Guillaume Strub
email: guillaume.strub@isl.eu.

Spilios Theodoulis
email: spilios.theodoulis@isl.eu.

Vincent Gassmann
email: vincent.gassmann@isl.eu.

Simona Dobre
email: simona.dobre@isl.eu.

Michel Basset
email: michel.basset@uha.fr

Pitch Axis Control for a Guided Projectile in a Wind Tunnel-based Hardware-In-the-Loop Setup

This article details the design of a pitch axis autopilot for an 80mm fin-stabilized, canard-guided projectile and its validation on a Hardware-In-the-Loop test setup. This setup is built around an autonomous projectile prototype, which is installed in the test section of a wind tunnel by the means of a 3-DoF gimbaled structure. The autopilot design is based on a family of linearized dynamic models of the projectile, whose parameters were estimated from experimental data. Several control approaches are considered. Using the H ∞ robust control framework, a full-order disturbance rejection controller is designed, taking into account the limited actuator dynamics. A fixed-order, fixed-structure controller of lower complexity with the same performance objective is also designed, using a nonsmooth H ∞ technique. The tracking performance of these controllers is improved with the addition of a feedforward controller. A final approach considers the disturbance rejection and reference tracking as a multi-objective problem, where the feedback and the feedforward controllers are designed in a single step. The performance of these approaches is then assessed and compared using numerical simulation as well as experimental results gathered from the Hardware-In-the-Loop setup.

in terms of accuracy, range, collateral damage and round expenditure). The development process for such weapons is based on the traditional missile design procedure, which can be long and costly. As the design of the control laws is usually done using model-based techniques, an accurate model of the projectile's dynamic behavior is crucial. The parameters of such models are traditionally quantified using various techniques such as semi-empirical aerodynamic prediction codes, wind tunnel testing, Computational Fluid Dynamics (CFD) predictions and/or free-flight tests. However, these techniques may lack precision for some airframe configurations or flight conditions and are usually expensive in terms of time and cost.

Experimental validation and testing of the designed control laws is usually done through free-flight tests or Hardware-In-the-Loop simulations on flight tables. The former technique requires a gun-hardened readyto-fly prototype and a sufficiently mature controller design, therefore it is unsuitable for initial testing. The latter technique consists in using a functional prototype with a 3-DoF actuation system and simulating the aerodynamic environment using a model. Although they are much more flexible and less expensive than free-flight tests, such simulations are limited in accuracy regarding the free-flight dynamic behavior.

The present paper proposes a novel, low-cost technique applicable to parameter estimation and control algorithm validation for guided projectiles, trying to overcome the shortcomings associated with the above two techniques. To this end, a fully functional guided projectile prototype based on low-cost, off-the-shelf hardware is installed in the test section of a wind tunnel by the means of a 3-DoF gimbaled support structure, enabling rotation on the roll, pitch and yaw axes. This setup, illustrated in Figure 1 and referred to as ACHILES (Automatic Control Hardware-In-the-Loop Experimental Setup), provides with realistic flight conditions in a controlled experimental environment and enables rapid algorithm development and testing through the use of industry-standard software.

To the author's knowledge, there are very few similar setups targeted at identification and control studies for guided projectiles and similar systems. In Hann et al. [START_REF] Hann | Minimal modeling of rocket roll response and aerodynamic disturbance in wind tunnel[END_REF], the parameters of a sounding rocket roll dynamics model are estimated from experimental data gathered using an actual prototype in a vertical wind tunnel. Fresconi [START_REF] Fresconi | Projectile Roll Dynamics and Control with a Low-Cost Maneuver System[END_REF] proposes a guided projectile prototype similar to our proposal, with an emphasis on the use of low-cost, off-the-shelf components and their resistance to high accelerations. Using aerodynamic data from PRODAS studies, a linear quadratic optimal roll controller is designed and tested on the projectile prototype in a wind tunnel. In both cases, the experimental setups in these studies only allows motion on the roll axis, whereas ACHILES allows the three angular degrees of freedom. In Patel et al. [START_REF] Patel | Closed-Loop Missile Yaw Control via Manipulation of Forebody Flow Asymmetrics[END_REF], a missile flow-control-based yaw actuator is devised. A closed-loop controller modulates flow effectors to produce a yawing moment by exploiting flow asymmetry around the missile forebody. Dynamic tests in a wind tunnel demonstrated the ability of the device to generate and maintain a range of yawing moments during high-incidence pitch sweeps.

Previous work by the authors [START_REF] Strub | Hardware-In-the-Loop Experimental Setup Development for a Guided Projectile in a Wind Tunnel[END_REF] focused on the identification of ACHILES' pitch axis dynamics. Using flight mechanics principles, a nonlinear pitch axis model is constructed, then linearized around a family of equilibrium points. The projectile is then stimulated around each operating point and the model parameters are estimated using the collected data, resulting in a family of linear models. Results have shown that parameters do not vary significantly along the flight envelope for a fixed airspeed, allowing a single-model uncertain representation of the pitch dynamics.

In this paper, this knowledge is leveraged so as to design a pitch axis autopilot. The main purpose of this control system is twofold: it has to ensure good tracking of the reference signal while minimizing the effects of disturbances on the system output. The H ∞ framework has been used successfully for missile flight control design [START_REF] Nichols | Gain scheduling for H ∞ controllers: a flight control example[END_REF][START_REF] Apkarian | Self-scheduled H ∞ control of missile via linear matrix inequalities[END_REF][START_REF] Theodoulis | Missile autopilot design: gain-scheduling and the gap metric[END_REF], which share several characteristics with the ACHILES projectile prototype. These techniques are well-suited to this task, as they take into account limiting factors, e.g. modeling uncertainty, operating condition variations, external disturbances or limited actuator and sensor bandwidth, while guaranteeing system stability and the desired performance. Moreover, recent advances in H ∞ control, such as nonsmooth H ∞ synthesis [START_REF] Apkarian | Nonsmooth H ∞ Synthesis[END_REF] and multi-objective control [START_REF] Gumussoy | Multiobjective Robust Control with HIFOO 2[END_REF][START_REF] Apkarian | Multi-Model, Multi-Objective Tuning of Fixed-Structure Controllers[END_REF], enable the design of low-complexity, high-performance autopilots. As it is desirable to have a low-order control system, three design approaches with decreasing complexity are proposed and compared using numerical simulation as well as Hardware-In-the-Loop experiments.

This paper is organized as follows: Section II presents the hardware and software components of the ACHILES experimental setup. In Section III, an uncertain model of the body dynamics is built using estimated models, and the actuator model is presented. Section IV details the design of the pitch autopilot, using three approaches with decreasing controller complexity. The robustness of the designed control laws is assessed in Section V and in Section VI the proposed controllers are implemented and compared on the projectile prototype. Fig. 2 Inside view of the ACHILES projectile prototype. The IMU, which is on the other side of the main electronics boar, is not pictured here.

II. Experimental Setup

The ACHILES projectile prototype is based on a fin-stabilized 80mm caliber shell with four nose-located canards, of which an inside view is given in Fig. 2. This missile-like structure has several advantages for identification and control investigations: as the frame is stabilized with tail fins, the stability only depends on the geometry of the aerodynamic surfaces, as opposed to a gyrostabilized projectile. Moreover, the crossaxis coupling as well as the mechanical complexity are greatly reduced since there is no need for a high spin rate. The projectile is installed in the test section of a subsonic wind tunnel by the means of a 3-DoF gimbaled structure as shown in Fig. 1. This structure enables rotation along the roll, pitch and yaw axes while restraining all linear motion.

The projectile is controlled using aerodynamic surfaces (canards), which provide good control authority.

The canards are independently driven by modified COTS R/C servomotors, which have been retrofitted with a custom control board in order to improve their linearity and dynamic behavior. A Microstrain 3DM-GX3 Inertial Measurement Unit (IMU) provides the control algorithms with an estimate of the projectile's angular attitude and velocities, and is based on a set of three accelerometers, magnetometers and gyrometers and an embedded sensor fusion algorithm.

In order to speed up algorithm development, a rapid prototyping environment based on MAT-LAB/Simulink has been developed. The algorithms are designed as Simulink diagrams, using standard blocks as well as custom blocks for interacting with the projectile's actuators and sensors. These diagrams are then converted to C code and compiled using the MATLAB Coder and Simulink Coder toolboxes. The resulting software is executed onboard the projectile, which features a realtime Linux distribution running on

δ u G A k b G B H B d G S y =               α q               δ y δ m + + Projectile Actuators Sensors G Fig. 3 Open-Loop plant model G.
an ARM-based Gumstix embedded computer. A more detailed review of the hardware components and the software stack is given in Ref. [START_REF] Strub | Hardware-In-the-Loop Experimental Setup Development for a Guided Projectile in a Wind Tunnel[END_REF].

III. Modeling

The open-loop pitch dynamics G of the ACHILES prototype can be represented as a series interconnection of the actuator model, the body model and the sensors model, as highlighted in Fig. 3. These models are detailed in the following paragraphs.

A. Actuators

The ACHILES actuators are based on Hobbyking HK47010 metal-geared servomotors, where the stock control board has been replaced with a custom microcontroller-based control board and an AS5045 12-bit magnetic absolute position sensor. The servos are linked to the control fins using a belt drive with reduction ratio k b = 16/18 ; the belt dynamics are not considered here for simplicity. As a complete control study has been conducted on the actuators, a model of their dynamic behavior can be built using this knowledge [START_REF] Strub | Pitch Axis Control for a Guided Projectile in a Wind Tunnel-based Hardware-In-the-Loop Setup[END_REF].

To this end, the complete servo model is represented as a block diagram in Fig. 4 and the corresponding transfer function is

G A = k pos K(k p s + k i ) τs 3 + (1 + Kk p )s 2 + K(k i + k pos k p )s + Kk pos k i ( 1
)
where the parameter values are given in Table 3 (see Appendix).

The open-loop servo dynamics are represented with a first-order model K/(1 + τs), where τ captures the mechanical time constant of the motor and the gear train. The implemented control system consists in a cascade of a proportional position controller k pos and a proportional-integral velocity controller k p + k i /s.

These parameters have been tuned using a nonsmooth H ∞ mixed-sensitivity design procedure [START_REF] Apkarian | Nonsmooth H ∞ Synthesis[END_REF], which matches the closed-loop response to a second-order system T A ref with bandwidth ω 0 = 50 rad/s and an optimal damping ζ = 0.78. As an additional constraint, the motor control voltage must remain under the supply voltage at all times, for any reference step up to ±60 • in order to avoid saturation.

