Carlos Armenta

Guasave Sinaloa

Sébastien Delprat

Prof. Dr Rudy R Negenborn

G Hawksley

ECMS for energy management of hybrid vessels via quasi-LPV control

Keywords: Optimal control, Linear Matrix Inequalities and SOS techniques Energy management, Marine vehicles, Optimal Control

His research activities are dedicated to vehicle control, and especially hybrid vehicle

Introduction

REDUCING energy consumption and emissions of the marine industry nowadays has become a priority. Powertrain hybridization is one of the approaches to achieve these goals [START_REF] Miyazaki | Hybrid Modeling of Strategic Loading of a Marine Hybrid Power Plant With Experimental Validation[END_REF]. A hybrid powertrain uses at least two energy sources, with at least one reversible one, su ch as a battery. During vessels operations, the propulsion power request needs to be split between the different energy converters (genset, engine, electric machine, fuel cell, etc.) and storage (battery, super-capacitor). An energy management algorithm (EMA) is adopted to compute this power split and to minimize a criterion such as CO2 emissions while managing the energy storage level (e.g., the battery state of charge (SOC)) and to consider several constraints induced by the powertrain architecture and component sizing [START_REF] Sciberras | Managing Shipboard Energy: A Stochastic Approach Special Issue on Marine Systems Electrification[END_REF]. If operational and environmental details of a mission are known a priori, the energy management problem can be formulated as an optimal control problem. Dynamic Programming (DP) [START_REF] Bai | Optimal Design of a Hybrid Energy Storage System in a Plug-In Hybrid Electric Vehicle for Battery Lifetime Improvement[END_REF][START_REF] Wang | Simulation of Energy Control Strategy for Hybrid Electric Vehicle Based on Modified Dynamic Programming[END_REF]) and Pontryagin's Minimum Principle (PMP) are widespread approaches to derive an optimal solution to hybrid powertrain energy management problems, even though alternative approaches based on the calculus of variations have been considered (East and Cannon, 2019). Dynamic Programming requires discretizing both the state and the time. Extensive search in the resulting grid allows approximating the optimal solution. This approach allows easily integrating state constraints, but it is only suitable for a limited number of states due to the so-called curse of dimensionality. Pontryagin's Minimum Principle (PMP) allows formulating optimality conditions in continuous-time, which boils down to consideration of a Boundary Value Problem (BVP) that can be solved using an appropriate solver. The resulting algorithm is, in general, faster than Dynamic Programming. In this work, PMP is considered since it allows deriving a real-time suboptimal, but efficient control algorithm referred to as the Equivalent Consumption Minimization Strategy (ECMS) [START_REF] Sampathnarayanan | An optimal regulation strategy with disturbance rejection for energy management of hybrid electric vehicles[END_REF][START_REF] Xie | MPC-informed ECMS based real-time power management strategy for hybrid electric ship[END_REF].

Both the offline PMP and the real-time ECMS compute the control as a solution to the instantaneous minimization of the Hamiltonian associated with the optimal control problem. This Hamiltonian is defined as the sum of the fuel consumption and the state dynamics multiplied by a co-state. Within the ECMS framework, this Hamiltonian is denoted as total equivalent consumption. PMP optimality conditions provide the optimal dynamics of the co-state, and the initial value of the co-state can only be computed in simulation over known missions. In real-time, adaptive-ECMS replaces this optimal dynamic by a real-time closed-loop control approach of the energy storage level (typically the SOC) [START_REF] Onori | Adaptive Equivalent Consumption Minimization Strategy for Hybrid Electric Vehicles[END_REF]. Two main problems arise: how to design the controller and how to prove the closed-loop stability. Various controllers have been investigated, such as PI, fuzzy, or predictive controllers, but only a few of them are provided with a closed-loop stability proof [START_REF] Sampathnarayanan | An optimal regulation strategy with disturbance rejection for energy management of hybrid electric vehicles[END_REF].

The contribution of this paper is to present a new controller that benefits from the ECMS framework and the underlying PMP optimal control theory. It is designed using the quasi-Linear Parameters Varying (LPV) approach, and it comprises a nonlinear state feedback control law coupled with a state observer. The Input-to-State Stability (ISS) of the closed loop is demonstrated [START_REF] Sontag | New characterizations of input-to-state stability[END_REF]. This paper is organized as follows. In Section II, the considered hybrid vessel is introduced, and the formulation of its optimal control using PMP is given. In Section III, the real-time powertrain control is introduced. The system dynamics are modeled using the quasi-LPV control framework [START_REF] Wang | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]) and a Parallel Distributed Control law along with a state observer is designed. Simulation results are provided in Section IV. Section V concludes this paper and provides directions for future research.

Hybrid vessel optimal control

Powertrain energy management can be studied using quasi-static models for energy converters. The only dynamics considered are then those of the energy storage [START_REF] Guzzella | Vehicle Propulsion Systems: Introduction to Modeling and Optimization[END_REF]. In this paper, we consider the simplified series hybrid vessel depicted in Fig. 1; which is made up of a traction motor, connected to the propeller; a battery pack, with a DC/DC converter; and a fuel cell, with auxiliaries. In order to formalize the dynamics, let w denote the power to be provided for the propulsion, y the power provided to/from the battery, u the fuel cell output power, and p the power consumed by the fuel cell auxiliaries. The mission profile ( ) wt is the DC power to be provided to the traction motor as a function of time t . It may be computed from a speed profile and a vessel model as described in [START_REF] Haseltalab | Multi-Level Predictive Control for Energy Management of Hybrid Ships in the Presence of Uncertainty and Environmental Disturbances[END_REF]. Alternatively, in this paper, we use data recorded on an existing sightseeing barge. The considered 6-days long mission profile is given in Fig. 2. For the considered optimal control problem, the mission is assumed to be of fixed length T and known of the optimization horizon   0,T . This study investigates a possible fuel cell retrofit of the barge such that it could be operated on hydrogen and electricity from the batteries.

Fig. 2: Considered mission profile

Notation: When there is no ambiguity, the time dependence is omitted. The partial derivative with respect to a variable is denoted using a subscript. For instance, x H denotes the partial derivative of H with respect to .

x

The power split is given by: w y u p = + -

Both the battery power y and the fuel cell output power u are limited:

( )   min max , y t y y  (2) ( )   min max , u t u u  (3)
Over an optimization horizon   0,T , the hydrogen consumption to be minimized is assumed to be a quadratic function of the fuel cell produced power u [START_REF] Tazelaar | Sizing Stack and Battery of a Fuel Cell Hybrid Distribution Truck[END_REF]:

( ) 0 T f m l u dt =  (4) ( ) 2 l u a bu cu = + + (5)
The fuel cell auxiliaries electric consumption is estimated using the following loss model [START_REF] Tazelaar | Sizing Stack and Battery of a Fuel Cell Hybrid Distribution Truck[END_REF]:

01 pu  =+ (6)
Combining ( 1)-( 3) and ( 6), the feasible set ( ) ( ) , u w u w   of the fuel cell output power can be determined:

( ) ( )

0 min max min 1 min , max , 1 wy u w u u     +- =     -   (7) ( ) ( ) 0 max min 1 max , max , 1 max wy u w u u     +- =     -   (8)
The battery coupled to the DC/DC converter is modeled using a simple voltage generator with a resistance in series [START_REF] Gao | Adaptive real-time optimal energy management strategy based on equivalent factors optimization for hybrid fuel cell system[END_REF][START_REF] Oncken | Real-Time Model Predictive Powertrain Control for a Connected Plug-In Hybrid Electric Vehicle[END_REF]:

2 4 2 batt E E Ry I R -- = (9)
The open circuit voltage E and the internal resistance R are assumed to be constant. The battery SOC dynamics is :

( ) ( ) , x t f u w = (10) With ( ) ( ) 2 4 , 2 batt It E E Ry f u w Q QR - -+ - == .
Two additional constraints on the SOC are considered:

( ) 0 0 xx = (11) ( ) T x T x = (12)
with 0

x and T x as the initial and final SOC, respectively. Equations ( 1)-( 12) represent the optimal control problem to be solved. Several approaches can be considered to solve it. With ( )

t 
the co-state, the Hamiltonian associated with the optimal control problem is formulated as:

( ) ( ) , , , H u w l f u w  =+ . ( 13
)
The optimal control policy can now be denoted by  :

( )

, uw  = (14) ( ) ( ) ( ) ( ) , , arg min , , u w u w w H w        = (15)
The optimal co-state dynamics is then:

0 x H  = -= . (16) Let us denote   , T Yx  =
, the BVP to be solved is:

( ) , , 0 T Y g w  =   (17) ( ) ( ) ( ) 0 , 0 d x x T = , (18)
where

( ) ( ) ( ) ( ) ( ) ( ) 0 0 , 0 , T d x x T x x x T x = - -
and the optimal state dynamics is:

( ) ( ) ( ) , , , g w f w w  = . ( 19
)
The BVP ( 17)-( 18) can be solved using a dedicated solver [START_REF] Kitzhofer | The New MATLAB Code bvpsuite for the Solution of Singular Implicit BVPs[END_REF] or using a simple bi-section search [START_REF] Armenta | Computational Reduction of Optimal Hybrid Vehicle Energy Management[END_REF].

01 x = , 0.2 T x =
. The initial co-state is ( )

0 -18.94  =
. Fig. 3 depicts the optimal control result. The obtained hydrogen consumption is 27.8 kg and the final SOC is 19.99 % . It should be noticed that the studied optimal control problem does not include any SOC limits. As a result, the SOC is allowed to reach negative values, and its corresponding fuel consumption will be interpreted as a lower bound as it does not have physical meaning (Interested readers may refer to state constrained PMP studied in [START_REF] Hermant | On the shooting algorithm for optimal control problems with state constraints[END_REF] or to PMP with state penalties [START_REF] Sanchez | Hybrid Vehicle Energy Management: Avoiding the Explicit Hamiltonian Minimization[END_REF]). This result emphasizes that real-time powertrain operations should not be performed using a constant co-state, but a SOC feedback controller.

