religious discourse works in a much richer and subtler fashion than by supposing that one has to instantiate physically what one … represents figurally." 24 But the liturgical translation of this statement is everything but easy, as the variety of attempts to name the persons of Trinity in non-patriarchal ways sufficiently indicates. 25 Eschatological bridal imagery is also traditionally steeped in a hierarchical worldview. The bride in Ephesians 5:21-33 is submitted to Christ as the woman is submitted to her husband. And in spite of the egalitarian way in which this passage is analysed by SCLM in the theological document,26 bridal imagery was not given liturgical expression. This imagery, however, is now offered to same-sex couples thanks the gender-neutral version of the 1979 marriage rite whose introduction states that the "joining of two people in a life of mutual fidelity signifies to us the mystery of the union between Christ and his Church." But the adaptation of the 1979 rite came out of the request for equal rites, that the Prayer Book liturgies be made available to gay and lesbian couples and that the new 2012 liturgy be made accessible to straight couples. The adaptation of the Prayer Book liturgy did not go much beyond using inclusive language. The only other change is the deletion of the reference to Cana in the introduction. 27 Given the theological work of SCLM and of the more recent task force on marriage, it would be very surprising if this adaptation (and indeed the 1979 rite) was not revised before long.

The present state of things reflects the superimposition of two different logics, one theological and one political, in the work that has brought the Episcopal Church to bless same-sex marriages. The secular campaign for marriage equality was paralleled in the Church by demands formulated in the language of rights, which was markably different from the theological language of the material produced by SCLM. Two distinct things in fact happened: the Episcopal Church increasingly positioned itself politically on the side of marriage equality in the US debate and it increasingly positioned itself theologically in favour of a profound rethinking of the significance, place and role in the Church of covenanted relationships.

The consultation on same-sex blessings that took place in June 2014 in Kansas City was organised by SCLM in preparation for the General Convention of 2015. The forceful demand by many participants for applying the term marriage to same-sex unions was part of an equality agenda; separate liturgies for gay and straight couples could never be equal. The reference to African American history of "separate but equal" education which turned out not to be equal was the explicit reference. The changing context of the legalization of same-sex marriage in an increasing number of states also provided for a practical objection to the continued avoidance of the term marriage. In these states, priests would marry same-sex couples on behalf of the state in the context of a liturgy which never used the word marriage: from a civil point of view, a marriage was being contracted, while from a religious point of view, something not called a marriage was celebrated simultaneously. The legalisation of same-sex marriage in all states following the decision of the Supreme Court, which was reached during General Convention in June 2015 provided an additional boost.

Discussions at the Kansas City consultation in 2014 and the in-depth conversations that I conducted with three members of my small group 28 one year later suggest that the demand for equal rites reflected a political necessity whose theological grounding was the respect for the dignity of all human beings created in the image of God. But my small group did not explicitly relate this to the theological and liturgical reflections of SCLM on the blessing of same-sex unions.

Two middle-aged gay and lesbian participants strikingly refer to the superimposition of the demand for equality and the theological work on same-sex relationships. Rev Tom expresses his personal preference for the term holy union (or holy covenant, as in the 2012 liturgy). "It was the secular LGBT community that decided that marriage was going to be the vehicle to get to more equity. Just like they chose the military service as the vehicle to get equity. Would I have chosen the military service as the vehicle? No. Would I have chosen marriage as the vehicle? No. But they did and here we are." 29 For Rev Tom, a staff consultant of SCLM, the marriage model is problematic because it is still loaded with patriarchal baggage which has traditionally made this institution something about property. He echoes many gays and lesbians who have been hesitant or have simply refused to call their unions marriages because they did not want to buy into bourgeois values and its patriarchal, property-based order. At the same time, Tom did not hesitate to get married when he was told by the Church Pension Fund that because he lived in a state which had legalised same-sex marriage, his male partner could only claim various spousal benefits if he was married to Tom. A civil union was no longer enough. Tom is now a married man who remains uneasy about the institution of marriage.

Joy shares some of Rev Tom's concerns. Unlike him, she has always thought of her union with her female partner as a marriage, but marriage defined as covenant and not as a patriarchal institution. "I want to see marriage transformed," she says. 30 Like Tom, she cannot be satisfied with the mere legal fact of marriage equality if it means gays and lesbians simply conforming to the traditional institution of marriage (which Joy describes as transactional rather than covenantal). However, she realises that her and her partner's view of marriage as a school of faith and a vocation is something that they do not share with a great number of people. She sees the theological work of the Church as essential to point the way towards a more profound view of marriage that liberates and encourages people to build a community of love that reaches beyond the couple's home. This work (which the SCLM understood as one of its most important tasks) 31 has started but it is still in its infancy and needs to be received widely. The theological document, she says, "may look alien to many people in our culture.

Even as a Church, as a whole, we're not there. I'd like us to move towards that. With baptism we've moved in a little time from a private ceremony celebrating the birth of a child to covenantal vows. I hope the same thing will happen with marriage." Marriage equality should not lead to gay and lesbian conformity to the bourgeois family model but should renew the meaning of marriage for all. And this means learning to be Church as a covenanted community, within which all types of covenanted commitments have their unique role to play.