K 1 + τs 1 s k p + k i s k pos δ u δ y + + δy - -
This controller is implemented on the custom servo control boards with the addition of a simple antiwindup scheme, which consists in limiting the integral error to the supply voltage. This simple modification avoids controller overshoot in the event of output saturation. Figure 5 shows the normalized actuator step response for various amplitudes, which presents a 5% response time of 100 ms. Note that for a step larger than the design value of 60 • , the response is no longer linear but does not present overshoot. Thanks to the chosen position sensor, the output fin position is accurate to ±0.44 • , with 0.078 • resolution and 0.027 • repeatability. The difference in time response among all four actuators is less than 10 ms, which is negligible with respect to the airframe dynamics.

B. Body Dynamics

The projectile dynamics are subject to the flight mechanics principles [START_REF] Zipfel | Modeling and Simulation of Aerospace Vehicle Dynamics[END_REF], which lead to a nonlinear parameter-dependent model. In the present case, this model is simplified thanks to the absence of linear motion and the following assumed hypotheses for the study of the pitch axis: the airspeed V is held constant and the other axes are locked at zero, therefore p = 0, r = 0, φ = 0 and ψ = 0, where ω = [p q r] is the body rate vector and φ and ψ denote respectively the body roll and yaw angles. Furthermore, in this experimental setup the altitude h is constant (ground level). Figure 6 presents the projectile in a pitching motion and the remaining parameters, which are the angle of attack α, which is equal to the pitch angle θ, the pitch rate q and the control fin deflection δ m .

The pitch axis nonlinear model is linearized around a familiy of equilibrium (or trim) points, resulting in a second-order quasi-LPV model. In order to account for aerodynamic disturbances and measurement noise, a second-order disturbance model is appended to the projectile quasi-LPV model with the same poles. For any given operating point ρ = [ ᾱ V] , this model can be written as:

ẋδ (t) = A(ρ)x δ (t) + B G (ρ)δ m,δ (t) + B H (ρ)d(t) y δ (t) = C(ρ)x δ (t) + D G (ρ)δ m,δ (t) + D H (ρ)d(t) (2)
For simplicity, the δ subscript will be omitted for the rest of the article. The system matrices are:

              A B G B H C D G D H               =                                    0 1 0 M αd M qα M qq M qδ M qd 1 0 0 0 0 1 0 M αd                                    (3)
In the above matrices, the parameters M qα , M qq and M qδ describe the dynamic behavior of the projectile and are related to the derivatives of the pitch aerodynamic coefficient C m at the considered operating point.

Parameters M αd and M qd describe the noise model and are scaled for a unity variance white noise signal d(t). These parameters have been estimated for different values of the trim angle of attack ᾱ from collected experimental data, using an estimation procedure based on numerical optimization [13] [14]. The estimated values of the parameters, and the corresponding uncertainty, are given in the appendix (Table 4).

Although the values of the dynamic model parameters (M qα , M qq and M qδ ) are dependent on the trim angle of attack ᾱ, the maximum variation does not exceed 45 % in the worst case. Furthermore, the corresponding second-order transfer function parameters (gain, corner frequency and damping) are shown in Table 5. The highest variation is on the damping ζ, which varies from 0.21 at ᾱ = 0 • to 0.13 at ᾱ = 7 • (±21.6% variation wrt the mean value), while the gain k and corner frequency ω 0 remain almost constant (less than 10% variation). Therefore, an adaptive or gain-scheduled controller may not be required for the constant speed case. The proposed approach consists in designing the autopilot for a nominal model, built using the mean estimated parameter values

θ = M qα M qq M qδ M αd M qd = -53.0712 -2.4470 38.8179 0.0399 2.3232 (4)
and in representing the deviations from this nominal model as uncertainty.

C. Sensors

The sensor model takes into account the bandwidth and noise contribution of the Inertial Measurement Unit. In this study, both effects are negligible and the IMU can be modeled by the unit transfer function: in terms of dynamics, the bandwidth of the sensors is much higher (over 200 Hz) than the system open-loop bandwidth (less than 2 Hz). The noise level has been assessed experimentally by comparing the variance of the pitch signal over a period of 10 seconds for two cases. In the first case, the projectile is at rest (zero airspeed), while in the second case the airspeed is raised to V = 25 m/s. As the variance in the latter case (4.10 -2 ) is two orders of magnitude higher than the former (3.10 -4 ), the noise contribution of the IMU can safely be neglected. Finally, a separate IMU model is neither necessary nor desirable since the body model is estimated from recorded IMU measurement, taking them into account the sensor dynamics.

IV. Autopilot Design

This section details the design of an angle of attack autopilot using the nominal model postulated in Section III, which is used throughout the pitch flight envelope and at a constant airspeed. The design goal is to obtain a low-order fixed-structure controller in order to simplify implementation as well as to prepare the ground for a future airspeed scheduling, while maintaining high performance levels.

G k d k δr d δ r α q k q + δ u - G P

A. Objectives and Limitations

The pitch axis autopilot aims at fulfilling several complementary objectives. The main goals are to augment the dynamic stability of the system and to track the reference AoA signal issued by guidance algorithms, while rejecting external disturbances such as wind gusts. The autopilot must also guarantee robust stability in the presence of uncertainty, such as neglected or unmodelled dynamics, or uncertainty on aerodynamic coefficients. Some of these goals are conflicting and lead the designer to a compromise, since improving performance degrades stability margins, and vice versa.

There are several limitations that must be taken care of during the design procedure. The resulting autopilot is to be implemented in Hardware-In-the-Loop experiments, where the control algorithms are executed on the embedded computer at an update rate F s = 100 Hz, and where the target software introduces a one-cycle controller delay [START_REF] Strub | Hardware-In-the-Loop Experimental Setup Development for a Guided Projectile in a Wind Tunnel[END_REF]. The actuators have limited bandwidth and a limited saturation-free operating range as detailed in Section III. All these considerations limit the achievable closed-loop bandwidth and require the designer to keep a sufficient delay margin in the closed-loop autopilot system.

In the present case, the objective is to attenuate the effects of disturbances on the output α and pro-

B. Rate Damping Controller

One of the first functions of the autopilot is to improve the pitch plane dynamic stability of the projectile.

In the field of missile and aircraft control design, this is accomplished by virtually increasing the aerodynamic damping derivative M qq through the use of a rate feedback loop [START_REF] Zipfel | Modeling and Simulation of Aerospace Vehicle Dynamics[END_REF], as presented in Fig. 7. In the present case, the open-loop nominal plant is largely underdamped with a damping ratio ζ G = 0.17. Using a pole placement approach, the feedback gain k q is selected such that the closed-loop damping approaches 0.7, while keeping the natural frequency of the actuator and body poles separated by a factor of at least three, as shown on Obviously, the rate damping controller can not constitute a tracking autopilot on its own, as any plant gain uncertainty will lead to a static error, and the system output is very sensitive to disturbances. Nevertheless, the system damping is vastly improved and there is an overall good agreement between the simulated and the experimentally observed responses. The difference between these two responses is caused by aerodynamic disturbances on the real system, which are not present in the simulation.

C. Angle of Attack Autopilot

The proposed angle of attack autopilot structure detailed throughout this section is illustrated in Fig. 10.

It is composed of two separate controller blocks, resulting in a two-degrees of freedom control structure.

The disturbance rejection feedback controller K FB acts on the tracking error signal e = α f -α and produces G the virtual control input δ r . The main role of K FB is to maintain closed-loop system stability and minimize the effects of the disturbance d on the system outputs α and q. Proper reference tracking is ensured by the feedforward controller K FF , which shapes the reference signal α r such that the achieved angle of attack α follows the response of a reference model T ref .

k d k δr d k q G P K FB K FF α r α + δ u - α f + e δ r -
In the following paragraphs, the feedback controller is designed using two H ∞ approaches, leading respectively to a full-order controller and a fixed-order fixed-structure controller for comparison purposes.

The feedforward controller is obtained first by closed-loop system inversion followed by a reduction step, and in a final step the design of both feedforward and feedback controllers is done simultaneously using a multi-objective approach.

Disturbance Rejection Controller

In the autopilot structure presented in Fig. 10, the feedback controller K FB is entirely responsible for rejection of external disturbances, as the feedforward controller K FF does not act on the tracking error. This paragraph focuses on the design of K FB , using modern robust H ∞ control techniques in a S/KS mixed- 11 Mixed S/KS sensitivity disturbance rejection problem in standard form.

G P r = 0 W 2 W 1 z 1 z 2 w = d K FB α - + v u z P Fig.
sensitivity setup [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF].

The H ∞ design problem is represented in the standard form interconnection, illustrated in Fig. 11. To this end, the design objectives and constraints are included in the augmented plant P, of which the inputs are the exogenous signal w = d (disturbance) and the control input u = δ r . The outputs of P are the performance

signals z = [z 1 z 2 ]
, which are to be minimized, and the measured variable v = α. The H ∞ control problem then consists in finding K FB , such that the system is nominally stable and

||T w→z || ∞ < γ, γ > 0 (5)
where T w→z is the transfer function from w to z, which is obtained by the lower linear fractional transformation (LFT) T w→z = F l (P, K FB ).

The S/KS mixed-sensitivity problem consists in shaping the closed-loop system sensitivity function S d→e , which is the transfer function between the disturbance d and the regulation error e, and K FB S d→e , which is the transfer function between the disturbance d and the control input u and corresponds to the control effort.

The body disturbance model H B acts as a low-pass filter on the disturbance signal d, which will be effectively rejected if the sensitivity function S d→e presents small gain at low frequencies. Another requirement is to limit the controller effort, as the actuators have limited bandwidth and are prone to saturation in case of high-amplitude input signals. Therefore, the controller should limit both the amplitude and the bandwidth of the control signal u. These requirements are implemented using the weighting filters W 1 and W 2 , which are here chosen as first-order systems of the form:

W i = s + ω i k i,HF s + k i,LF ω i (6)
The inverse of these weighting filters roughly correspond to the desired shape of the closed-loop transfer functions, and therefore the parameters specify the shape of the inversed filter, where k i,HF is its gain at high frequencies, k i,LF is its gain at low frequencies and ω i its -3dB bandwidth frequency.

The parameters of the filter W 1 acting on S d→e are selected such that its inverse exhibits a high-pass behavior. More specifically, the low-frequency gain k 1,LF must be low to reduce steady-state error but must be greater than zero in order to avoid numerical issues in the resolution of the H ∞ problem. The highfrequency gain k 1,HF and the bandwidth ω 1 define the tolerated disturbance amplitude and bandwidth. The weighting filter W 2 acts as a roll-off filter to limit the control gain after a certain frequency, which should be lower than the actuators bandwidth. The low-frequency gain k 2,LF must be greater than one, to allow controller action in the low frequencies, and the high-frequency gain k 2,HF is subject to the same numerical requirements as k 1,LF . The selection of the weighting filter parameters is a process subject to trial-and-error,

where the designer has to make a trade-off between effectiveness of disturbance rejection, control signal energy and open-loop stability margins. The finally selected filter parameters are summarized in Table 1, and the resulting performance transfer function T w→z is:

T w→z =               S d→e • W 1 K FB • S d→e • W 2               (7)
As a first approach, the H ∞ problem of Eq. ( 5) is first solved using classical techniques (solving a Riccati equation [START_REF] Doyle | State-space solutions to standard H 2 and H ∞ control problems[END_REF] or an LMI formulation [START_REF] Gahinet | A linear matrix inequality approach to H ∞ control[END_REF]), resulting in a full-order controller structure. The high-frequency modes are subsequently eliminated using model reduction techniques, and the controller is adjusted so that the lowest frequency pole is moved to zero, thereby eliminating steady-state error. In the end, a fifth-order controller K FB,FO is obtained. However, full-order controllers are usually not suitable for implementation on low-power control electronics, and are difficult to integrate in a future gain-scheduling scheme. Here, this controller serves as a performance standard for comparison against reduced-complexity controllers.