Real-time control

During real-time operation, the future values of the mission profile and the mission length are unknown. The considered energy management consists of minimizing the hydrogen consumption while regulating the SOC nearby a reference setpoint. This reference setpoint can be either a constant or may follow a discharge profile when the battery is large enough to ensure the vessel operation in pure electric mode for a period of time long enough (plugin operations). If additional information on the mission is available, then more elaborated strategies can be adopted. It should be noticed that the considered problem involves not only the state-of-charge (SOC) tracking but also the energy management (i.e. ensuring energy-efficient powertrain operations). In this work, these two aspects are tackled (i) using the quasi-LPV literature for the controller synthesis and for demonstrating the closed-loop stability and performances (ii) using the Hamiltonian minimization (based on optimal policy (14)) within the control scheme to ensure an efficient energy management.

Control structure

Due to the control saturation ( 2  have been computed for the parameters given in Table 1. They are depicted with a green and red line in Fig. 4. be the tracking error. The dynamic to be controlled is:

( ) ( ) ( ) ( ) , ref e t g t w t x  =- (20)
To design a controller for (20), first, we consider the time derivative of (20) as follows:

( ) ( ) ( )

( ) ( ) ( ) min max 0 ,, t sat v d w w       =    , (23)
where

( ) ( ) ( ) , , min , max , sat a b c b a c = .
In practice, (23) should be implemented using an anti-windup scheme.

The following state space representation is considered:

( ) ( ) 00
,, 10 00

w ee g w g w vw ee            = +  +                    . ( 24
)
The derivative of the mission profile w is in general not available. It will be treated as a norm bounded disturbance.

Controller design

Let us introduce a more compact notation for the system (24) :

0 1 , C = dw = .
The quasi-LPV provides a methodology to stabilize a wide range of nonlinear systems in a systematic way. According to Lemma 1 of [START_REF] Van Keulen | Solution for state constrained optimal control problems applied to power split control for hybrid vehicles[END_REF] [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]:

( ) ( ) ( ) 2 1 i h i i X AX h z B v t Dd AX B v t Dd x CX =  = + + = + +    =   (26) with , 0 i i z B  =   ( ) 2 1 21 zz hz zz - = - , ( ) ( ) 21 1 h z h z =- .
There exists an extensive literature on quasi-LPV control to deal with systems with the form of ( 26). A very simple control law is used but of course, many other alternatives are available (robust control, non-quadratic stability (Abdelkrim et al., 2019), etc.).

In order to stabilize the system (26) let us consider a PDC control law [START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]. It has the following form:

( ) 2 1 î i i v h z F X = =  , ( 27
)
with i F as the gains to be determined and X an estimate of the state X computed using a linear observer of the form:

( ) ( ) ˆˆ,  B v L x X X w x x CX   = + + -    =  (28)
with L as the observer gain. Let ô eX X

=be the observer error whose dynamics is

( ) ( ) , oo d A LC e e D w  =- + . ( 29
)
Considering ( 26) and ( 29) , we ( ) ( )

22 11 ij ij ij h h z h z G d G d X X D X D == = + = +  (30)
Where In the particular case ( ) 0 dt = , computing the gain j F and L for the closed loop (30) stabilization using Linear Matrix Inequalities (LMI) is straightforward using classical matrix manipulations [START_REF] Boyd | Fast Optimal Energy Management With Engine On/Off Decisions for Plug-in Hybrid Electric Vehicles[END_REF][START_REF] Tanaka | Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach[END_REF]. Then, our goal is to prove the ISS stability of the system (30) in the presence of a disturbance d . Proposition. Let us consider the system dynamics (26) with the nonlinear control law (27) and observer (28), then, the origin 0 X = of the closed-loop dynamics (30) is ISS with a decay rate  and input constraint v   , for a scalar 0   and initial condition 0 X , if there exist matrices 1 0

P  , 2 0 P  , L , i M ,   1, 2,..., , ir
M I  - - -  -              (31)
where ( ) Proof. Considering ( ) 0 dt = , let us prove that the origin 0 X = of the system is asymptotically stable with a decay rate  . Taking a quadratic Lyapunov function as:

11 11 1 1 ij S A B P M P  -- + + = + , 1 12 1 ij SB M P - - = , ( ) ( )
( )

1 2 0 0 TT P X X PX X X V P    =  =   .
Let min P  and max P  be the smallest and largest eigenvalue of P . The proof of the stability condition ( 31) is trivial and is given in [START_REF] Yoneyama | Output stabilization of Takagi-Sugeno fuzzy systems[END_REF]. Since the system is asymptotically stable with a decay rate  , we have:

( ) 2 min hh T T T P GP X PG P X X X    -  + - . (32)
Now, let us consider

( ) 0 dt  . If there exist -functions i  ,   1, 2,3, 4 i  such that ( ) ( ) ( ) 12 n X X X V X      and ( ) ( ) ( ) 34 , d VXd X
  -+ then the system is ISS [START_REF] Sontag | New characterizations of input-to-state stability[END_REF]. The first condition is fulfilled, since Remark. From the first inequality in (31), matrices i M ,   1, 2 , i  can be obtained. In the second inequality, by fixing negative poles for the linear observer,  can be chosen large enough such that (31) holds. The proposition above implies that the LMI conditions allow designing the observer and the controller separately and moreover the origin 0 X = of the closed-loop system (30) is ISS. Finally, the control design procedure is : Step 1 compute the function g and its limits

Simulation results

In this section, a simulation of the hybrid powertrain, shown in Fig. 1, is carried out using the quasi-LPV ECMS to obtain the control signal. From the PMP optimality conditions ( 16),  should be kept as close to 0 as possible. In practice, a compromise has to be found between the SOC error and the hydrogen consumption. Several simulations have been conducted over the mission profile depicted in Fig. 2 In order to assess the effectiveness of our approach it is compared against the adaptive-ECMS strategy from [START_REF] Onori | Adaptive Equivalent Consumption Minimization Strategy for Hybrid Electric Vehicles[END_REF], where the costate is computed as: ( ) ( )

( 1) 0.5 ( ) ( 1)

p ref c x kk x k    + -+ - += (36)
We chose 100 p c = and a costate update period of 50 s. Fig. 6 depicts the comparison of the control input () ut and the battery power profile () yt ; it is shown how in the adaptive-ECMS strategy the control signal presents a bang-bang behaviour, whereas in the quasi-LPV ECMS the control signal is smoother; the corresponding evolution of the SOC is presented in Fig. 7. The obtained hydrogen consumption is 33.6 kg for our approach and 41.68 kg for the adaptive one.

Conclusions

A controller that benefits from the ECMS and quasi-LPV framework has been proposed. The controller synthesis is reduced to a set of LMI conditions to be solved. The closed loop ISS has been demonstrated.

Preliminary results have been presented to illustrate the effectiveness of our approach. Two design parameters  (related to a constraint on the costate derivative) and  (decay rate) allows tuning the closed loop dynamics.  control the tradeoff between fuel consumption and state of charge regulation. Future work will be devoted to more in dept analysis of the control performances both in terms of fuel consumption and SOC reference tracking.

Fig. 1 :

 1 Fig. 1: Powertrain architecture

Fig. 3 :

 3 Fig. 3: Optimal control results for ( ) 01 x = and ( ) 0.2 xT =

  )-(3) , not all the co-state values are of interest for SOC control. Let us restrict the co-state to  , with  small negative constant. As an example,

Fig. 4 :

 4 Fig. 4: State dynamics ( ) , gw 

  LMI (31).Step 3 : design a linear observer (28) by pole placement. The whole control scheme to be implemented comprises the observer (28) , the control law (27), the change of variable (23), and finally the Hamiltonian minimization (15).

Fig. 5 :

 5 Fig. 5: Hydrogen consumption and () RMS  as a function of  .

Fig. 6 :

 6 Fig. 6: Real time control law (simulation result).

Fig. 7 :

 7 Fig. 7: SOC profiles comparison (simulation result).

Table 1 :

 1 Parameters values

		Parameter		Value
	(	min yy max ,	)	(	30 -	kW	,30	kW	)
	(	min uu ,	max	)	(	0	kW	,15	kW	)
	(	E	,, R	Q	)	( 105.3 , 0.985 V	, 2813  m	Ah	)
			(	,, abc	)	(	6 7.48 10 ,6.97 10 ,7.40 10 6 --  	7 -	)
			(	01 , 	)			(	5 6.3 10 ,0.125	)

-

As an illustration, let us consider a powertrain whose parameters are shown in Table

1

. The BVP has been solved for ( )

  ,

	z g =	( ) , w  	is upper bounded. In order to formulate the
	system dynamic as a quasi-LPV model, let us consider a non-linear sector as the measured and controlled output	 z z z 12 , 		with 12 zz    and x 0

  F and larger (resp. smaller) amplitude of the control signal  in response to the exogenous signal w . The reference SOC signal

					Considering the bounds	z	[ 4.696 10  -	78 ] , 3.2021 10 ---	, the two controller
	gains are computed choosing a decay rate		1 10 =	7 -	and the initial condition 0 [0 0.425] T X =	; whereas the
	observer gain is obtained by poles placement with poles 10 -and 11 -. When solving the LMI conditions of
	proposition 1, the value of  allows reaching different closed loop dynamics. Larger (resp. lower)  values lead
	to larger (resp. lower) control gains 1,2
							ref xt varies from 100% down to a specified value with ( )
	a rate limitation of	0.75% -	/	h	.

Acknowledgements

This work was supported in by the European Interreg Program through the Implentation of Hybrid Ship project ISHY) of the EU Interreg 2 seas program.


Michel Balazard

On the infimum of the absolute value of successive derivatives of a real function defined on a bounded interval

Keywords: Chebyshev polynomials, Legendre polynomials, extremal problems, inequalities for derivatives MSC classification: 26D10, 41A10

A study of the greatest possible ratio of the smallest absolute value of a higher derivative of some function, defined on a bounded interval, to the L p -norm of the function.