*

Will the blessing of same-sex marriages as marriages in the Episcopal Church facilitate the process of renewing the meaning of marriage for all? This has undoubtedly been the hope of SCLM and of others in the leadership of the Episcopal Church. One must note, however, that this attempt at renewal is entering a new operating mode. Up to the last General Convention, reflection was conducted largely without the language of marriage. The prohibition had positive sides in that it forced groping for some other language that has proved fruitful. I am not only talking of the work of SCLM but of the work accomplished beforehand by various rank and file gay and lesbian Episcopalians. SCLM's work was collective in more ways than one. One of the Commission's starting points was revewing the many liturgies that had been created unofficially by or for same-sex couples, for unofficial use in local parish contexts, often with the approval of the bishop. Joy and her partner wrote their own commitment ceremony in 1999, when using the word marriage was not an option in their diocese. Parts of this liturgy inspired the rite produced by SCLM. Joy's words provide some food for thought as to how some gays and lesbians have been forced to reimagine marriage in a very deep way.

[Marriage] was always the word that meant something to us because it had a lot of resonance in a lot of ways. When we had our wedding (in 1999) we were forbidden from using that word so we had to think again and then the biblical language of covenant became meaningful. … I now refer to us as married but when I use that word, it has covenant very much embedded in it, in a way that it might not have if we 30 Interview, 29th June 2015. 31 Interview with Ruth Meyers, 7th September 2015.

had simply been able to use the word marriage and not have to think more deeply. So in some ways the Church saying 'you can't use that word' was a benefit to us. It required us to reflect and think more deeply. I won't say it was a good thing the Church did that, but as our President said last week,32 'God works in mysterious ways.'

The transformation of the meaning of marriage for Episcopalians will now have to take place in the open, calling a spade a spade, in a new post-2015 political situation. The dynamics of this transformative process will probably be different and should be watched with renewed interest by all participants in the sexuality debate in the Anglican Communion and beyond.

  22 "Faith, Hope and Love", 34.23 Interview with Ruth Meyers, 7 th September 2015. The 2015 General Convention has set the process of liturgical revision into motion by requesting the SCLM to present "a plan for the comprehensive revision of the current Book of Common Prayer" to the 2018 General Convention (Resolution A169).24 Eugene Rogers, op. cit., 235.25 See Esther Mclntosh, "The Possibility of a Gender-Transcendent God: Taking Macmurray Forward", Feminist Theology 15.2 (January 2007): 236-255. McIntosh's failure to identify or indeed put forward herself non-patriarchal ways of naming God in the liturgy that are at once poetic, recognisably Christian and fully satisfactory from a feminist point of view is telling.

See my own papers "Homosexualité, loi naturelle et rapport à l'accomplissement eschatologique de l'humain: Réflexions sur un aspect négligé du débat anglican sur l'homosexualité", Istina vol. LVIII no.

(April-June 2013) and "The Difficult Attempt to Separate Reproduction from Sexuality in Anglican Moral Theology", Reproduktionen, Salecina-Beiträge zur Gesellschafts-und Kulturkritik, ed. Gisela Engel and Nicole C. Karafyllis (Berlin: Trafo Verlag, 2005), 103-14. 2 "Creation-as-moral-package" as James Alison calls it(Faith Beyond Resentment: Fragments Catholic and Gay [London: Darton, Longman and Todd, 2001], 98-99).

James Alison, op. cit., p. 103.

See St Augustine's view of intercourse within marriage that is not exclusively for the sake of procreation in The Good of Marriage, X. 10-11.

See John Boswell, The Marriage of Likeness: Same-Sex Unions inPre-Modern Europe (London : Fontana Press, 1995), 119-120.

As distinct from gay theology, following the distinction made by Elizabeth Stuart in Gay and Lesbian Theologies: Repetitions with Critical Difference (Aldershot: Ashgate, 2003).

Vasey followed David Greenberg's analysis in The Construction of Homosexuality (Chicago: Chicago University Press, 1988).

Eugene F. Rogers, "Sanctification, Homosexuality and God's Triune Life", Theology and Sexuality:Classic and Contemporary Readings, ed. Eugene F. Rogers (Oxford: Blackwell, 2002), 221.

Eugene F. Rogers, op. cit., 223.

Eugene F. Rogers, op. cit., 224.

Rogers makes great use of Rowan Williams's famous essay "The Body's Grace".

This giving of oneself could already be found in the vows of the two liturgies designed for same-sex couples by the Anglican diocese of New Westminster in Canada in2000 and 2003.

The brackets refers to an alternative text for the vows which deletes the references to the grace of God, the love of Christ and the Spirit's help to make it easier for couples in which one of the partners is not a Christian to use the liturgy.

This detail was reported to me by Ruth Meyers, chair of SCLM.

This is the case in all versions of the English and American Books of Common Prayer.

Thomas E. Breidenthal, Christian Households: The Sanctification of Nearness (Eugene, Oregon :Wipf and Stock, 2004).

"Faith, Hope and Love: Theological Resources for Blessing Same-Sex Relationships", I Will Bless You and You Will Be a Blessing: Resources for the Witnessing and Blessing of a Lifelong Covenant in a Same-Sex Relationship (New York: Church Publishing, 2012), 48.

"Faith, Hope and Love", 48.

The chair of SCLM explained that initially the Commission had even kept the term 'sponsors' and only replaced it by 'presenters' when they realised that it was misunderstood as patronizing by some same-sex couples who were given one first draft of the text for feeback. (Interview with Ruth Meyers, 7th September

2015).20 The 1979 Book of Common Prayer provides for the possibility of a presentation, but the parallel with Baptism is tenuous, especially because the presenters do not make any vows (distinct from the general affirmation of support given by the assembly) to love and support the couple. 21 "Faith, Hope and Love", 49.

"Faith, Hope and Love", 48.

After the Charleston Church shooting on 17 th June 2015, Barack Obama referred to the failure of the killer to deepen racial divisions in the United States. Instead, some national coming together had taken place, showing the power of God's grace to bring about good out of evil.