In the second approach, a fixed-order, fixed-structure technique is employed, resulting in a reduced controller order and hence a much simpler structure, which can be imposed by the designer. In guided projectile control design, PI-type structures have been used successfully in stabilizing and reference tracking controllers [START_REF] Sève | Pitch/yaw channels control design for a 155mm projectile with rotating canards, using a H ∞ loop-shaping design procedure[END_REF]. The PID controller provides additional damping through the derivative term, however the derivative must be approximated using a low-pass filter such that the controller is proper and so as to limit the high-frequency gain. The fixed-structure controller K FB,FS to be synthesized here is then of the form:

K FB,FS = k p + k i 1 s + k d s T f s + 1 ( 8
)
with parameters k p , k i , k d , and T f to be determined. The H ∞ problem is then non-convex, and the nonsmooth H ∞ synthesis method of Apkarian and Noll [START_REF] Apkarian | Nonsmooth H ∞ Synthesis[END_REF] is employed to compute the fixed-order, fixedstructure controller K FB,FS .

Reference Tracking Controller

The above designed feedback controllers are focused on disturbance rejection. Even if the closed-loop system has no steady-state error, the time response does not conform to the specified requirements because the complete system should behave like the reference model T ref .

A straightforward approach for designing the feedforward filter K FF consists in connecting the reference model and the inverse of the closed-loop transfer function T α f →α in series [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF]:

K FF = T ref T -1 α f →α (9)
Obviously, the resulting system is non-proper and cannot be directly implemented on the real system.

The approach employed to circumvent this limitation consists in approximating the magnitude frequency response of the ideal feedforward on a specific frequency band with a fixed-order transfer function. As the ideal feedforward filter behaves as a derivative filter at high frequencies, it is desirable to use a strictly proper transfer function (i.e. that has more poles than zeros), so as not to excite high-frequency dynamics of the closed-loop system. The frequency band must also be chosen with care as the high-frequency dynamics of the ideal feedforward filter may not be relevant and usually impose a higher-order transfer function and high gains. The selection of these parameters therefore results in a trade-off between model-matching accuracy (minimizing the difference between the reference model and the closed-loop behavior) and controller/actuator effort. However, this inverse-based design procedure may result in a too high model-matching error for low-order systems, and the designer's actions are quite limited. Moreover, while the disturbance rejection controller satisfies the constraints on the actuator control signal, the closed-loop system with feedforward may violate these specifications. It is more desirable to design the feedback and the feedforward controllers in a single step, resulting in a 2-DoF controller such that all constraints are applied during synthesis, as presented in the next subsection.

Multi-objective Synthesis

A naive approach to the design of the 2-DoF controller would be to add the feedforward controller and associated model-matching and controller effort constraints to the mixed-sensitivity problem of Fig. 11. The exogenous input vector w then consists in the reference input r as well as the disturbance input v. However, it is not possible to define the weighting filters W i for individual transfers, as all inputs are used during synthesis. For example, the performance signal relative to the model-matching error would include transfers from both inputs, when only the reference is relevant. This limits the designer's actions and leads to an unsatisfactory result.

As the simultaneous feedback and feedforward design problem addresses different objectives, a better formulation lies in multi-objective methods [START_REF] Apkarian | Multi-Model, Multi-Objective Tuning of Fixed-Structure Controllers[END_REF][START_REF] Scherer | Multiobjective output-feedback control via LMI optimization[END_REF]. In these frameworks, the different requirements are specified in multiple w j → z j channels for performance assessment, where the exogenous inputs w j and performance outputs z j are vectors. The resulting T w j →z j transfers specify these requirements independently of each other, as opposed to the case discussed previously. The synthesis procedure then consists in finding K such that the nominal system is stable and all requirements are satisfied simultaneously:

                                   T w 1 →z 1 T w 2 →z 2 . . . T w j →z j                                    ∞ < γ, γ > 0 (10)
In the present case, two w j → z j channels are considered. The first channel, w The roll-off filter W 22 acts on the transfer T r→u and is specified in the same way as in the disturbance rejection case. However, the high-frequency limitation mostly impacts the feedforward filter and its bandwidth may be much larger than in the disturbance rejection case, as it will not compromize stability. The numerical values of the parameters of W 21 and W 22 are summarized in Table 1, and the performance transfer function Fig. 15 Open-loop gain and phase margins for the disturbance rejection controllers (blue: full-order controller K FB,FO , red: fixed-structure controller K FB,FS , green:fixed-structure controller K FB,MO in the multi-objective case).

G K FB K FF r y + e u - G K FB K FF r y + + e + u -
T w→z is:

T w→z =                                    S d→e • W 11 0 K FB • S d→e • W 12 0 0 (T ref -T r→α ) • W 21 0 K FB • S r→e • W 22                                    (11)
The multi-objective synthesis problem of Eq. ( 10) is expressed in the standard form in Fig. 13 for the two-channel case. The 2-DoF controller to be synthesized is composed of the feedback controller K FB,MO and the feedforward controller K FF,MO , where the structure of K FB,MO is defined in Eq. ( 8). In the first approach, the feedforward controller K FF,MO is designed in filter-form, as in Fig. 10. However, an injection-form feedforward, as illustrated in Fig. 14, appears to achieve a similar level of performance with a lower-order structure and the design procedure behaves better numerically. In the end, the best performing structure is the first-order lead filter

K FF,MO = k FF,MO s + z FF,MO s + p FF,MO (12)
of which an equivalent series form is given by:

K FF,MO = 1 + K FF,MO K -1 FB,MO (13)

D. Controller Synthesis and Simulation Results

Disturbance Rejection Controller

The results of the feedback controller synthesis, using the three presented methods, are summarized in Table 2. In all cases, the minimum γ is less than 1.4, and the gain and phase margin requirements are met.

The open-loop margins, shown in Fig. 15, have been computed for two loop opening sites: in the outer loop, at the output of K FB and in the inner rate damping loop, at the actuator input δ u .

The sensitivity transfer functions with respect to the disturbance input, as well as the controller Bode diagram and the closed-loop system response to a unit disturbance step are illustrated in Fig. 16. In these plots, the solid red and blue lines correspond respectively to the 1-DoF full-order and fixed-structure controllers K FB,FO and K FB,FS , and the green line corresponds to the feedback controller K FB,MO computed using the multi-objective procedure. In Fig. 16a and 16b, the dashed black line corresponds to the inverse of the weighting filters.

As specified in Section IV C, all feedback controllers were synthesized against the same requirements.

The full-order controller K FB,FO corresponds to the best achievable result, as there are no constraints on its structure. Nevertheless, it can be seen that both fixed-structure controllers exhibit similar performance, despite a less complex structure. As expected, the low-frequency error sensitivity S d→e tends to zero at low frequencies and does not exceed the upper limit specified in W 11 , and the control sensitivity K FB S d→e tends to unity at low frequencies and zero at high frequencies. However, the controllers do not roll-off at high frequencies as the effect of the disturbance d is already filtered out by the disturbance model H B . Therefore, the spectral content of reference signal r must be limited in order to limit the injection of high-frequency content in the feedback loop. This is guaranteed by the roll-off action of the feedforward controller.

Finally, Fig. 16d presents the closed-loop system response to a unity disturbance step against the openloop damped system, of which the response is represented by a black dashed line. The maximum deviation due to the disturbance is less than 0.45 • , and the system reaches steady-state after about 1 second, with no static error.

Reference Tracking

The closed-loop transfer functions relative to the reference tracking objective, as well as the feedforward controller Bode diagrams and the autopilot step response, are illustrated in Fig. 17. These plots correspond to the before three considered controller structures: first the 1-DoF full-order and fixed-structure feedback controllers K FB,FO and K FB,FS with inverse-based feedforward controllers K FF,FO and K FF,FS , respectively in solid blue and red, and second the fixed-structure feedback and feedforward controllers K FB,MO and K FF,MO computed using 2-DoF multi-objective synthesis in solid green.

In Fig. 17a and Fig. 17b, the model-matching error e MM = T ref -T r→α and control sensitivity transfer T r→u are compared to the respective design constraints 1/W 21 and 1/W 22 . In these figures, both inverse-based feedforward controllers violate the model-matching constraint but meet the control sensitivity requirement within a good margin. It must however be noted that these constraints are only effective during multiobjective synthesis, and the results obtained using inverse-based feedforward controllers are here presented only for comparison. injection feedforward controller K FF,MO is used for comparison. As expected, all ideal inverse-based feedforward controllers have a similar shape, except in the high frequencies. The controllers obtained through approximation exhibit an identical response, while the multi-objective feedforward controller rolls off at higher frequencies, which is the best trade-off between the model-matching and the control sensitivity constraints respectively enforced with W 21 and W 22 .

Finally, the time-domain performance is assessed in Fig. 17d by comparing the system response to a unit reference step against the response of the reference model T ref .

Of the three designed autopilots, only the multi-objective controller follows the reference model specification, with a response time of 0.64 ms.

The two controllers obtained using the two-step approach achieve respective response times of 0.91 ms and 0.96 s for the full-order and the reduced-order controllers. In all three cases, the closed-loop response does not present steady-state error.

The inverse-based feedforward controllers K FF,FO and K FF,FS were designed according to Section IV C 2, 18 Body dynamics model with multiplicative uncertainty. using 2-pole, 1-zero transfer functions to approximate Eq. ( 9) for frequencies ranging from 0.1 rad/s to 10 rad/s. As observed in Fig. 17c, the obtained filters fit the ideal inverse-based feedforward in the specified frequency band. These specifications were found using trial-and-error and result in a trade-off between controller complexity, model-matching error and controller effort. If a better conformance to the design constraints is required, the inverse-based design method is not as effective as multi-objective 2-DoF design, as in the latter case the requirements are taken into account during the synthesis of both controllers, while in the former case they are verified a posteriori.

Ḡq ∆ I w I δ m 1 s α + q + Fig.