Introduction

Let n be a positive integer, I = [a, b] a bounded segment of the real line, of length L = b -a. Define D n (I) as the set of real functions f defined on I, with successive derivatives f (k) defined and continuous on I for 0 k n -1, and f (n) defined on I = ]a, b[. We will use the notation This problem has been posed by Kwong and Zettl in their 1992 Lecture Notes [START_REF] Kwong | Zettl -Norm inequalities for derivatives and differences[END_REF] (see Lemma 1.1, p. 6). They give upper bounds for C * (n, p, I), but their reasoning and results are erroneous. In her 1993 PhD Thesis [START_REF] Huang | On extremal properties of algebraic polynomials[END_REF], Huang has pointed out that this problem is equivalent to a classical problem in the theory of polynomial approximation: that of determining the minimal L pnorm of a monic polynomial of given degree on a given bounded interval. Our purpose in this text is to give a new proof of the equivalence, and to list the consequences of the known results about this extremal problem for the evaluation of C * (n, p, I).

m n (f ) = inf a<t<b |f (n) (t)|.

First observations

Homogeneity

Defining g(u) = f (a + uL) for f ∈ D n (I) and 0 u 1, one has

g ∈ D n ([0, 1]) ; g (n) (u) = L n f (n) (a + uL) (0 < u < 1) ; g p = L -1/p f p .
Hence, C * (n, p, I) = C * (n, p, [0, 1])

L -n-1/p , (1)
and one is left with determining C * (n, p, [0, 1]) = C(n, p), or in fact C * (n, p, I) for any fixed, chosen segment I. We will see that I = [-1, 1] is particularly convenient.

An extremal problem

One has

C * (n, p, I) = sup{m n (f )/ f p , f ∈ D n (I), m n (f ) = 0} = sup{m n (f )/ f p , f ∈ D n (I), m n (f ) = λ} (for every λ > 0) = λ/D * (n, p, λ, I),
where Also, since a derivative has the intermediate value property (cf. [START_REF]Mémoire sur les fonctions discontinues[END_REF], pp. 109-110), the inequality m n (f ) λ > 0 implies that f (n) has constant sign on I, so that

D * (n, p, λ, I) = inf{ f p , f ∈ D n (I), m n (f ) = λ} = inf{ f p , f ∈ D n (I), m n (f ) λ},
D * (n, p, λ, I) = inf{ f p , f ∈ D n (I), f (n) (t) λ for a < t < b}.
Thus, determining C * (n, p, I) is equivalent to minimizing f p for f ∈ D n (I) with the constraint f (n) (t) λ > 0 for a < t < b. We will denote this extremal problem by E * (n, p, λ, I).

The relevance of monic polynomials

Let P n be the set of monic polynomials of degree n, with real coefficients, identified with the set of the corresponding polynomial functions on I, which is a subset of

D n (I). Since m n (f ) = n! for f ∈ P n , one has D * (n, p, n!, I) D * * (n, p, I), (2)
where

D * * (n, p, I) = inf{ Q p , Q ∈ P n }.
A basic fact in the study of the extremal problem E * (n, p, λ, I) is that ( 2) is in fact an equality.

Let us review the history of Proposition 1. For p = ∞, it is a corollary to a theorem of S. N. Bernstein from 1937. Denoting by E k (f ) the distance (for the uniform norm on I) between f and the set of polynomials of degree at most k, he proved in particular that [START_REF] Bernstein | Extremal properties of polynomials and the best approximation of continuous functions of a single real variable. part i[END_REF], p. 48, inequalities (47bis)-(48bis)). Proposition 1 follows by taking f 1 (x) = x n and f 0 (x) = λf (x), where f is a generic element of D n (I) such that f (n) (t) n! for a < t < b, and λ > 1, then letting λ → 1.

E n-1 (f 0 ) > E n-1 (f 1 ) (f 0 , f 1 ∈ D n (I)), provided that the inequality f (n) 0 (ξ) > |f (n) 1 (ξ)| is valid for every ξ ∈ I (cf.
This theorem of Bernstein was generalized by Tsenov in 1951 to the case of the L p -norm on I, where p 1 (cf. [START_REF] Tsenov | On a question of the approximation of functions by polynomials[END_REF], Theorem 4, p. 477), thus providing a proof of Proposition 1 for p 1. The case 0 < p < 1 was left open by Tsenov.

The study of the extremal problem E * (n, p, λ, I) was one of the themes of the 1993 PhD thesis of Xiaoming Huang [START_REF] Huang | On extremal properties of algebraic polynomials[END_REF]. In Lemma 2.0.7, pp. 9-10, she gave another proof (due to Saff) of Proposition 1 in the case p = ∞. For 1 p < ∞, she gave a proof of Proposition 1 which is unfortunately incomplete (cf. [START_REF] Huang | On extremal properties of algebraic polynomials[END_REF], pp. 28-30). Again, the case 0 < p < 1 was left open.

We present now a self-contained proof of Proposition 1, valid for 0 < p ∞. As it proceeds by induction on n, we will need the following classical-looking division lemma, for which we could not locate a reference (compare with [START_REF] Whitney | Differentiability of the remainder term in Taylor's formula[END_REF] or [START_REF] Schoenfeld | On the differentiability of indeterminate quotients[END_REF]).

Proposition 2 Let n 2 and f ∈ D n (I). Let c ∈ [a, b]. Put g(x) =    f (x) -f (c) x -c (x ∈ I, x = c) f ′ (c) (x = c). (4)
Then g ∈ D n-1 (I). For every x ∈ ]a, b[ , one has

g (n-1) (x) = f (n) (ξ) n , where ξ ∈ ]a, b[. Proof Since f ′ is continuous, one has g(x) = 1 0 f ′ c + t(x -c) dt (x ∈ I).
Using the rule of differentiation under the integration sign, one sees that g is n -2 times differentiable on I, with

g (n-2) (x) = 1 0 t n-2 f (n-1) c + t(x -c) dt (x ∈ I).
As f (n-1) is continuous on I, this formula yields the continuity of g (n-2) on I.

The function g is n times differentiable on I \ {c} (this set is just I if c = a or c = b),
being a quotient of n times differentiable functions, with non-vanishing denominator. In the case a < c < b, we have now to check that g is n -1 times differentiable at the point c.

The function f (n-1) being continuous on I and differentiable at the point c, there exists a function ε(h), defined and continuous on the segment [a -c, b -c] (the interior of which contains 0), vanishing for h = 0, such that

f (n-1) (c + h) = f (n-1) (c) + hf (n) (c) + hε(h) (a c + h b).
Hence,

g (n-2) (x) = 1 0 t n-2 f (n-1) c + t(x -c) dt = 1 0 t n-2 f (n-1) (c) + t(x -c)f (n) (c) + t(x -c)ε t(x -c) dt = f (n-1) (c) n -1 + f (n) (c) n (x -c) + (x -c) 1 0 t n-1 ε t(x -c) dt
When x tends to c, the last integral tends to 0, so that the function g (n-2) is differentiable at the point c, with

g (n-1) (c) = f (n) (c) n • If x ∈ I \ {c}
, one may use the general Leibniz rule and Taylor's theorem with the Lagrange form of the remainder in order to compute g (n-1) (x) :

g (n-1) (x) = d n-1 dx n-1 f (x) -f (c) • 1 x -c = f (x) -f (c) • (-1) n-1 (n -1)! (x -c) n + n-1 k=1 n -1 k f (k) (x) • (-1) n-1-k (n -1 -k)! (x -c) n-k = (n -1)! (c -x) n f (c) -f (x) - n-1 k=1 f (k) (x) k! (c -x) k = (n -1)! (c -x) n • f (n) (ξ) n! (c -x) n (
= f (n) (ξ) n • ✷
In the next proposition, we stress the main element of our proof of Proposition 1, namely the fact that the condition f (n) n!, for some f ∈ D n (I), implies that the absolute value of f dominates the absolute value of some monic polynomial of degree n.

Proposition 3 Let n 1 and f ∈ D n (I) such that f (n) (x) n! for every x ∈ ]a, b[ .
Then there exists a monic polynomial P of degree n, with all its zeros in I, such that the inequality |f (x)| |P (x)| is valid for every x ∈ I.

Moreover, if |f (x)| = |Q(x)| for every x ∈ I, where Q is a monic polynomial of degree n with real coefficients, then f (x) = Q(x) for every x ∈ I.

Proof

The assertion about the zeros may be obtained a posteriori, by replacing the zeros of P by their projections on I. The following proof leads directly to a polynomial P with all zeros in I.

We use induction on n.

For n = 1, the function f is continuous on [a, b], differentiable on ]a, b[ , with f ′ (x) 1 for a < x < b. If f (a) 0, one has, for a < x b, f (x) = f (a) + (x -a)f ′ (ξ) (where a < ξ < x), thus f (x) x -a. Hence, one has |f (x)| |x -a| for every x ∈ I. If f (b) 0, one proves similarly that |f (x)| |x -b| for every x ∈ I. If f (a) < 0 < f (b), there exists c ∈ ]a, b[ such that f (c) = 0. One has then, for every x ∈ I, f (x) = f (x) -f (c) = (x -c)f ′ (ξ) (where a < ξ < b).
Hence |f (x)| |x -c| for every x ∈ I, and the result is proven for n = 1.

Let now n 2, and suppose that the result is valid with n -

1 instead of n. Let f ∈ D n (I) such that f (n) (x) n! for every x ∈ ]a, b[ .
If f vanishes at some point c ∈ I, it follows from Proposition 2 that the function g defined on I by

g(x) =    f (x) x -c (x ∈ I, x = c) f ′ (c) (x = c),
belongs to D n-1 (I) and that, for every x ∈ ]a, b[ , one has

g (n-1) (x) = f (n) (ξ) n ,
where ξ ∈ ]a, b[, thus g (n-1) (x) (n -1)!. By the induction hypothesis, there exists a monic polynomial Q of degree n -1, with all its roots in I, such that |g(x)| |Q(x)| for every x ∈ I. Hence, one has the inequality |f (x)| |P (x)| for every x ∈ I, where P (x) = (x -c)Q(x) is a monic polynomial of degree n, with all its roots in I.

If f > 0, it reaches a minimum at some point c ∈ I. Again, it follows from Proposition 2 that the function g defined on I by (4) satisfies the required hypothesis for degree n -1. Thus there exists a monic polynomial Q of degree n -1, with all its roots in I, such that |g(x)| |Q(x)| for every x ∈ I. Hence, one has the inequality In other words, if 0 < p < ∞, the extremal problem E * (n, p, n!, I) has exactly the same solutions (value of the infimum and extremal functions) as the problem E * * (n, p, I) obtained by considering only monic polynomials of degree n, which one may even take with all their roots in I.

f (x) -f (c) = |f (x) -f (c)| |P (x)| (x ∈ I), where P (x) = (x -c)Q(x). It follows that |f (x)| = f (x) f (c) + |P (x)| > |P (x)| (x ∈ I) If f < 0,
For p = ∞, our reasoning does not prove that an extremal function for E * (n, p, n!, I) (if it exists) must be a polynomial. This is true anyway, as proved by Huang in [START_REF] Huang | On extremal properties of algebraic polynomials[END_REF], pp. 10-13.

Extremal polynomials

One may now use the results of the well developed theory of the extremal problem E * * (n, p, I) for polynomials. Thus, since the integral

b a |(x -x 1 ) • • • (x -x n )| p dx (x 1 , . . . , x n ∈ I) (or the value max x∈I |(x -x 1 ) • • • (x -x n )|
) is a continuous function of (x 1 , . . . , x n ), the compactness of I n yields the existence of an extremal (polynomial) function for E * * (n, p, I), hence for E * (n, p, n!, I).

It is a known fact that the polynomial extremal problem E * * (n, p, I) has a unique solution for all p ∈ ]0, ∞], but there is no proof valid uniformly for all values of p.

• For p = ∞, uniqueness was proved by Young in 1907 (cf. [START_REF] Young | General theory of approximation by functions involving a given number of arbitrary parameters[END_REF], Theorem 5, p. 340)) and follows from the general theory of uniform approximation (cf. [START_REF] Rivlin | An introduction to the approximation of functions[END_REF], Theorem 1.8, p. 28).

• For 1 < p < ∞, as proved by Jackson in 1921 (cf. [START_REF]On functions of closest approximation[END_REF], §6, pp. 121-122), this is a consequence of the strict convexity of the space L p (I).

• For p = 1, this is also due to Jackson in 1921 (cf. [START_REF] Jackson | Note on a class of polynomials of approximation[END_REF], §4, pp. 323-326).

• For 0 < p < 1, the uniqueness of the extremal polynomial was proved in 1988 by Kroó and Saff (cf. [START_REF] Kroó | On polynomials of minimal L q -deviation, 0 < q < 1[END_REF], Theorem 2, p. 184). Their proof uses the uniqueness property for p = 1 and the implicit function theorem.

We will denote by T n,p,I the unique solution of the extremal problem E * * (n, p, I). Uniqueness gives immediately the relation

T n,p,I (a + b -x) = (-1) n T n,p,I (x) (x ∈ R).
Another property of these polynomials is the fact that all their roots are simple. For p = 1, this fact was proved by Korkine and Zolotareff in 1873 (cf. [START_REF] Korkine | Sur un certain minimum[END_REF], pp. 339-340), before their explicit determination of the extremal polynomial (see §5.4 below), and their proof extends, mutatis mutandis, to the case 1 < p < ∞. For p = ∞, this is a property of the Chebyshev polynomials of the first kind (see §5.2 below). Lastly, for 0 < p < 1, this was proved by Kroó and Saff in [START_REF] Kroó | On polynomials of minimal L q -deviation, 0 < q < 1[END_REF], p. 187.

Define T n,p = T n,p,[-1,1] , and write n = 2k + ε, where k ∈ N and ε ∈ {0, 1}. It follows from the mentioned results that

T n,p (x) = x ε (x 2 -x n,1 (p) 2 ) • • • (x 2 -x n,k (p) 2 ) (x ∈ R), (6)
where

0 < x n,1 (p) < • • • < x n,k (p) 1.
Kroó, Peherstorfer and Saff have conjectured that all the x n,k are increasing functions of p (cf. [START_REF] Kroó | On the zeros of polynomials of minimal L p -norm[END_REF], p. 656, and [START_REF] Kroó | On polynomials of minimal L q -deviation, 0 < q < 1[END_REF], p. 192).

Results on C(n, p)

The case n = 1

The value n = 1 is the only one for which C(n, p) is explicitly known for all p.

Proposition 4 One has C(1, p) = 2(p + 1) 1/p for 0 < p < ∞, and C(1, ∞) = 2.

Proof

By ( 6), one has T 1,p (x) = x, so that, for 0 < p < ∞,

D * * (1, p, [-1, 1]) = 1 -1 |t| p dt 1/p = 2/(p + 1) 1/p , and, by (3)
, C(1, p) = 2 1+1/p /D * * (1, p, [-1, 1]) = 2(p + 1) 1/p . ✷
Note that the Lemma 1.1, p. 6 of [START_REF] Kwong | Zettl -Norm inequalities for derivatives and differences[END_REF], asserts that C(1, p) 2 • 3 1/p for p 2, and that bound is < 2(p + 1) 1/p for p > 2.

The case p = ∞

This is the classical case, solved by Chebyshev in 1853 by introducing the polynomials T n defined by the relation T n (cos t) = cos nt (now called Chebyshev polynomial of the first kind): the unique solution of the extremal problem

E * * (n, ∞, [-1, 1]) is 2 1-n T n .
Let us record a short proof of this fact.

Take I = [-1, 1] and suppose that P is a monic polynomial of degree n satisfying the inequality P ∞ 2 1-n T n ∞ = 2 1-n . Then, for λ > 1 the polynomial

Q λ = λ2 1-n T n -P is of degree n, with leading coefficient λ -1. Moreover, it satisfies (-1) k Q λ (cos kπ/n) = λ2 1-n -(-1) k P (cos kπ/n) > 0 (k = 0, . . . , n)
By the intermediate value property, Q λ has at least n distinct roots, hence exactly n, and these roots, say x 1 , . . . , x n , have absolute value not larger than 1. Hence,

|Q λ (x)| = (λ -1) |(x -x 1 ) • • • (x -x n )| (λ -1)(1 + |x|) n (x ∈ R).
When λ → 1, Q λ (x) tends to 0 for every real x, which means that P = 2 1-n T n .

One deduces from this theorem the value of C(n, ∞). One has

D * * (n, ∞, [-1, 1]) = max |x| 1 2 1-n T n (x) = 2 1-n , hence C(n, ∞) = 2 n • n!/D * * (n, ∞, [-1, 1]) = 2 2n-1 n! (7)
(compare with the upper bound C(n, ∞) 2 n(n+1)/2 n n of [START_REF]Dieudonné -Foundations of modern analysis[END_REF], 3 (a), p. 185). This result is essentially due to Bernstein (1912, cf. [1], p. 65).

Qualitatively, the result expressed by [START_REF]On functions of closest approximation[END_REF] was nicely described by Soula in [START_REF] Soula | Sur une inégalité verifiee par une fonction et sa dérivée d'ordre n[END_REF], p. 86, as follows.

Bernstein's principle: the minimum of the absolute value of the n-th derivative of an n times differentiable function and the maximum of the absolute value of the n-th derivative of an analytic function have similar orders of magnitude.

The case p = 2

In this case, the extremal problem E * * (n, 2, [-1, 1]) is an instance of the general problem of computing the orthogonal projection of an element of a Hilbert space onto a finite dimensional subspace. Here, the Hilbert space is L 2 (-1, 1), the element is the monomial function x n , and the subspace is the set of polynomial functions of degree less than n. The solution follows from the theory of orthogonal polynomials: the extremal polynomial for E * * (n, 2, [-1, 1]) is

2 n (n!) 2 (2n)! P n (x) (|x| 1),
where P n is the n-th Legendre polynomial, defined by

P n (x) = 1 2 n n! d n dx n (x 2 -1) n .
Hence,

D * * (n, 2, [-1, 1]) = 2 n (n!) 2 (2n)! P n 2 = 2 n (n!) 2 (2n)! 2 2n + 1 ,
(see [START_REF] Whittaker | Watson -A course of modern analysis[END_REF], §15•14, p. 305) and

C(n, 2) = 2 n+ 1 2 • n!/D * * (n, 2, [-1, 1]) = (2n)! n! √ 2n + 1, (8)
a result given by Soula in 1932 (cf. [START_REF] Soula | Sur une inégalité verifiee par une fonction et sa dérivée d'ordre n[END_REF], pp. 87-88).

The case p = 1

The problem E * * (n, 1, [-1, 1]) was solved by Korkine and Zolotareff in [START_REF] Korkine | Sur un certain minimum[END_REF]: the extremal polynomial is 2 -n U n (x), where U n is the n-th Chebyshev polynomial of the second kind, defined by the relation U n (cos t) = sin(n + 1)t/ sin t.

Therefore, one has

D * * (n, 1, [-1, 1]) = 2 -n 1 -1 |U n (x)| dx = 2 -n π 0 |U n (cos t)| sin t dt = 2 -n π 0 |sin(n + 1)t| dt = 2 -n π 0 sin u du = 2 1-n , and
C(n, 1) = 2 n+1 • n!/D * * (n, 1, [-1, 1]) = 2 2n n!. (9)

Bounds for C(n, p)

We begin with a simple monotony result.

Proposition 5 For every positive integer n, the function p → C(n, p) is decreasing on the interval 0 < p ∞.

To prove the second inequality of (10), we just compute Q p p when Q(t) = (t -1/2) n :

1 0 |t -1/2| np dt = 2 (1/2) np+1 np + 1 • ✷
For 0 < p < 1, we can also prove the following result.

Proposition 7 Let n be a positive integer, and p such that 0 < p < 1. One has

1 C(n, p) 2 2n n! 1 2 (8/π) 1/p .

Proof

The first inequality is just C(n, 1) C(n, p).