V. Robustness Analysis

An important part of the H ∞ robust design process is to check whether the design specifications are respected even in the presence of uncertainty. This uncertainty in the plant may arise from many sources, such as neglected or unmodelled dynamics, uncertain parameters estimates or changes in operating conditions. To this end, the first step consists in modeling the uncertainty in the plant. In this section, the uncertain plant model is constructed using a multiplicative unstructured uncertainty representation, based on the difference between the nominal model and the family of observed models, as presented in Section III B. In a second step, the robust stability of the autopilot is assessed: the system must remain stable for all perturbations in the uncertainty set.

A. Uncertainty Modeling

The controllers were designed using a nominal "virtual" dynamics model based on mean estimated parameter values, as proposed in Section III B. In this paragraph, the proposed uncertain model takes into account the operating point variations as well as the uncertainty on estimated parameters.

In the body dynamics model of Eqs (2)-(3), the pitch rate q is the derivative of the angle of attack α. As the uncertain parameters only affect q, this model can be rewritten as a SISO uncertain pitch rate dynamics second-order weight w I satisfying Eq. ( 16).

model G q , followed by an integrator, as illustrated in Fig. 18. A commonly used uncertainty representation is the multiplicative input uncertainty [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF], where the perturbed plant form is:

G q (s) = Ḡq (s) [1 + w I (s)∆ I (s)] |∆ I ( jω)| ≤ 1, ∀ω (14)
where Ḡq is the nominal pitch rate dynamics model, ∆ I is any stable transfer function which is less than 1 in magnitude at each frequency. The multiplicative uncertainty weight w I is a minimum-phase rational transfer function that bounds the relative difference between the nominal plant and the set of all observed realizations.

This set contains models estimated from experimental data as well as bootstrapped models, which were used to compute parameter uncertainties in Table 4. For each operating point, the bootstrapped models are estimated from fictive data, built from the nominal response and a random permutation of the model residues [START_REF] Walter | Identification of Parametric Models from Experimental Data[END_REF]. In the end, the set of observed models contains the 8 measured models and 30 bootstrapped models for each operating points, for a total of 248 models. At each frequency, the uncertainty radius I (ω) is:

I = max G q (s) G q (s) -Ḡq (s) G q (s) (15)
and the uncertainty weight w I satisfies

|w I ( jω)| ≥ I (ω), ∀ω (16)
In the present case, w I is selected as a second-order transfer function, i.e. the same order as the system.

Figure 19 presents the obtained multiplicative weight against the relative error and uncertainty radius as per Eq. ( 15).

B. Robust Stability

This subsection focuses on the stability of the proposed autopilots in the presence of uncertainty. To this end, the closed-loop plant is rewritten using the uncertain dynamics model given in Section V A. As the actual autopilots are to be implemented as a discrete-time control system, an additional 20 ms delay is inserted before the actuator input to account for the control computation delay and the zero-order hold, which arise from the discretization of the controllers.

In order to assess robust stability, the closed-loop system with uncertainty can be represented in the generic M -∆ structure of Fig. 20, where M is the transfer function from the output to the input of the perturbation ∆ I . As the nominal system M(s) and the perturbation ∆ I (s) are stable, the robust stability condition is, in the SISO case [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF]

|M( jω)| < 1, ∀ω (17)
which is a necessary and sufficient stability condition. The complete autopilot in M -∆ form is represented in Fig. 21, and the transfer function M(s) for each feedback controller designed in Section IV is represented in Fig. 22. Note that the feedforward controller as well as the disturbance model are not included, as they are not part of any feedback loop. In all three cases, the closed-loop system is robustly stable to model uncertainty since the magnitude of the M transfers remain under 1 at all frequencies. The overall autopilot is robustly stable even if the feedback controllers were designed without taking the actual control delay into account.

VI. Experimental Results

This section deals with the validation of the multi-objective autopilot proposed in Section IV on the ACHILES test setup. The two other controllers will not be considered in the following paragraphs, as they exhibit higher complexity and lower performance than the multi-objective autopilot. The constraints related to the implementation on the embedded computer are first discussed, then the obtained experimental results are related to numerical simulation results and discussed.

A. Hardware-In-the-Loop Implementation

As presented in Section II, the ACHILES test setup constitutes a novel means of testing projectile control laws using real hardware in a restricted, controlled environment. In this setup, the projectile attitude and angular rates are measured by means of an IMU and the control algorithms are implemented on the embedded computer, both of which being discrete-time systems. In order to implement the above mentioned control laws, it is first necessary to convert them in an equivalent discrete-time representation.

The general form of the discrete-time autopilot is given in Fig. 23, where the controlled process G is a continuous-time system and the controller is executed in discrete timesteps. In the herein followed design by emulation method, the controller is first designed in the continuous domain, then discretized using an appropriate transform [START_REF] Stevens | Aircraft Control and Simulation[END_REF].

G(s) zoh K FB (z) K FF (z) T y(t) r(kT ) + u(kT ) u (
In the present case, the sampling period T = 10 ms, and Fig. 24 compares the discrete equivalents of K FB obtained with a zero-order hold on the input and using the bilinear transformation. The best discretization is obtained by the bilinear transformation, as the former method is suited for a piecewise-constant input over a sampling period, which is not the case in a feedback control setup. The discretized controllers K FB , K FF as well as the rate damping gain k q are then implemented in the Simulink diagram of Fig. 25, using the ACHILES custom blocks for interacting with the actuators and sensors.

B. Results

In Section IV, the three proposed autopilot structures have been designed using a "virtual" nominal model, based on the mean value of the estimated parameters. In this section, these autopilots have been implemented on the ACHILES test setup, as well as in the nonlinear simulator illustrated in Fig. 25c. This simulator is based on the nonlinear pitch axis model presented in Ref. [START_REF] Strub | Hardware-In-the-Loop Experimental Setup Development for a Guided Projectile in a Wind Tunnel[END_REF] and which led to the q-LPV model of Eqs. ( 2)-( 3). The dependency of the model parameters on the angle of attack is modeled using lookup tables with the estimated parameter values of Table 4.

During the HIL experiments, the airspeed was set to the same value as for the system identification data collection experiments, i.e. V = 25 m/s.

Disturbance Rejection Performance

In the autopilot synthesis model, the disturbance d is entering the open-loop system G, where it is filtered by the disturbance model H B . However, this disturbance signal cannot be influenced or measured directly in the case of HIL testing. As a result, in order to assess the disturbance rejection performance, an artificial disturbance step of known amplitude is injected at the plant input, i.e. at the k q summation point in Fig. 10, for a constant reference angle of attack.

Figure 26 presents disturbance rejection results, where the 4 • disturbance step is injected while the system is at steady state. This is repeated for three different values of the reference angle of attack, so as to assess the effect of an operating point variation on the disturbance rejection performance.

In Fig. 26a, the observed response follows the simulation closely, and the disturbance is attenuated by more than 40% and rejected in less than 0.9 s. This is coherent with the results presented in Section IV and Fig. 17c. Figure 26b presents the control signal δ u , where the peaks due to the injection of the disturbance are clearly visible. The steady-state difference between the simulated and the observed control signals is mainly due to uncertainty on the estimated DC gain, which does not impact performance thanks to the integrator in the feedback controller. As a final comment, both experimental signals exhibit visible superimposed oscillations. The exact source of these oscillations may emerge from an unmodeled nonlinearity (e.g. backlash in the servo-canard linkage), a flexible mode of the support system, interaction of the horizontal support rods and the tail fins, and/or to aerodynamic unsteadiness around the control surfaces and tail fins.

Reference Tracking Performance

The performance of the multi-objective autopilot is finally assessed and compared to the simulation using the nonlinear model, as well as the target reference model T ref .

In the present case, the input signal is a series of steps with different amplitudes, so as to exploit the whole projectile flight envelope.

Figure 27 presents the results of two reference tracking experiments using the multi-objective feedback controller with no feedforward controller in the first case (Figs. 27a and27b), and with the associated feedforward filter in the second case (Figs. 27c and27d).

In the latter case, the experimental and simulated responses are compared to the reference response using T ref , represented as a green dashed-dotted line in Fig. 27c. The three responses are almost undistinguishable, and the same level of performance as stated in Section IV is found. Figure 27d presents the corresponding control signal δ u , which shows initial peaking on reference step, but with an amplitude not higher than the steady-state value.

In the absence of the feedforward filter, as highlighted by the former case, the reference signal no longer rolls off at high frequencies and the injection of these dynamics create visible overshooting and oscillations in the angle of attack output, as shown in Fig. 27a. Correspondingly, the control input δ u exhibits considerable peaking, as illustrated in Fig. 27b. The amplitude and "sharpness" of these peaks exceed the actuator design constraints, resulting in actuator output saturation. Although not necessary for pure disturbance rejection, the feedforward controller is mandatory for respecting the reference tracking specifications.

VII. Conclusion

In this article, a complete pitch axis autopilot design is carried out for the ACHILES projectile prototype.

The ACHILES experimental setup is a novel benchmark for projectile control studies, where a projectile prototype is installed in a wind tunnel by the means of a 3-DoF gimbal. First, a plant model was built upon prior system identification studies, which yielded a family of linear models. The nominal synthesis model was then built using the mean values of estimated model parameters, and the difference modeled as multiplicative uncertainty.

In a second step, two autopilot design approaches have been considered. The considered autopilot structure consists in a feedback disturbance rejection controller and a feedforward reference tracking controller.

In the first approach, these two controllers have been designed in separate steps, where the feedback controller was obtained using H ∞ synthesis techniques and the feedforward controller was designed using an inverse-based approximation. The second approach is based on recent multi-objective H ∞ synthesis methods, where both controllers are tuned in a single step. Compared to the first two-step approach, this latter technique yields fixed-structure controllers with slightly better performance and good conformance to the design specification, which greatly eases the designer's task.

As a robust control approach was followed for the autopilot design, the third step consisted in verifying the robust stability of the designed controllers. Using multiplicative unstructured uncertainty and an M -∆ representation, the three controllers were proved to be robustly stable against parameter uncertainty for the

MK

  αd , M qd Body disturbance model parameters k d , k δ r Input scaling factors k q Rate damping gain M qα , M qq , M qδ Body dynamic model parameters ω 0 , ζ Reference model bandwidth, rad/s, and damping ratio FF , K FF Feedforward (reference tracking) controller, filter and injection forms T ref Reference model Synthesis-related symbols I. Introduction An important trend in the field of guided ammunition research is the development of low-cost gunlaunched guided projectiles that aim at overcoming the limitations of traditional artillery ammunition (e.g.

Fig. 1

 1 Fig. 1 Overview of the ACHILES Hardware-In-the-Loop experimental setup, including the guided projectile prototype, the 3-DoF support structure and the wind tunnel.

Fig. 4 Fig. 5

 45 Fig. 4 Servo controller structure.

Fig. 6

 6 Fig. 6 Projectile side view and parameters.

Fig. 7

 7 Fig. 7 Rate feedback loop, forming the damped open-loop plant G P .

  vide accurate reference angle of attack tracking. The closed-loop system should behave like a second-order low-pass reference model T ref with bandwidth ω 0 = 6 rad/s and damping ζ = 0.78. These performance requirements must not compromize stability, therefore the autopilot should comply with gain margin (GM) and phase margin (PM) requirements, where typical values in the aircraft and missile community are GM > 6 dB and PM > 45 • .