To prove the second inequality, let r and s such that 1 < s < 2 and r -1 + s -1 = 1. Define

I 1 (s) = 1 -1 dt (1 -t 2 ) s/2 I 2 (s) = 1 -1 |t| (s-1)/s dt √ 1 -t 2 •
The integrals I 1 (s) and I 2 (s) may be computed, using the eulerian identity

1 0 t x-1 (1 -t) y-1 dt = B(x, y) = Γ(x)Γ(y) Γ(x + y) (x > 0, y > 0).
The results are

I 1 (s) = 2 1-s Γ(1 -s 2 ) 2 Γ(2 -s) I 2 (s) = Γ 1 -1 2s Γ( 1 2 ) Γ( 3 2 -1 2s ) •
Now, let Q ∈ P n and put p ′ = p/r. By Hölder's inequality, one has

1 -1 |Q(t)| p ′ dt √ 1 -t 2 1 -1 |Q(t)| p ′ r dt 1/r 1 -1 dt (1 -t 2 ) s/2 1/s = Q p ′ p I 1 (s) 1/s .
It was proved by Kroó and Saff (cf. [START_REF] Kroó | On polynomials of minimal L q -deviation, 0 < q < 1[END_REF], pp. 182-183) that

2 (n-1)p ′ 1 -1 |Q(t)| p ′ dt √ 1 -t 2 1 -1 |T n (t)| p ′ dt √ 1 -t 2 = π 0 |cos nu| p ′ du = π 0 |cos u| p ′ du = 1 -1 |t| p ′ dt √ 1 -t 2 1 -1 |t| 1/r dt √ 1 -t 2 (one has p ′ = p/r < 1/r) = I 2 (s).
Therefore, with I = [-1, 1],

Q p 2 1-n I 2 (s) 1/p ′ I 1 (s) -1/p ′ s = 2 1-n A(s) 1/p (1 < s < 2), (11)
where A(s) = I 2 (s) s/(s-1) I 1 (s) -1/(s-1) .

Hence The result follows. ✷

A(s) = 2 Γ 1 -1 2s s Γ( 1 2 ) s Γ(2 -s) Γ(1 -s 2 ) 2 Γ 3 2 -1

An open question

Finally, observing that

C(n, 2) ∼ 2 π • 2 2n n! (n → ∞),
(an exercise on Stirling's formula from ( 8)), we ask the following question.

Is it true that, for every p > 0, the quantity 2 -2n C(n, p)/n! tends to a limit when n tends to infinity?

,

  Let p be a positive real number, or ∞. The problem addressed in this article is that of determining the best constantC * = C * (n, p, I) in the inequality m n (f ) C * f p (f ∈ D n (I)),with the usual convention when p = ∞, here: f ∞ = max |f |.

  the last equality being true since D * (n, p, µ, I) = µ λ D * (n, p, λ, I) D * (n, p, λ, I) if µ λ.

Proposition 1

 1 For all n, p, I, one has D * (n, p, n!, I) = D * * (n, p, I). It follows from this proposition that C * (n, p, I) = n!/D * * (n, p, I) and, by (1), C(n, p) = L n+1/p n!/D * * (n, p, I).

  where ξ belongs to the open interval bounded by c and x)

  the reasoning is similar by considering a point c ∈ I where f reaches a maximum. Let us prove the last assertion. The hypothesis |f | = |P | is equivalent to the equality f 2= P 2 , that is (f -P )(f + P ) = 0. The set E = {x ∈ I, f (x) + P (x) = 0} has empty interior, since f (n) (x) + P (n) (x) = 0 on every open subinterval of E, whereas f (n) (x) + P (n) (x)2n! on I. The set I \ E is therefore dense in I ; its elements x all verify f (x) = P (x), hence f = P on I by continuity. ✷ Proposition 1 is an immediate corollary of Proposition 3: by taking f and P as stated there, one has |f (x)| |P (x)| for every x ∈ I, so that b a |f (x)| p dx b a |P (x)| p dx, (5) for every p > 0 (for p = ∞: max |f | max |P |). Moreover, if p < ∞, equality in (5) implies that |f | = |P | on I, hence f = P .

2s s 1 /(s- 1 )( 1 1 • 1 )Proposition 8

 111118 < s < 2). Putting f (s) = ln Γ(s), one has ln A(s) = ln 2 + sf (1 -1/2s) + sf (1/2) + f (2 -s) -2f (1 -s/2) -sf (3/2 -1/2s) s -When s tends to 1, the last fraction tends to = ln π -3 ln 2,with the usual notationf ′ = Γ ′ /Γ = ψ. It follows that A(s) → π 4 (s → 1).Together with[START_REF] Kwong | Zettl -Norm inequalities for derivatives and differences[END_REF], this gives the inequalityD * * (n, p, [-1, 1]) 2 1-n (π/4)1/p and (3) now implies C(n, p) 2 2n-1 n!(8/π) 1/p . ✷ We now prove an inequality involving three values of the function C. Let p, q, r be positive real numbers such that Let m and n be positive integers. Then, C(m + n, p) (3), one has to prove that D * * (m + n, p, I) D * * (m, q, I) • D * * (n, r, I),where I is a segment of the real line.In fact, if P ∈ P m and Q ∈ P n , thenP Q ∈ P m+n henceD * * (m + n, p, I) p I |P (t)Q(t)| p dt I |P (t)| q dt p/q • I |Q(t)| r dt p/rby the definition of D * * (m + n, p, I) and Hölder's inequality. The greatest lower bound of the last term, when P runs over P m and Q runs over P n , is D * * (m, q, I) p • D * * (n, r, I) p .

BALAZARD, MichelInstitut de Mathématiques de Marseille (I2M) CNRS, Aix Marseille Université Marseille, France e-mail address: balazard@math.cnrs.fr

Proof

Let I = [0, 1]. Equivalently, we will see that the function p → D * * (n, p, I) is increasing. This is due to the fact that, for a fixed f ∈ L ∞ (I) such that |f | is not equal almost everywhere to a constant, the function p → f p is increasing (a consequence of Hölder's inequality). Thus, for every Q ∈ P n and 0 < p < p ′ ∞,

✷ In particular, ( 7) and ( 9) yield the inequalities

The next proposition implies that the limit of C(n, p) when p tends to 0 is (2e) n n!.

Proposition 6 For every positive integer n and every positive real number p, one has

Proof

Equivalently, we will prove that

where

, where 0 x 1 , . . . , x n 1. One has

attains its minimal value, namely -1 -ln 2, when x = 1/2. This implies the first inequality of [START_REF] Kroó | On polynomials of minimal L q -deviation, 0 < q < 1[END_REF].


Ioan Doré Landau
email: ioan-dore.landau@gipsa-lab.grenoble-inp.fr

Aurelian Constantinescu

Marouane Alma

Adaptive regulation -Rejection of unknown multiple narrow band disturbances

Keywords: direct adaptive control, internal model principle, Youla-Kucera parametrization, adaptive disturbance rejection, multiple narrow band disturbances

The paper presents a methodology for feedback adaptive control of active vibration systems in the presence of time varying unknown multiple narrow band disturbances. A direct adaptive control scheme based on the internal model principle and the use of the Youla-Kucera parametrization is proposed. This approach is comparatively evaluated with respect to an indirect adaptive control scheme based on the estimation of the disturbance model. The evaluation of the methodology is done in real time on an active suspension system and on an active vibration control system using an inertial actuator.

I. INTRODUCTION

One of the basic problems in control is the attenuation (rejection) of unknown disturbances without measuring them. The common framework is the assumption that the disturbance is the result of a white noise or a Dirac impulse passed through the "model of the disturbance". While in general one can assume a certain structure for such "model of disturbance", its parameters are unknown and may be time varying. This will require to use an adaptive approach. To be more specific, the disturbances considered can be defined as "finite band disturbances". This includes single or multiple narrow band disturbances or sinusoidal disturbances. Furthermore for robustness reasons the disturbances should be located in the frequency domain within the regions where the plant has enough gain (see explanation in section III).

Solutions for this problem, provided that an "image" of the disturbance can be obtained by using an additional transducer, have been proposed by the signal processing community and a number of applications are reported ( [START_REF] Elliott | Active noise control. Noise / News International[END_REF], [START_REF] Elliott | Performance of feedforward and feedback systems for active control[END_REF], [START_REF] Beranek | Noise and Vibration Control Engineering: Principles and Applications[END_REF], [START_REF] Fuller | Active Control of Vibration[END_REF]). However, these solutions (inspired by Widrow's technique for adaptive noise cancellation ( [START_REF] Widrow | Adaptive Signal Processing[END_REF])) ignore the possibilities offered by feedback control systems and require an additional transducer. The principle of this signal processing solution for adaptive rejection of unknown disturbances is that a transducer can provide a measurement, highly correlated with the unknown disturbance. This information is applied to the control input of the plant through an adaptive filter (in general a Finite Impulse Response -FIR) whose parameters are adapted such that the effect of the disturbance upon the output is minimized. The disadvantages of this approach are:

• It requires the use of an additional transducer.

• Difficult choice for the location of this transducer (it is probably the main disadvantage). • It requires the adaptation of many parameters. To achieve the rejection of the disturbance (at least asymptotically) without measuring it, a feedback solution can be considered. As mentioned earlier, the common framework is the assumption that the disturbance is the result of a white noise or a Dirac impulse passed through the "model of the disturbance" 1 . Several problems have been considered within this framework leading to adaptive feedback control solutions:

1) Unknown plant and disturbance models ([14]).

2) Unknown plant model and known disturbance ( [START_REF] Sun | Adaptive control with asymptotic tracking performance and its application to an electro-hydraulic servo system[END_REF], [START_REF] Zhang | Direct adaptive control for tonal disturbance rejection[END_REF]). 3) Known plant and unknown disturbance model ( [START_REF] Bodson | Adaptive algorithms for the rejection of sinusosidal disturbances with unknown frequency[END_REF], [START_REF] Ben Amara | Adaptive sinusoidal disturbance rejection in linear discrete-time systems -Part I: Theory[END_REF], [START_REF] Ben Amara | Adaptive sinusoidal disturbance rejection in linear discrete-time systems -Part II: Experiments[END_REF], [START_REF] Valentinotti | Adaptive Rejection of Unstable Disturbances: Application to a Fed-Batch Fermentation[END_REF], [START_REF] Marino | Robust adaptive compensation of biased sinusoidal disturbances with unknown frequency[END_REF], [START_REF] Ding | Global stabilization and disturbance suppression of a class of nonlinear systems with uncertain internal model[END_REF], [START_REF] Gouraud | Design of robust and frequency adaptive controllers for harmonic disturbance rejection in a single-phase power network[END_REF], [START_REF] Hillerstrom | Rejection of periodic disturbances with unknown period -a frequency domain approach[END_REF], [START_REF] Landau | Adaptive narrow band disturbance rejection applied to an active suspension -an internal model principle approach[END_REF]). The present paper will focus on the last case, since this is the situation encountered in many applications. Among the various approaches considered for solving this problem, the following ones may be mentioned:

1) Use of the internal model principle ( [START_REF] Francis | The internal model principle of control theory[END_REF], [START_REF] Johnson | Theory of disturbance-accomodating controllers[END_REF], [START_REF] Bengtsson | Output regulation and internal models -a frequency domain approach[END_REF], [START_REF] Tsypkin | Stochastic discrete systems with internal models[END_REF], [START_REF] Valentinotti | Adaptive Rejection of Unstable Disturbances: Application to a Fed-Batch Fermentation[END_REF], [START_REF] Ben Amara | Adaptive sinusoidal disturbance rejection in linear discrete-time systems -Part I: Theory[END_REF], [START_REF] Ben Amara | Adaptive sinusoidal disturbance rejection in linear discrete-time systems -Part II: Experiments[END_REF], [START_REF] Gouraud | Design of robust and frequency adaptive controllers for harmonic disturbance rejection in a single-phase power network[END_REF], [START_REF] Hillerstrom | Rejection of periodic disturbances with unknown period -a frequency domain approach[END_REF], [START_REF] Landau | Adaptive narrow band disturbance rejection applied to an active suspension -an internal model principle approach[END_REF]). 2) Use of an observer for the disturbance ( [START_REF] Marino | Robust adaptive compensation of biased sinusoidal disturbances with unknown frequency[END_REF], [START_REF] Ding | Global stabilization and disturbance suppression of a class of nonlinear systems with uncertain internal model[END_REF]). 3) Use of the "phase-locked" loop structure considered in communication systems ( [START_REF] Bodson | Adaptive algorithms for the rejection of sinusosidal disturbances with unknown frequency[END_REF], [START_REF] Bodson | Rejection of periodic distrubances of unknown and timevarying frequency[END_REF]). Of course, since the parameters of the disturbance model are unknown, all these approaches lead to an adaptive implementation which can be of direct or indirect type.

From the user point of view and taking into account the type of operation of existing adaptive disturbance compensation systems one has to consider two modes of operation of the adaptive schemes:

• Self-tuning operation (the adaptation procedure starts either on demand or when the performance is unsatisfactory and the current controller is frozen during the estimation/computation of the new controller parameters). Indirect adaptive control scheme for rejection of unknown disturbances

q B / A -d 1 / S 0 u(t) + + - y(t) Q A R 0 q B -d - -+ N / D p p p (t) 1 w(t) d(t) Plant Model Model
Adaptation Algorithm Ĉontroller Fig. 2. Direct adaptive control scheme for rejection of unknown disturbances

• Adaptive operation (the adaptation is performed continuously and the controller is updated at each sampling).

Using the internal model principle, the controller should incorporate the model of the disturbance ( [START_REF] Francis | The internal model principle of control theory[END_REF], [START_REF] Johnson | Theory of disturbance-accomodating controllers[END_REF], [START_REF] Bengtsson | Output regulation and internal models -a frequency domain approach[END_REF], [START_REF] Tsypkin | Stochastic discrete systems with internal models[END_REF]). Therefore the rejection of unknown disturbances raises the problem of adapting the internal model of the controller and its re-design in real-time.

One way for solving this problem is to try to estimate in real time the model of the disturbance and re-compute the controller, which will incorporate the estimated model of the disturbance (as a pre-specified element of the controller). While the disturbance is unknown and its model needs to be estimated, one assumes that the model of the plant is known (obtained for example by identification). The estimation of the disturbance model can be done by using standard parameter estimation algorithms (see for example [START_REF] Landau | Techniques de modélisation récursive pour l'analyse spectrale paramétrique adaptative[END_REF], [START_REF] Ljung | System Identification -Theory for the User[END_REF]). This will lead to an indirect adaptive control scheme. The principle of such a scheme is illustrated in figure 1. The time consuming part of this approach is the redesign of the controller at each sampling time. The reason is that in many applications the plant model can be of very high dimension and despite that this model is constant, one has to re-compute the controller because a new internal model should be considered. This approach has been investigated in [START_REF] Bodson | Adaptive algorithms for the rejection of sinusosidal disturbances with unknown frequency[END_REF], [START_REF] Gouraud | Design of robust and frequency adaptive controllers for harmonic disturbance rejection in a single-phase power network[END_REF], [START_REF] Hillerstrom | Rejection of periodic disturbances with unknown period -a frequency domain approach[END_REF].

However, by considering the Youla-Kucera parametrization of the controller (known also as the Q-parametrization), it is possible to insert and adjust the internal model in the controller by adjusting the parameters of the Q polynomial (see figure 2). It comes out that in the presence of unknown disturbances it is possible to build a direct adaptive control scheme where the parameters of the Q polynomial are directly adapted in order to have the desired internal model without recomputing the controller (polynomials R 0 and S 0 in figure 2 remain unchanged). The number of the controller parameters to be directly adapted is roughly equal to the number of parameters of the denominator of the disturbance model. In other words, the size of the adaptation algorithm will depend upon the complexity of the disturbance model.

This paper focuses on the direct feedback adaptive control for the case of unknown and time-varying frequency narrow band disturbances. The direct adaptive control scheme to be presented( [START_REF] Landau | Adaptive narrow band disturbance rejection applied to an active suspension -an internal model principle approach[END_REF]) takes advantage of the Youla-Kucera parametrization for the computation of the controller. For evaluation purposes (complexity and performance) an indirect adaptive control scheme based on the Internal Model Principle will be also presented.

The paper is organized as follows. Section II is dedicated to a brief review of the plant, disturbance and controller representation as well as of the Internal Model Principle. Some robustness issues are addressed in section III. The direct and the indirect adaptive control schemes for disturbance rejection are presented in sections IV and V, respectively. The application to an active suspension system, including the real-time results, is presented in VI. The application to the active vibration control system using an inertial actuator, including real time results is presented in section VII . Some concluding remarks are given in section VIII.

II. PLANT REPRESENTATION AND CONTROLLER

STRUCTURE

The structure of a linear time invariant discrete time model of the plant (used for controller design) is:

G(z -1 ) = z -d B(z -1 ) A(z -1 ) = z -d-1 B * (z -1 ) A(z -1 ) , (1)
with: d = the plant pure time delay in number of sampling periods

A = 1 + a 1 z -1 + • • • + a n A z -n A ; B = b 1 z -1 + • • • + b n B z -n B = q -1 B * ; B * = b 1 + • • • + b n B z -n B +1 ,
where A(z -1 ), B(z -1 ), B * (z -1 ) are polynomials in the complex variable z -1 and n A , n B and n B -1 represent their orders 2 . The model of the plant may be obtained by system identification. Details on system identification of the models considered in this paper can be found in [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF], [START_REF] Constantinescu | Commande robuste et adaptative d'une suspension active[END_REF], [START_REF] Landau | Direct controller order reduction by identification in closed loop[END_REF], [START_REF] Landau | A methodology for the design of feedback active vibration control systems[END_REF], [START_REF] Karimi | Design and optimization of restricted complexity controllers -benchmark[END_REF], [START_REF] Constantinescu | Direct controller order reduction by identification in closed loop applied to a benchmark problem[END_REF].

Since in this paper we are focused on regulation, the controller to be designed is a RS-type polynomial controller ( [START_REF] Landau | Adaptive control[END_REF], [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF]) -see also figure 5.

The output of the plant y(t) and the input u(t) may be written as:

y(t) = q -d B(q -1 ) A(q -1 ) • u(t) + p 1 (t) ; (2)
S(q -1 ) • u(t) = -R(q -1 ) • y(t) , (3)
where q -1 is the delay (shift) operator (x(t) = q -1 x(t + 1)) and p 1 (t) is the resulting additive disturbance on the output of the system. R(z -1 ) and S(z -1 ) are polynomials in z -1 having the orders n R and n S , respectively, with the following expressions: 4)

R(z -1 ) = r 0 + r 1 z -1 + . . . + r n R z -n R = R ′ (z -1 ) • H R (z -1 ) ;(
S(z -1 ) = 1 + s 1 z -1 + . . . + s n S z -n S = S ′ (z -1 ) • H S (z -1 ) , (5)
where H R and H S are pre-specified parts of the controller (used for example to incorporate the internal model of a disturbance or to open the loop at certain frequencies).

We define the following sensitivity functions:

• Output sensitivity function (the transfer function between the disturbance p 1 (t) and the output of the system y(t)):

S yp (z -1 ) = A(z -1 )S(z -1 ) P(z -1 ) ; (6)
• Input sensitivity function (the transfer function between the disturbance p 1 (t) and the input of the system u(t)):

S up (z -1 ) = - A(z -1 )R(z -1 ) P(z -1 ) , (7)
where

P(z -1 ) = A(z -1 )S(z -1 ) + z -d B(z -1 )R(z -1 ) = A(z -1 )S ′ (z -1 ) • H S (z -1 ) + z -d B(z -1 )R ′ (z -1 ) • H R (z -1 ) (8)
defines the poles of the closed loop ( roots of P(z -1 )). In pole placement design, P(z -1 ) is the polynomial specifying the desired closed loop poles and the controller polynomials R(z -1 ) and S(z -1 ) are minimal degree solutions of [START_REF] Bodson | Adaptive algorithms for the rejection of sinusosidal disturbances with unknown frequency[END_REF] where the degrees of P, R and S are given by:

n P ≤ n A + n B + d -1, n S = n B + d -1 and n R = n A -1.
Using the equations ( 2) and

(3), one can write the output of the system as:

y(t) = A(q -1 )S(q -1 ) P(q -1 ) • p 1 (t) = S yp (q -1 ) • p 1 (t) . ( 9
)
For more details on RS-type controllers and sensitivity functions see [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF].

Suppose that p 1 (t) is a deterministic disturbance, so it can be written as

p 1 (t) = N p (q -1 ) D p (q -1 ) • δ (t) , (10)
where δ (t) is a Dirac impulse and N p (z -1 ), D p (z -1 ) are coprime polynomials in z -1 , of degrees n N p and n D p , respectively. In the case of stationary disturbances the roots of D p (z -1 ) are on the unit circle. The energy of the disturbance is essentially represented by D p . The contribution of the terms of N p is weak compared to the effect of D p , so one can neglect the effect of N p .