Imaginary Axis (s - 1 )Fig. 8

 18 Fig. 8 Pole loci for the open-loop plant G P . Black crosses mark the location of open-loop G plant poles. Bold blue crosses mark the location of rate damped plant poles.

Fig. 8 .

 8 The virtual control input δ r and disturbance input d are then respectively scaled by the gains k δ r and k d such that the DC gain between these inputs and the angle of attack output is equal to unity. The resulting plant will be further referenced to as the damped open-loop plant G P and the variable to be controlled is the angle of attack α. The obtained gain values for the rate damping controller are k δ r = 1.538, k d = 19.49 and k q = 0.1914.

Figure 9

 9 Figure 9 presents simulated and experimental responses of the closed loop against the open-loop re-

  Fig. 9 Open-loop and rate damping controller step responses. Dashed blue line: simulated response, solid red line: measured response.

Fig. 10

 10 Fig. 10 Angle of attack autopilot structure.

Fig. 12

 12 Fig.12Inverse of the weighting filters W i and associated parameters.

Fig. 13

 13 Fig. 13 Multi-objective two-degrees of freedom synthesis diagram in standard form.

1 → z 1 ,

 11 deals with the disturbance rejection requirements, which were specified in Section IV C 1. The reference tracking requirements, namely a model-matching constraint with respect to T ref and a high-frequency roll-off constraint on K FB S r→e , which limits the injection of high-frequency dynamics in the plant from the reference input, are specified in the second channel w 2 → z 2 , where w 2 = r. These requirements are enforced by the means of the respective weighting filters W 21 and W 22 . The model-matching filter W 21 is a constant gain and acts on the model-matching error e MM = r(T ref -T r→α ). The gain of W 21 defines the maximum allowable difference between the target system and the closed-loop transfer function, which is the highest at medium frequencies.

Fig. 14

 14 Fig. 14 Two-degrees of freedom control structures. Top: feedforward in series, or filter form. Bottom: feedforward in injection form.

11 Magnitude 12 Magnitude

 1112 Fig.[START_REF] Doyle | State-space solutions to standard H 2 and H ∞ control problems[END_REF] Closed-loop transfer functions and unit step response for the disturbance rejection controllers (blue: fullorder controller K FB,FO , red: fixed-structure controller K FB,FS , green:fixed-structure controller K FB,MO in the multiobjective case).

Figure 17c presents 21 Magnitude

 21 Figure17cpresents the Bode diagrams for the three feedforward controllers against the ideal, inverse-

5 Fig. 19

 519 Fig.[START_REF] Scherer | Multiobjective output-feedback control via LMI optimization[END_REF] Relative error for all observed realizations (dotted lines). Dashed line: uncertainty radius I . Solid line:

Fig. 20

 20 Fig. 20 Generic M -∆ structure for robust stability analysis.

Fig. 21 Fig. 22

 2122 Fig. 21 Complete autopilot in M -∆ form.

Fig. 23 Fig. 24

 2324 Fig. 23 General form of the discrete-time autopilot.

Fig. 25

 25 Fig. 25 Simulink diagrams used for autopilot validation. The "Open-loop plant" block in (a) references to diagram (b) for HIL experiments and to diagram (c) for numerical nonlinear simulations.

  Fig. 26 Disturbance rejection performance results, obtained with 4 • disturbance steps injected during steady-state. Dashed blue line: nonlinear simulation. Solid red line: experimental results

  Fig. 27 Full flight envelope reference tracking results. Dashed blue line: nonlinear simulation. Solid red line: experimental results.

Table 1

 1 Parameters of the weighting filters W i j used in 1-DoF and 2-DoF controller synthesisW 1 = W 11 W 2 = W 12

	k 11,LF	0.001	k 12,LF	1
	k 11,HF	0.6	k 12,HF	0.001
	ω 11	3 rad/s	ω 12	15 rad/s
	W 21		W 22
	k 21	20	k 22,LF	1
			k 22,HF	0.001
			ω 22	150 rad/s
		Table 2 Feedback controller synthesis results
		Full-order	Fixed-structure	Multi-objective
	γ	1.26	1.38	1.36
	Gain margin a	14.40 dB	13.08 dB	13.69 dB
	Phase margin a	79.50 •	83.08 •	81.22 •
	Gain margin b	11.88 dB	12.93 dB	12.22 dB
	Phase margin b	49.01 •	44.60 •	45.79 •
	a at the outer loop

b at the actuator input

Acknowledgments

The authors wish to thank the French Ministry of Defence Procurement Agency (DGA, Direction Générale de l'Armement) for its financial support.

The authors also wish to thank Denis Bidino, Kevin Meder and all the staff at the ISL workshop for the mechanical design and manufacturing of the ACHILES test setup, along with Emmanuel Roussel and Joseph Juncker for their precious help during the development of the embedded electronics and software.

whole flight envelope.

The fourth and final step consisted in the validation of the multi-objective controller on the experimental setup and comparison against non-linear simulations, in order to assess the disturbance rejection and reference tracking performance. There is a very good agreement between the experimental and the simulation results, and both conform to the design specifications.

In future developments, the approaches proposed herein will be extended to integrate the projectile's yaw axis, in order to develop a coupled pitch-yaw autopilot.

Appendix


Corinne Berland
email: c.berland@esiee.fr

Jean-François Bercher
email: jf.bercher@esiee.fr

Olivier Venard
email: o.venard@esiee.fr

Jean-François Bercher

C Berland Lamips

J.-F Bercher

Adaptive Gain and Delay Mismatch Cancellation for LINC Transmitter

Keywords: Radio transmitters, power amplifiers, adaptive signal processing, delay filters, delay estimation, nonlinear distortion

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

The high data rate demand for data transmissions in radiocommunication systems implies the use of efficient modulation schemes for increasing the spectral efficiency of these systems. The use of non constant envelope modulations induces stringent constraints on the linearity and the efficiency of the transmitter. Thus, the choice of the transmit architecture is realized considering the feasibility of the analog front end and the additive digital signal processing necessary for achieving desired performances [START_REF] Berland | Digital signal processing techniques to compensate for RF imperfections in advanced transmitter architectures[END_REF]. Among all the appropriate transmitter architectures, we find the LINC solution which is based on the separation of any modulated signal into two constant envelope phase modulated signals. These two signals follow two parallel paths and are recombined after efficient power amplification. However, this architecture is sensitive to mismatch between the two transmit paths: gain impairment and differences in propagation delays. The gain imbalance is a well known drawback of this topology and some solutions for solving this problem are presented in [START_REF] Zhang | Gain/phase imbalance-minimization techniques for LINC transmitters[END_REF][START_REF] Zhang | Gain and phase error-free LINC transmitter[END_REF][START_REF] Garcia | Nonlinear distortion cancellation using LINC transmitters in OFDM systems[END_REF][START_REF] Chandrasekaran | Adaptive algorithms for calibrating a LINC amplifier[END_REF]. In addition to this gain mismatch, we demonstrate in this paper that the delay mismatch between the two paths has to be taken into account. It does not only degrade the output performances of the transmitter but also corrupts the gain correction by adding a time dependent perturbation on the phase.

In the first part of this paper, after a brief introduction to the LINC architecture and the impact of the delay mismatch on output performances, we will present a solution for delay adjustment based on the minimization of the quadratic error between the desired output signal and the emitted one. In the second part of this article, we will demonstrate the usefulness of this correction in addition to a gain mismatch correction between the two paths

LINC transmitter

Principle of the architecture

The principle of the LINC transmitter [START_REF] Cox | Linear Amplification with Nonlinear Components[END_REF] consists of rewriting any modulated signal as the sum of two constant envelope modulated signals, denoted by s 1 (t) and s 2 (t).

s(t) = s 1 (t) + s 2 (t).
After the frequency translation and efficient amplification stages, these two signals are added, with a combiner, so as to recover the original modulated signal. The generation of the two constant envelope phase modulated signals is based on the following development: let us consider the band-limited modulated signal expressed using its complex representation,

s(t) = a(t) exp jφ (t) , (1)
with max [a(t)] = A m and a(t) = A m cos (θ (t)).

At the output of the SCS (Signal Component Separator), two phase modulated signals, s 1 (t) and s 2 (t), are generated with:

s 1 (t) = exp j(φ (t)-θ (t)) , s 2 (t) = exp j(φ (t)+θ (t)) . (2)
These two signals can also be expressed as:

s 1 (t) = 1 2 [s(t) -s e (t)] s 2 (t) = 1 2 [s(t) + s e (t)] (3)
with

s e (t) = js(t) A 2 m a 2 (t) -1. (4)
This component separator can be implemented using either a digital processing unit or an analog approach as demonstrated in [START_REF] Shi | Voltage-Translinear Based CMOS Signal Component Separator Chip for Linear LINC Transmitters[END_REF] and [START_REF] Gerhard | LINC digital component separator for single and multicarrier W-CDMA signals[END_REF].

The model of the LINC transmitter developed in this article, Fig. 1, is realized with Agilent ADS. This model takes into account quantization effects, with 8-bit DACs at a sample rate of 8F s . The output of these DACs are filtered with 5th order Butterworth filters before the quadrature modulation and the amplification. For the purpose of the study, RF delays are added in each path to model the desynchronization between the two signals. The feedback loop is implemented with a quadrature demodulator and two 8-bit ADCs at a similar sampling rate of 8F s .

Impact of delay mismatch

In this paper, we consider the performance of a transmitter according to the 3GPP standard using the 16QAM modulation for HSDPA with a chip rate of 3.84 Mcps [START_REF]Universal Mobile Telecommunications System (UMTS); User Equipment (UE) radio transmission and reception[END_REF]. Among all the scenarios specified in this standard, we will focus on the most stringent performance specifications. The output RF spectrum emissions are specified in term of Adjacent Channel Leakage power Ratio (ACLR). The specification of the ACLR is 33 dB at ±5 MHz from the assigned channel frequency and 43 dB at ±10 MHz. The EVM has to remain lower than 17.5 %. In the simulation model implemented, without any desychronization, the EVM value is 0.3 % and ACLR at ±5 MHz and ±10 MHz are respectively 43 dB and 53 dB.

To evaluate the impact of delay mismatch between the two paths, we introduce a differential delay. The results of simulations are given in Table 1, where T s is the duration of an emitted symbol at the output of the modulator (this is in fact the chip time of the standard) and is equal to T chip = T s = 1/3.84e6 ≈ 260 ns. From these results, it is obvious that both EVM and ACLR are impacted: T s /32 ≈ 8 ns seems to be the limit of acceptance for the delay, without any margin for other impairments in this architecture. Therefore delay compensation is mandatory.