Internal Model Principle:

The effect of the disturbance given in [START_REF] Constantinescu | Direct controller order reduction by identification in closed loop applied to a benchmark problem[END_REF] upon the output:

y(t) = A(q -1 )S(q -1 ) P(q -1 ) • N p (q -1 ) D p (q -1 ) • δ (t) , (11)
where D p (z -1 ) is a polynomial with roots on the unit circle and P(z -1 ) is an asymptotically stable polynomial, converges asymptotically towards zero if and only if the polynomial S(z -1 ) in the RS controller has the form:

S(z -1 ) = D p (z -1 )S ′ (z -1 ) . (12)
In other terms, the pre-specified part of S(z -1 ) should be chosen as

H S (z -1 ) = D p (z -1
) and the controller is computed using [START_REF] Bodson | Adaptive algorithms for the rejection of sinusosidal disturbances with unknown frequency[END_REF], where P, D p , A, B, H R and d are given 3 . Using the Youla-Kucera parametrization (Qparametrization) of all stable controllers ([4], [START_REF] Tsypkin | Stochastic discrete systems with internal models[END_REF]), the controller polynomials R(z -1 ) and S(z -1 ) get the form:

R(z -1 ) = R 0 (z -1 ) + A(z -1 )Q(z -1 ) ; (13) S(z -1 ) = S 0 (z -1 ) -z -d B(z -1 )Q(z -1 ) . (14)
The (central) controller (R 0 , S 0 ) can be computed by poles placement (but any other design technique can be used).

Given the plant model (A, B, d) and the desired closed-loop poles specified by the roots of P one has to solve:

P(z -1 ) = A(z -1 )S 0 (z -1 ) + z -d B(z -1 )R 0 (z -1 ) . (15)
Equations ( 13) and ( 14) characterize the set of all stabilizable controllers assigning the closed loop poles as defined by P(z -1 ) (it can be verified by simple calculations that the poles of the closed loop remain unchanged). For the purpose of this paper Q(z -1 ) is considered to be a polynomial of the form:

Q(z -1 ) = q 0 + q 1 z -1 + . . . + q n Q z -n Q . ( 16
)
To compute Q(z -1 ) in order that the controller incorporates the internal model of the disturbance one has to solve the diophantine equation:

S ′ (z -1 )D p (z -1 ) + z -d B(z -1 )Q(z -1 ) = S 0 (z -1 ) , (17)
where D p (z -1 ), d, B(z -1 ) and S 0 (z -1 ) are known and S ′ (z -1 ) and Q(z -1 ) are unknown. Equation ( 17) has a unique solution for

S ′ (z -1 ) et Q(z -1 ) with: n S 0 ≤ n D p + n B + d -1, n S ′ = n B + d -1, n Q = n D p -1 .
One sees that the order n Q of the polynomial Q depends upon the structure of the disturbance model.

III. ROBUSTNESS CONSIDERATIONS

As it is well known, the introduction of the internal model for the perfect rejection of the disturbance (asymptotically) will have as effect to raise the maximum value of the modulus of the output sensitivity function S yp . This may lead to unacceptable values for the modulus and the delay margins if the controller design is not appropriately done (see [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF]). As a consequence, a robust control design should be considered assuming that the model of the disturbance is known, in order to be sure that for all situations an acceptable modulus margin and delay margin are obtained.

On the other hand at the frequencies where perfect rejection of the disturbance is achieved one has S yp (e -jω ) = 0 and

S up (e -jω ) = A(e -jω ) B(e -jω ) . (18)
Equation (18) corresponds to the inverse of the gain of the system to be controlled. The implication of equation ( 18) is that cancellation (or in general an important attenuation) of disturbances on the output should be done only in frequency regions where the system gain is large enough. If the gain of the controlled system is too low, |S up | will be large at these frequencies. Therefore, the robustness vs additive plant model uncertainties will be reduced and the stress on the actuator will become important. Equation ( 18) also implies that serious problems will occur if B(z -1 ) has complex zeros close to the unit circle (stable or unstable zeros) at frequencies where an important attenuation of disturbances is required. It is mandatory to avoid attenuation of disturbances at these frequencies.

Since on one hand we would not like to react to very high frequency disturbances and on the other hand we would like to have a good robustness it is often wise to open the loop at 0.5 f s ( f s is the sampling frequency) by introducing a fixed part in the controller H R (q -1 ) = 1 + q -1 (for details see [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF] and section II).

IV. DIRECT ADAPTIVE CONTROL FOR DISTURBANCE

ATTENUATION

The objective is to find an estimation algorithm which will directly estimate the parameters of the internal model in the controller in the presence of an unknown disturbance (but of known structure) without modifying the closed loop poles. Clearly, the Q-parametrization is a potential option since modifications of the Q polynomial will not affect the closed loop poles. In order to build an estimation algorithm it is necessary to define an error equation which will reflect the difference between the optimal Q polynomial and its current value.

In [START_REF] Tsypkin | Stochastic discrete systems with internal models[END_REF], such an error equation is provided and it can be used for developing a direct adaptive control scheme. This idea has been used in [START_REF] Valentinotti | Adaptive Rejection of Unstable Disturbances: Application to a Fed-Batch Fermentation[END_REF], [START_REF] Ben Amara | Adaptive sinusoidal disturbance rejection in linear discrete-time systems -Part I: Theory[END_REF], [START_REF] Ben Amara | Adaptive sinusoidal disturbance rejection in linear discrete-time systems -Part II: Experiments[END_REF], [START_REF] Landau | Adaptive narrow band disturbance rejection applied to an active suspension -an internal model principle approach[END_REF]. Using the Q-parametrization, the output of the system in the presence of a disturbance can be expressed as:

y(t) = A(q -1 )[S 0 (q -1 ) -q -d B(q -1 )Q(q -1 )] P(q -1 ) • N p (q -1 ) D p (q -1 ) • δ (t) = S 0 (q -1 ) -q -d B(q -1 )Q(q -1 ) P(q -1 ) • w(t) , (19)
where w(t) is given by (see also figure 2):

w(t) = A(q -1 )N p (q -1 ) D p (q -1 ) • δ (t) = A(q -1 ) • y(t) -q -d • B(q -1 ) • u(t) . (20)
In the time domain, the internal model principle can be interpreted as finding Q such that asymptotically y(t) becomes zero. Assume that one has an estimation of Q(q -1 ) at instant t, denoted Q(t, q -1 ). Define ε 0 (t + 1) as the value of y(t + 1) obtained with Q(t, q -1 ). Using [START_REF] Hillerstrom | Rejection of periodic disturbances with unknown period -a frequency domain approach[END_REF] one gets:

ε 0 (t +1) = S 0 (q -1 ) P(q -1 ) •w(t +1)- q -d B * (q -1 ) P(q -1 ) Q(t, q -1 )•w(t) . ( 21
)
One can define now the a posteriori error (using Q(t + 1, q -1 )) as:

ε(t + 1) = S 0 (q -1 ) P(q -1 ) • w(t + 1) - q -d B * (q -1 ) P(q -1 ) Q(t + 1,q -1 ) • w(t) . ( 22
)
Replacing S 0 (q -1 ) from the last equation by [START_REF] Fuller | Active Control of Vibration[END_REF] one obtains

ε(t + 1) = [Q(q -1 ) -Q(t + 1,q -1 )] • q -d B * (q -1 ) P(q -1 ) • w(t) + v(t + 1) , (23)
where v(t) = S ′ (q -1 )D p (q -1 ) P(q -1 )

• w(t) = S ′ (q -1 )A(q -1 )N p (q -1 )

P(q -1 ) • δ (t)
is a signal which tends asymptotically towards zero. Define the estimated polynomial Q(t, q -1 ) as: Q(t, q -1 ) = q0 (t) + q1 (t)q -1 + . . . + qn Q (t)q -n Q and the associated estimated parameter vector : θ (t) = [ q0 (t) q1 (t) . . . qn Q (t)] T . Define the fixed parameter vector corresponding to the optimal value of the polynomial Q as: θ = [q 0 q 1 . . . q n Q ] T . Denote:

w 2 (t) =
q -d B * (q -1 ) P(q -1 )

• w(t) [START_REF] Landau | Adaptive control[END_REF] and define the following observation vector:

φ T (t) = [w 2 (t) w 2 (t -1) . . . w 2 (t -n Q )] . (25)
Equation [START_REF] Landau | Direct controller order reduction by identification in closed loop[END_REF] becomes

ε(t + 1) = [θ T -θ T (t + 1)] • φ (t) + v(t + 1) . (26)
One can remark that ε(t) corresponds to an adaptation error ( [START_REF] Landau | Adaptive control[END_REF]).

From equation ( 21) one obtains the a priori adaptation error:

ε 0 (t + 1) = w 1 (t + 1) -θ T (t)φ (t) , with
w 1 (t + 1) = S 0 (q -1 ) P(q -1 ) • w(t + 1) ; ( 27
)
w 2 (t) = q -d B * (q -1 ) P(q -1 ) • w(t) ; ( 28
)
w(t + 1) = A(q -1 ) • y(t + 1) -q -d B * (q -1 ) • u(t) , (29)
where B(q -1 )u(t + 1) = B * (q -1 )u(t).

The a posteriori adaptation error is obtained from [START_REF] Landau | Adaptive narrow band disturbance rejection applied to an active suspension -an internal model principle approach[END_REF]:

ε(t + 1) = w 1 (t + 1) -θ T (t + 1)φ (t) .
For the estimation of the parameters of Q(t, q -1 ) the following parameter adaptation algorithm is used ( [START_REF] Landau | Adaptive control[END_REF]):

θ (t + 1) = θ (t) + F(t)φ (t)ε(t + 1) ; (30) ε(t + 1) = ε 0 (t + 1) 1 + φ T (t)F(t)φ (t) ; (31)
ε 0 (t + 1) = w 1 (t + 1) -θ T (t)φ (t) ; ( 32)

F(t + 1) = 1 λ 1 (t)   F(t) - F(t)φ (t)φ T (t)F(t) λ 1 (t) C) + φ T (t)F(t)φ (t)   . ( 33
) 1 ≥ λ 1 (t) > 0; 0 ≤ λ 2 (t) < 2 (34)
where λ 1 (t), λ 2 (t) allow to obtain various profiles for the evolution of the adaption gain F(t) (for details see [START_REF] Landau | Adaptive control[END_REF], [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF] and section VI).