Delay Adjustment

In this section we present a delay identification algorithm in order to cancel mismatch, and so, improve the performance of the transmitter.

Synchronisation algorithm principle

Let us consider that s 1 (t) is delayed by τ 1 whereas s 2 (t) is delayed by τ 2 , with τ 2 -τ 1 = ∆ . At the output of the transmitter, considering these delays, the original modulated signal s(t) is modified to give z(t):

z(t) = z 1 (t) + z 2 (t) = s 1 (t -τ 1 ) + s 2 (t -τ 2 ) (5)
This can be rewritten as:

z(t) = 1 2   s(t -τ 1 ) + s(t -τ 2 ) 1   + 1 2   s e (t -τ 2 ) -s e (t -τ 1 ) 2   . (6)
The first term of this expression corresponds to the linear interpolation of s(t) at the time t -τ 1 + ∆ 2 , the second term is the first order derivative of s e (t) at the time t -τ 1 + ∆ 2 . In order to compensate for these delays, a feedback loop associated with an adaptive algorithm is one solution. The aim of the algorithm is to reduce the quadratic distance between the original signal, s(t), and the emitted one, z(t).

Two solutions for the implementation can be explored. The first is to advance directly the emitted signal by values µ 1 = τ 1 and µ 2 = τ 2 so that the various delays are canceled: this is a correction algorithm. This approach is similar to the one presented in [START_REF] Garcia | An adaptive digital method of imbalances cancellation in LINC transmitters[END_REF] for gain correction. One of the drawbacks of this solution is the need for an additive SCS in the feedback loop. Moreover, this solution is restricted to real time correction that precludes any monitoring capabilities.

The second solution resolves the values of τ 1 and τ 2 : this is the identification procedure. Once the identification is done, the correction could be applied, if necessary. This implementation allows both correction and monitoring. Finally, the identification procedure presents the advantage of avoiding the use of an additive SCS. Advancing or delaying the signal is of course only possible if the transmitter introduces a delay between the emitted signal and the output of the modulator with a buffer. Moreover, the solution for delaying or advancing signals by fractional amounts relies on the use of interpolation filters which allow to adjust the delay without modifying the implementation of the filter [START_REF] Farrow | A continuous variable digital delay element[END_REF], [START_REF] Candan | An Efficient Filtering Structure For Lagrange Interpolation[END_REF]. The choice of this interpolator has to be studied further because it impacts directly the value of the interpolated signal EVM. For this reason, we choose to implement a fifth order Lagrange interpolator using the farrow structure [START_REF] Bercher | Envelope and phase delays correction in an EER radio architecture[END_REF].

Derivation of the delay identification algorithm

Let us now study the formulation of the algorithm. If we denote e(t) = z(t)s(t) the error function we want to minimize, the gradient algorithm consists of iterating the searched value as follows:

µ(k + 1) = µ(k) -γ(k) ∂ J(µ) ∂ µ µ=µ(k)
with J(µ) = E |e(t)| 2 being the mean square error and γ the adaptation step. Rewriting the expression of the derivative of the mean square error brings to:

∂ J(µ) ∂ • = ∂ E |e(t)| 2 ∂ • = 2E ℜ ∂ e * (t) ∂ • e(t)
For the identification of the delays introduced by the transmitter, the algorithm will resolve µ 1 and µ 2 values such as:

s 1 (t -µ 1 ) = z 1 (t) s 2 (t -µ 2 ) = z 2 (t) (7)
This procedure is illustrated Fig. 2. Considering the error to be of the form:

e(t) = z(t) -s 1 (t -µ 1 ) -s 2 (t -µ 2 ),
the derivative of the mean square error can be written as:

∂ J(µ 1 ,µ 2 ) ∂ µ i = ∂ E[|e(t)| 2 ] ∂ µ i = -2E ℜ e(t) ∂ s * i (t-µ i ) ∂ µ i (8)
The final formulation of the gradient algorithm is, with i = 1, 2,

µ i (k + 1) = µ i (k) + 2γ(k)ℜ e(t) ∂ s * i (t -µ i ) ∂ µ i µ i =µ i (k)
.

This procedure then consists of finding the values of these two delays µ 1 and µ 2 . Once these delays are identified, the + -

+ Modulator Signal Component Separator Delay Adjustment Transmitter Transmitter Delay τ 1 Delay τ 2 s 1 (t) s 2 (t) s 1 (t -τ 1 ) s 2 (t -τ 2 ) z(t) s 1 (t -µ 1 ) + s 2 (t -µ 2 ) µ 1 , µ 2 Correction
Fig. 2 Identification architecture : delay µ 1 and µ 2 are adjusted to cancel the quadratic error. After convergence, the correction is calculated and applied.

correction would have to be applied to the transmit path. It is worth noting that the algorithm not only identifies differential delay mismatch but also the global delay introduced by the analog front end. If the global delay introduced in the transmitter is acceptable for the application, the correction may be only differential and operated on a single path instead of being absolute for the two paths.

Simulation results

The convergence time for the identification procedure is illustrated Fig. 3, for the same test cases as the ones given in Table 1, with a pseudo-random sequence at the input of the modulator. In the simulation scenarios, the differential delay, ∆ , was only applied on τ 1 . Both delays µ 1 and µ 2 are iden- tified and the results take into account both, global propagation time linked to the baseband filters and the differential delay mismatch. After 300T c ≈ 100 µs the convergence is obtained. The correction is then applied (by delaying or advancing one path relative to the other) and this allows, for all cases, the reduction of the EVM to 0.6 % with an ACLR at ±5 MHz higher than 43 dB. The output spectrum after correction is shown in Fig. 4. The dramatic improvement of the performance compared to the values given in Table 1 highlights the effectiveness of this delay equalization algorithm.

Gain and delay adjustment

In this section, we will now present the gain identification algorithm and demonstrate the necessity to realize a delay identification before applying a gain correction procedure.

s 2 (t) G 1 s 1 (t -τ 1 ) G 2 s 2 (t -τ 2 ) z(t) K 1 s 1 (t -µ 1 ) + K 2 s 2 (t -µ 2 ) µ 1 , µ 2 K 1 , K 2 Correction
Fig. 5 Gain and Delay mismatch cancellation architecture : delay µ 1 and µ 2 are identified (K 1 = K 2 = 1), corrected. The gain mismatch identifi- cation is realized in a second step. After convergence, the gain correction is applied. choose to apply for the gain correction the same identification procedure as presented previously for the delay adjustment. Indeed, direct correction solutions as in [START_REF] Garcia | An adaptive digital method of imbalances cancellation in LINC transmitters[END_REF] are only valid if we consider that the SCS in the return path gives at its output z i (t) ∝ s i (t): inspection of the formula suggests that this assumption is not valid for all types of signals. The output signal z(t), see Fig. 5, has now the following form, with G 1 and G 2 the two complex gains:

z(t) = G 1 s 1 (t) + G 2 s 2 (t).
The objective of the algorithm is to find K 1 and K 2 so that we minimize the quadratic error:

J(K 1 , K 2 ) = E |K 1 s 1 (t) + K 2 s 2 (t) -z(t)| 2 .
This gives, for i = 1, 2:

ℜ (K i (k + 1)) = ℜ (K i (k)) + 2γ k (k)ℜ[e(t)s * i (t)],
and

ℑ (K i (k + 1)) = ℑ (K i (k)) + 2γ k (k)ℜ[ j.e(t)s * i (t)].
As already highlighted in the introduction, the gain correction cannot be correctly achieved without a preliminary synchronization between the reference and the emitted signal. We introduced a gain and phase mismatch of 0.5 dB and 5 o in the transmitter Fig. 1 with no additional differential delays in both transmit paths than the one introduced by the analog filters. The ACLR is about 30 dB at 5 MHz and the EVM is approximately 5.9 %. Without any preliminary identification of the processing delay in the transmit path, the gain correction algorithm fails to converge. This is due to the fact that the algorithm tries to cancel, with the two complex gain phase values, a delay comprising the analog front end time processing and the phase value of both complex gains. If we simply identify this processing delay (ideal case here) and add it in the simulation before gain identification, the ACLR is higher than 43 dB and the EVM lower than 0.6 % after the identification and correction of those two complex gains. This demonstrates the usefulness of the delay synchronization before the gain identification procedure even if there is no differential delay mismatch between the two paths. The algorithm is implemented as follows, see Fig. 5:

-Identification of the delays -Correction of the delays -Identification of the complex gains -Correction of the complex gains

In our simulation environment, we choose to apply the delay correction after a convergence time of 500T s.

Results of simulation

Further simulations were realized including both delay mismatch and gain impairment. We introduced a differential delay mismatch of T s/32 ≈ 9 ns between the two paths: this corresponds to a mismatch of 10 % of the propagation time between the two analog filters. A comparison of results was realized between the performance before the correction, after only a simple non differential delay identification (theoretical value taken for processing time in our simulated analog front end) and finally with the differential delay adjustment procedure. Results given in Table 2 demonstrate the efficiency of a differential delay correction as this allows a reduction of about 15 dB on ACLR and lowers the EVM to less than 1 %. It should be pointed out that for greater mimatch (above 1 dB and 15 o ) the algorithm exhibits poorer results in term of ACLR and EVM.

Conclusions

This paper addresses the sensitivity of the LINC transmitter architecture to impairments between the two transmit paths.

In Addition to gain mismatch, we consider the impact of mismatch in analog processing delays and we evaluate its effects on the transmitter performances. A solution to identify delays in each transmit path is proposed. This allows the identification of the global processing delay in the transmitter and also the correction of the differential delay impairments between the two paths. As the loop delay impacts the convergence of the gain correction solution, a two step solution is adopted: first an identification of paths processing time with a correction of the differential delay and then an identification and a correction of the gain mismatches. The proposed approach allows drastic improvement for both EVM and ACLR: in our test case it reduces the EVM from 14 % to 1 % and improves the ACLR at ±5 MHz by more than 20 dB (from 18 dB to 39 dB for instance in the worst test case).

Corinne

Fig. 3

 3 Fig. 3 Identification of µ 1 and µ 2 normalized to the symbol time, for ∆ = [T s /64, T s /32, T s /16, T s /8]

Fig. 4

 4 Fig.[START_REF] Garcia | Nonlinear distortion cancellation using LINC transmitters in OFDM systems[END_REF] Output spectrum after identification and correction of the delays, for ∆ = [T s /64, T s /32, T s /16, T s/8]

4. 1 τ 1 , Gain G 1 Delay τ 2 , Gain G 2 s 1

 111221 Gain adjustment algorithmDue to the non-linear operation on the modulated signal for the generation of the two constant envelope signals, we+ -(t)

  Berland was born in France, in 1966. She is engineer of ESIEE Paris(1989) and received the Doctorat degree in electronics at the University of Marne La Vallée in 2001. She worked until 1998 in Alcatel Business Systems. She is now Associate Professor in the Electronics Systems department at ESIEE Paris and joins the LaMIPS, in Caen, for her research activities. Her main interests are transceiver architecture for radiocommunication systems and RF/analog integrated circuits. Jean-François Bercher was born in Marseille, France, in 1965. He is maître es Sciences et Techniques in biomedical engineering (1988) and engineer of the Institut National Polytechnique de Grenoble (1990). He received the Doctorat degree in physics at the Laboratoire des Signaux et Systèmes, Université de Paris-Sud, Orsay, in 1995. He is now with the University Paris-Est, LabInfo-IGM, and is Associate Professor in the Telecommunication Department at ES-IEE Paris. His main interests are statistical radiotechnique and applications of information theory to signal processing and telecommunications. Olivier Venard was born in France, in 1961. He is an associate professor in Signal Processing and Telecommunication in the Telecommunication Department at ESIEE Paris, University Paris-Est. Since 1997, he has been working in the field of signal processing, digital communication and digital software and hardware architecture.

  Simulation model of the transmitter. This model includes an I/Q modulator signal component separator for the generation of the two constant envelope modulated signals. Both signals are converted into analog using 8-bit DAC and translated to RF with an I/Q modulator. The sum of the two signals is operated at the output of the PAs and a feedback loop (quadrature demodulation and 8-bit ADCs) is implemented for applying correction algorithms.

	Bits 16QAM modulator	s(t)	Signal Component s 1 (t) Separator s 2 (t)	Identification and correction Algorithm	I Q I Q	8 bits 8 bits	DAC DAC DAC DAC	RF I/Q Modulator Modulator RF I/Q	PA 1 PA 2	+ z 1 (t) z 2 (t)	z(t)
							ADC	Quadrature
					8 bits		ADC	Demodulator
	Fig. 1

Table 1

 1 Impacts of differential delay mismatches on the EVM and on the spectrum.

	Delay EVM (%) ACLR @ ±5 MHz
	T s /64	1.6 %	38 dB
	T s /32	3.2 %	32 dB
	T s /21	5 %	28 dB
	T s /16	6.8 %	26 dB
	T s /13	8.8 %	24 dB
	T s /10	10.7 %	22 dB
	T s /9	12.1 %	21 dB
	T s /8	13.5 %	20 dB

Table 2

 2 Algorithms performances with and without differential delay adjustment, with ∆ = τ 1 -τ 2 = T s /32

	∆ G	∆ φ	Before EVM (%) ACLR EVM (%) Without Diff Delay Adj ACLR	With Delay Adj EVM (%) ACLR
	0.1dB	2 o	4.2 %	27 dB	3.7 %	28 dB	0.7 %	43 dB
	0.2dB	5 o	6.9 %	23 dB	3.7 %	28 dB	0.8 %	41 dB
	0.5dB	5 o	6.9 %	23 dB	3.8 %	28 dB	0.7 %	41 dB
	0.6dB	8 o	10.8 %	20 dB	3.8 %	28 dB	0.8 %	40 dB
	0.8dB 10 o	14.1 %	18 dB	3.8 %	28 dB	0.9 %	39 dB

Acknowledgment

The authors would like to thank Prof P. Gamand and S. Kowlgi Srinivasan for their contributions for this paper.


Corinne Berland
email: c.berland@esiee.fr

Jean-Franc ¸ois Bercher
email: jf.bercher@esiee.fr

Olivier Venard
email: o.venard@esiee.fr

Lamips

Gain and Delay Mismatches Cancellation in LINC and Polar Transmitters

This article presents the application of a gradient algorithm on impairments correction for polar and LINC transmitters (LInear amplification with Non linear Component). The two aspects of the approach are presented: identification and correction. The large improvements obtained using such solutions is demonstrated for both transmitters.

I. INTRODUCTION

Current research in transceivers architecture is focused on the reduction of the overall power consumption of the system. For this reason, architectures such as polar [START_REF] Kahn | Single-sideband transmission by envelope elimination and restoration[END_REF] and LINC transmitters [START_REF] Cox | Linear amplification with nonlinear components[END_REF] are appealing alternatives to the traditional cartesian transmitter. The differences between these solutions can be simply viewed as different methods of writing the modulated signal. Indeed, the polar architecture is based on the polar representation of the signal, with its magnitude ρ(t) and its phase φ(t), whereas the LINC solution consists in rewriting the signal as the sum of two constant envelope signals. These different ways of expressing the signal influence the architecture of the transmitter. In the case of polar solutions, the envelope signal and the phase signal are differently processed before being recombined at the output stage of power amplifier. For the LINC architecture, the two constant envelope signals are processed and amplified separately before being added.

The main drawback of these solutions is their sensitivity to mismatches between the two paths. In both cases, the output signal is significantly deteriorated, in terms of EVM (Error Vector Magnitude) and ACLR (Adjacent Channel Leakage Ratio) when a delay mismatch between paths and/or complex gain mismatch occur.

In this paper, we propose an adaptive algorithm for identifying and correcting delay and gain mismatches for these two transmitters.

We illustrate the issues and performances of proposed algorithms on a 16QAM modulation with a chip rate of 3.84 Mcps. This corresponds to the 3GPP standard. We mainly focus the analysis on the EVM (Error Vector Magnitude) and on the ACLR (Adjacent Leakage Channel Ratio) at ±5 MHz from the assigned channel frequency. The specifications are an EVM lower than 17.5% and an ACLR higher than 33 dB [START_REF]Universal Mobile Telecommunications System (UMTS); User Equipment (UE) radio transmission and reception (FDD)[END_REF].

In a first section, we introduce the general principle of identification and direct correction by an adaptive algorithm. These approaches are then applied to the polar transmitter in section III and to the LINC transmitter in section IV

II. ADAPTIVE ALGORITHM

A. Introduction to gradient algorithm

Let us consider a system, with input x(t) and output y(t). The output of the system is a function f of the input, characterized by k parameters σ 1 , . . . , σ k . The definition of function f is such that f (x(t), 0, . . . , 0) = s(t). For instance, in case of a pure delay, we have f (x, τ 1 ) = x(t -τ 1 ), and in case of a gain, we could have f (x, g 1 ) = G 1 x(t) = e g1 x(t). We will also denote by d(t) a desired signal. The objective is to identify these parameters so that the output of the system becomes similar to the desired one. This is realized through the identification of parameters called σ i such as we minimize the following mean square error:

J(σ 1 , . . . , σ k ) = E |f (x(t), σ 1 , . . . , σ k ) -d(t)| 2 , (1)
where E [•] is the statistical expectation operator. The solution can be obtained with a descent algorithm. This algorithm consists in iterating for i = 1 . . . n:

σ i (n + 1) = σ i (n) -γ i (n) ∂J(σ 1 , . . . , σ k ) ∂σ i σi=σi(n) (2)
where γ i is the adaptation step. The gradient is computed according to

∂J(σ 1 , . . . , σ k ) ∂σ i = ∂E |e(t)| 2 ∂σ i = 2E ℜ ∂e * (t) ∂σ i e(t) (3)
The theoretical algorithm is obtained with the derivatives in [START_REF]Universal Mobile Telecommunications System (UMTS); User Equipment (UE) radio transmission and reception (FDD)[END_REF]. In practice, we use a stochastic gradient algorithm that involves the instantaneous expression of the gradient rather than the statistical average. The formulation is finally

σ i (n + 1) = σ i (n) -2γ i (n) ℜ ∂e * (n) ∂σ i e(n) σi=σi(n) . (4)

B. Identification procedure

The resolution of the minimization problem and the correction of the imperfections can be formulated in two ways. The first solution is to identify these defaults and then correct them. The identification process, Fig. 1, consists in modifying the ideal signal x(t) with parameters µ 1 , . . . , µ k and then comparing it with the emitted one z(t) = f (x(t), τ 1 , . . . , τ k ).

+ -

Modulator RF Transmitter x(t) z(t)
f (x(t), τ1, . . . , τk) f (x(t), µ1, . . . , µk) µ1, . . . , µk The desired signal is here z(t) and the parameters σ i = µ i . The equation ( 2) becomes, for i = 1 . . . n:

µ i (n + 1) = µ i (n) -γ i (n) ∂J(µ 1 , . . . , µ k ) ∂µ i µi=µi(n) (5)
Once the τ i values are identified, the correction can be applied, if necessary. Hence, the identification implementation allows both correction an monitoring.

C. Correction procedure

The other approach consists in directly correcting the signal before the emission, according to Fig. 2. The desired signal is here x(t). This gives according to (2) the following iterative formulation:

∆ i (n + 1) = ∆ i (n) -γ i (n) ∂J(∆ 1 , . . . , ∆ k ) ∂∆ i ∆i=∆i(n) . (6)
This correction procedure avoids the intermediate step of the identification.

One has to pay attention to the fact that this algorithm needs, for the calculation of the instantaneous derivative of the gradient, to know perfectly the output signal. It implies that the correction algorithm cannot be applied when the transformation applied to the signal is non linear.

+ - E Modulator RF Transmitter z(t) f (x(t), ∆1 -τ1, . . . , ∆k -τk)) x(t) x(t) ∆1, . . . , ∆k

D. Application to desynchronisation

In the specific case of desynchronisation, the implementation of such algorithms requires a block that controls the advance and the delay of the signal. For this, the transmitter has to introduce a buffer between the emitted signal and the output of the modulator. Moreover, the solution for delaying or advancing signals by fractional amounts relies on the use of interpolation filters which allow to adjust the delay without recomputing the coefficients of the filter [START_REF] Farrow | A continuous variable digital delay element[END_REF]. We choose to implement a fifth order Lagrange interpolator using the Farrow structure, as we did in [START_REF] Bercher | Envelope and phase delays correction in an EER radio architecture[END_REF].

III. APPLICATION TO THE POLAR TRANSMITTER

A. Polar architecture and simulation model

Polar architecture is a regarded as an efficient solution for SDR transmitter [START_REF] Groe | Polar transmitters for wireless communications[END_REF]. This architecture, based on the Kahn's Envelope Elimination and Restoration (EER) procedure, consists in amplifying the phase modulated signal with an efficient amplifier while the envelope is restored through the voltage supply of this PA (power amplifier). The simulation model used in this paper is shown Fig. 3, with a sampling period T = T s /12, with T s being the duration of an emitted symbol. With this model, the output EVM is on the order of 0.3% and the ACLR is 53 dB.

This architecture is sensitive to delay mismatches between the amplitude path and the phase path [START_REF] Markovic | Feasibility of EER transmitters for 3GPP applications[END_REF]. For example a delay mismatch of 0.1T s between the two paths yields to an EVM of 5.5% and to an ACLR of 28 dB.

B. Delay cancellation

Let us consider the complex representation of the ideal signal: x(t) = ρ(t) exp(jφ(t)). This baseband signal is processed by the RF transmitter and results in the signal at the antenna denoted z(t), Fig. 4. The envelope of the signal is delayed by τ 1 whereas the phase of the signal is delayed by τ 2 . This gives for the emitted signal

ρ(n) φ(n) RF transmitter x(t) z(t) ρ(t -τ1) φ(t -τ 2 )
z(t) = ρ(t -τ 1 ) exp(φ(t -τ 2 )).
For this application, we can apply the correction algorithm without the preliminary identification step. This solution reduces the computation time. We also compute the mean square error only with the in-phase component of the complex signal (using both in phase and quadrature components implies higher computation load with little gain in performance). Following ( 1) and ( 6), we find that the mean square error is

J(∆ 1 , ∆ 2 ) = E |ρ(t) cos(φ(t)) -ρ(t 1 ) cos(φ(t 2 ))| 2 , (7)
and the updating rules for ∆ 1 and ∆ 2 are

∆ 1 (n + 1) = ∆ 1 (n) + γ 1 (n) dρ(u) du t1 cos φ(t 2 ) e(n) ∆ 2 (n + 1) = ∆ 2 (n) + γ 2 (n)ρ(t 1 ) d cos φ(u) du t2 e(n)
where we noted t = nT , t 1 = nT + ∆ 1 -τ 1 and t 2 = nT + ∆ 2 -τ 2 , T being the sampling period. This algorithm is implemented and the simulation results are presented Table I. These results show a huge improvement in the EVM which falls from 38% to 0.3%, and in the ACLR which rises from 13 dB to 51dB. The convergence time at 90% of the final value, t c , is also evaluated and for our test cases is under 150T s . These results are detailed in [START_REF] Bercher | Envelope and phase delays correction in an EER radio architecture[END_REF].

C. Delay cancellation with gain identification

However, the results indicated above were obtained without considering the gain and phase offset introduced by the power amplifier. Indeed, from the transmitter point of view, the phase of the emitted signal is none of concern and the magnitude is controlled within a specified range (few dB). Since our algorithm relies on the comparison between the emitted signal and the ideal one, an uncertainty about the phase and the amplitude of the emitted signal impacts directly the performances. This is reported on Table II, on the (B) columns, for different scenarios for the gain and phase mismatches with τ 1 = 0.95 T s and τ 2 = 0.45 T s . To enhance performances we add an identification loop dedicated to this complex gain according to the principle Fig. 1. The corresponding mean square error is then

J(K, ξ) = E |Kρ(t)cos(φ(t) + ξ) -Gρ(t 1 ) cos(φ(t 2 ) + Θ)| 2 ,
with G exp(jθ) the complex gain representative of the incertitude in the return path and K exp(jξ) the unknown gain. The implementation of the identification algorithm was elaborated from the cartesian expression of the gain K exp(jξ) = K r + jK i and leads to the following equations:

K r (n + 1) = K r (n) + γ 3 (n)ℜ(X * (t)e(n)) K i (n + 1) = K i (n) + γ 3 (n)jℜ(X * (t)e(n))
The associated results are given in Table II, under the columns (A). As shown in Table II, we obtain a large improvement. The EVM falls to about 1.5% for the worst case compared to 11% without gain identification and the ACLR becomes higher than 49 dB. The only concern is a slower convergence which is the price to pay to account for these additionnal mismatches.

IV. APPLICATION TO THE LINC TRANSMITTER

A. LINC architecture and simulation model

The LINC solution is based on the separation of any modulated signal into two constant envelope (phase modulated) signals, Fig. 5. These two signals are processed along two parallel paths and are recombined after efficient power amplification. However, this architecture is sensitive to mismatches between the two transmit paths: gain impairment and differences in propagation delays. The gain imbalance is a well known drawback of this architecture and some solutions for solving this problem are presented in [START_REF] Zhang | Gain/phase imbalanceminimization techniques for LINC transmitters[END_REF], [START_REF] Garcia | Nonlinear distortion cancellation using LINC transmitters in OFDM systems[END_REF]. In addition to this gain mismatch, the delay mismatch between the two paths has to be taken into account. It does not only degrade the output performances of the transmitter but also corrupts the gain correction by adding a time dependent perturbation on the phase. The simulation model implemented is similar to the one used for the polar model, Fig. 3. It is constituted of the main path with two transmitters in parallel and a return path for correction. It takes into account quantization effects, with 8-bit DACs and 8-bit ADCs at a sample rate of 8F s as detailed in [START_REF] Berland | Adaptive gain and delay mismatch cancellation for LINC transmitter[END_REF]. With the simulation model, the EVM value is 0.3% and ACLR is 43 dB.

B. Delay and Gain identification and correction

The principle of the LINC transmitter consists in rewriting any modulated signal as the sum of two constant envelope modulated signals, denoted by s 1 (t) and s 2 (t):

s(t) = s 1 (t) + s 2 (t).
These two phase modulated signals, s 1 (t) and s 2 (t), are generated according to [START_REF] Cox | Linear amplification with nonlinear components[END_REF]:

s 1 (t) = exp j (φ (t) -θ (t)),
s 2 (t) = exp j (φ (t) + θ (t)). Let us consider that s 1 (t) is delayed by τ 1 whereas s 2 (t) is delayed by τ 2 , with τ 2 -τ 1 = ∆. At the output of the transmitter, considering these delays, the original modulated signal s(t) is modified into:

z(t) = s 1 (t -τ 1 ) + s 2 (t -τ 2 ). (8)
This delay mismatch impacts both EVM and ACLR. For a differential delay of ∆ = T s /32, the EVM reaches 3.2% and the ACLR is 32 dB, for ∆ = T s /10 the EVM is near the limit with 10.7% and the ACLR is out of specification with 22 dB.

The synchronisation has to be recovered.

Applying the correction algorithm to this architecture implies that we can extract the values of s 1 (t) and s 2 (t) from z(t) with an additional SCS. But, as the operation is non linear, we cannot recover the expected signals. For this reason, only the identification approach is possible.

The equations for the identification of τ 1 and τ 2 , according to Fig. 1 are, for i = 1, 2:

µ i (k + 1) = µ i (k) + 2γ(k)ℜ e(n) ∂s * i (t -µ i ) ∂µ i µi=µi(k)
, with

e(n) = z(nT ) -s 1 (nT -µ 1 ) -s 2 (nT -µ 2 ).
This algorithm allows the reduction of the EVM below 0.6% and the increase of the ACLR above 43 dB for differential delays up to T s /8 after identification and then correction of the signals.

In addition to the delay correction, the gain impairments between the two paths have to be cancelled. The output signal for gain mismatches is:

z(t) = G 1 s 1 (t) + G 2 s 2 (t),
with G 1 and G 2 two complex gains. Using the identification procedure the error function is:

e(t) = K 1 s 1 (t) + K 2 s 2 (t) -z(t)
and the recursive equations are, for i = 1, 2:

ℜ (K i (k + 1)) = ℜ (K i (k)) + 2γ k (k)ℜ(e(n)s * i (t)), ℑ (K i (k + 1)) = ℑ (K i (k)) + 2γ k (k)ℜ(j.e(n)s * i (t)).
The gain correction cannot be correctly achieved without a preliminary synchronization between the reference and the emitted signal, even if there is no differential delays. The delays of the analog signal processing have to be identified otherwise the algorithm diverges. For this reason, the procedure adopted consists in: identifying the delay mismatch, correcting it, identifying the gain mismatch and finally correcting this gain mismatch.

Further simulations were realized including both delay mismatch and gain impairment. We introduced a differential delay mismatch of T s /32 between the two paths. This corresponds to a mismatch of 10% of the propagation time between the two baseband analog filters. A comparison of the results was realized between the performances before the correction, after only a simple non differential delay identification and finally with the differential delay adjustment procedure. Results given in Table III demonstrate the efficiency of a differential delay correction as it allows a increase of about 15 dB on ACLR and lowers the EVM to less than 1%. It should be pointed out that for greater mimatches (above 1 dB and 15 o ) the algorithm exhibits, of course, poorer results in terms of ACLR and EVM. V. CONCLUSION This paper addresses the correction of gain and delays mismatches in polar transmitter and LINC transmitter. A gradient algorithm is used in both cases, either through an identification procedure or a direct correction principle. The efficiency of the approaches is demonstrated for the two transmitters. The achieved performance exhibits dramatic improvements, with an EVM lower than 1% and an ACLR at 5MHz higher than 39 dB.

Fig. 1 .

 1 Fig. 1. Identification implementation

Fig. 2 Fig. 3 .

 23 Fig. 2. Correction implementation

Fig. 4 .

 4 Fig. 4. EER transmitter impairments

Fig. 5 .

 5 Fig. 5. LINC Transmitter impairments

TABLE I ALGORITHM

 I PERFORMANCES (CONVERGENCE TIME, tc, EVM AND ACLR) FOR DELAY CORRECTION IN EER TRANSMITTER

	τ 1	τ 2	tc	EVM (%)	ACLR @5 M Hz
			Opt	Initial	After	Initial	After
	0.2Ts	1.1Ts	150Ts	38%	0.3%	13 dB	51 dB
	0.45Ts	0.95Ts	130Ts	25%	0.3%	13 dB	51 dB
	0.5Ts	0.12Ts	110Ts	20%	0.3%	15 dB	51 dB
	0.1Ts	0.2Ts	80Ts	5.5%	0.3%	28.5 dB	51 dB

TABLE II ALGORITHM

 II PERFORMANCES (CONVERGENCE TIME, tc, EVM AND ACLR) WITH GAIN IDENTIFICATION (A) AND WITHOUT IDENTIFICATION (B) FOR τ 1 = 0.45 Ts AND τ 2 = 0.95 Ts

	Gain	phase	tc	EVM (%)	ACLR @5 M Hz
	mismatch	mismatch	A	B	A	B	A
	3 dB	45 o	300 Ts	11%	1.5%	27 dB	49 dB
	1 dB	30 o	250 Ts	9%	0.7%	30 dB	50 dB
	0.5 dB	10 o	230 Ts	3.4%	0.6%	38 dB	50 dB
	0.2 dB	2 o	100 T s	1.2%	0.6%	47 dB	50.5 dB

TABLE III ALGORITHM

 III PERFORMANCES WITH AND WITHOUT DIFFERENTIAL DELAY ADJUSTMENT, WITH ∆ = τ 1 -τ 2 = Ts/32

			Before	W/o Delay Adj	W Delay Adj
	∆G	∆φ	EVM	ACLR	EVM	ACLR	EVM	ACLR
	0.1dB	2 o	4.2%	27dB	3.7%	28dB	0.7%	43dB
	0.2dB	5 o	6.9%	23dB	3.7%	28dB	0.8%	41dB
	0.5dB	5 o	6.9%	23dB	3.8%	28dB	0.7%	41dB
	0.6dB	8 o	10.8%	20dB	3.8%	28dB	0.8%	40dB
	0.8dB	10 o	14.1%	18dB	3.8%	28dB	0.9%	39dB