In order to implement this methodology for disturbance rejection (see figure 2), it is supposed that the plant model

z -d B(z -1 ) A(z -1 )
is known (identified) and that it exists a controller

[R 0 (z -1 ), S 0 (z -1 )] which verifies the desired specifications in the absence of the disturbance. One also supposes that the degree n Q of the polynomial

Q(z -1 ) is fixed, n Q = n D p -1,
i.e. the structure of the disturbance is known.

The following procedure is applied at each sampling time for adaptive operation:

1) Get the measured output y(t + 1) and the applied control u(t) to compute w(t + 1) using ( 29). 2) Compute w 1 (t + 1) and w 2 (t) using ( 27) and [START_REF] Marino | Robust adaptive compensation of biased sinusoidal disturbances with unknown frequency[END_REF] with P given by [START_REF] Ficocelli | Adaptive regulation of mimo linear systems against unknown sinusoidal exogenous inputs[END_REF]. 3) Estimate the Q-polynomial using the parametric adaptation algorithm (30) - [START_REF] Zhang | Direct adaptive control for tonal disturbance rejection[END_REF]. 4) Compute and apply the control (see figure 2):

S 0 (q -1 ) • u(t + 1) = -R 0 (q -1 ) • y(t + 1) -Q(t,q -1 ) • w(t + 1) . (35)
For the self tuning operation of the adaptive scheme, the estimation of the Q-polynomial starts once the level of the output is over a defined threshold. A parameter adaptation algorithm ( 30)- [START_REF] Zhang | Direct adaptive control for tonal disturbance rejection[END_REF] with decreasing adaption gain is used and the estimation is stopped when the adaption gain is below a pre-specified level 4 . During estimation of the new parameters, the controller is kept constant. The controller is updated once the estimation phase is finished. For a stability analysis of this scheme see [START_REF] Landau | Adaptive narrow band disturbance rejection applied to an active suspension -an internal model principle approach[END_REF].

V. INDIRECT ADAPTIVE CONTROL FOR DISTURBANCE

ATTENUATION

Indirect adaptive control for the attenuation of unknown disturbances consists in two steps: (1) Identification of the disturbance model; (2) Computation of a digital controller using the identified disturbance model as internal model. Details on this approach can be found in [START_REF] Landau | Adaptive narrow band disturbance rejection applied to an active suspension -an internal model principle approach[END_REF]. 4 The magnitude of the adaptation gain gives an indication upon the variance of the parameter estimation error -see for example [START_REF] Landau | Adaptive control[END_REF].

A. The active suspension

The structure of the system (the active suspension) used in this paper is presented in figure 3. Two photos of the system are presented in figure 4 (Courtesy of Hutchinson Research Center and Laboratoire d'Automatique de Grenoble). It consists of the active suspension, a load, a shaker and the components of the control scheme. The mechanical construction of the load is such that the vibrations produced by the shaker, fixed to the ground, are transmitted to the upper side of the active suspension. The active suspension is based on a hydraulic system allowing to reduce the overpressure at the frequencies of the vibration modes of the suspension.

The controller will act upon the piston (through a power amplifier) in order to reduce the residual force. The sampling frequency is 800Hz. The equivalent control scheme is shown in figure 5. The system input, u(t) is the position of the piston (see figures 3, 5), the output y(t) is the residual force measured by a force sensor. The transfer function (q -d 1 C D ), between the disturbance force, u p , and the residual force y(t) is called primary path. In our case (for testing purposes), the primary force is generated by a shaker controlled by a signal given by the computer. The plant transfer function (q -d B A ) between the input of the system, u(t), and the residual force is called secondary path. The input of the system being a position and the output a force, the secondary path transfer function has a double differentiator behavior.

The control objective is to reject the effect of unknown narrow band disturbances on the output of the system ( the residual force). The system has to be considered as a "black box".

B. Results obtained on the active suspension

For the active suspension the disturbance will be a timevarying frequency sinusoid, so we shall consider n D p = 2 and

n Q = n D p -1 = 1.
The identification procedure in open and closed-loop operation for the active suspension is discussed in detail in [START_REF] Landau | Direct controller order reduction by identification in closed loop[END_REF], [START_REF] Landau | A methodology for the design of feedback active vibration control systems[END_REF], [START_REF] Karimi | Design and optimization of restricted complexity controllers -benchmark[END_REF]. The frequency characteristic of the identified The frequency characteristic of the identified secondary path model (closed-loop identification), is presented also in figure 6. This model has the following complexity: n B = 14, n A = 16, d = 0. The identification has been done using as excitation of the piston a PRBS (Pseudo Random Binary Sequence) with frequency divider p = 4 (for details on the PRBS signals see [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF]). There exist several very low damped vibration modes on the secondary path, the first one being at 31.8Hz with a damping factor 0.07. The identified model of the secondary path has been used for the design and implementation of the controller. The central controller (without the internal model of the disturbance) has been designed using the pole placement method and the secondary path identified model. The resulting nominal controller has the following complexity: n R = 14, n S = 16 and it satisfies the imposed robustness constraints on the sensitivity functions(for the design methodology see [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF]) 5 .

In order to evaluate the performances of direct and indirect methods in real time, time-varying frequency sinusoidal 5 Any design method allowing to satisfy these constraints can be used. For both direct and indirect adaptive control methods, two protocols have been defined.

• Protocol 1 : Self-tuning operation

The system operates in closed loop with a frozen controller. As soon as a change of the disturbance is detected (by measuring the variance of the residual output), the estimation algorithm is started with the last frozen controller in operation. When the algorithm converges (a criterion has to be defined -see below), a new controller is computed and applied to the system. The adaptation algorithm is stopped and one waits for a change of frequency.

• Protocol 2 : Adaptive operation

The estimation algorithm works permanently (once the loop is closed) and the controller is recomputed at each sampling. The adaptation gain in this case does not tend asymptotically to zero. • Start up: For comparison purpose the system is started in open-loop for both protocols. After 5 seconds (4000 samples) a sinusoidal disturbance of 32Hz is applied on the shaker. The model of the disturbance is estimated and an initial controller is computed (same initial controller for both direct and indirect adaptive control). In the case of the self-tuning operation the adaptation algorithm is stopped while in the case of the adaptive operation the adaptation algorithm continues to be active. After the start up ends, every 15 seconds (8000 samples) sinusoidal disturbances of different frequency are applied (32Hz, 25Hz,32Hz,47Hz,32Hz).

a) Protocol 1 : Self-tuning operation. Real time experimental results: The measured residual force obtained in self-tuning operation with the direct adaptation method is presented in figure 7 and with the indirect adaptation method in figure 8 . We note in general a faster convergence speed of the direct adaptive control scheme compared to the indirect For the self-tuning protocol, the spectral densities of the residual force obtained in open and in closed loop, respectively, using the direct adaptation scheme (after the algorithm converges) are presented in figure 9. The results are given for the three frequencies used: 25, 32 and 47 Hz. We remark that the attenuations are larger than 49 dB for all the frequencies. Similar results are obtained with the indirect adaptation algorithm. For details see [START_REF] Constantinescu | Commande robuste et adaptative d'une suspension active[END_REF].

In self-tuning operation, one uses an adaptation gain F(t) with variable forgetting factor, with λ 0 = 0.97 and the initial forgetting factor λ 1 (0) = 0.97 (the forgetting factor is given by λ 1 (t) = λ 0 λ 1 (t -1) + 1 -λ 0 , with 0 < λ 0 < 1). For the variable forgetting factor the adaptation gain tends asymptotically towards zero. The convergence criterion has been fixed as a threshold on the trace value of the adaptation gain matrix. For details see [START_REF] Constantinescu | Commande robuste et adaptative d'une suspension active[END_REF]. The detection of a change of frequency is done using the variance of the measured residual force computed on a sliding window of 50 samples.

b) Protocol 2 : Adaptive operation. Real time experimental results: The measured residual force obtained in adaptive operation is presented in figure 10 for the direct adaptation method and in figure 11 for the indirect adaptation method. An adaptation gain with variable forgetting factor combined with a constant trace ( [START_REF] Landau | Adaptive control[END_REF], [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF]) has been used in order to be able to track automatically the changes of disturbance characteristics. The low level threshold of the trace has been fixed at 3 • 10 -9 for the direct algorithm and at 5 • 10 -7 for the indirect one (note that in the indirect adaptive scheme there are more parameters to estimate than in the direct adaptive scheme). The attenuation obtained with the indirect adaptive scheme in adaptive operation is less good than in the self tuning operation and less good than the one obtained with the direct adaptive scheme. We note that the direct adaptive control scheme in adaptive operation gives better results than in self tuning operation (compare figures 7 and 10).

The spectral densities of the residual force for the direct adaptive scheme (after the algorithm converges) are similar with those obtained in self-tuning operation (see [START_REF] Constantinescu | Commande robuste et adaptative d'une suspension active[END_REF]).

According to the real time results presented above, one can conclude that the direct adaptive control scheme gives better results than the indirect adaptive control scheme, from the point of view of the convergence speed and performance. In addition the direct adaptation scheme is much simpler than the indirect one in terms of number of operations. is closed, the adaptation algorithm works permanently and the controller is updated (direct approach) at each sampling instant. After 5 seconds a sinusoidal disturbance of 25 Hz (constant frequency) is applied on the shaker. From 10 to 15 seconds a chirp between 25 and 47 Hz is applied. After 15 seconds a 47 Hz (constant frequency) sinusoidal disturbance is applied and the tests are stopped after 18 seconds. The time-domain results obtained in open and in closed-loop (direct adaptive control) are presented in figure 12. We can remark that the performances obtained are very good. d) Adaptation transients for direct adaptive control: Figure 13 illustrates the adaptation transients on the input and output when a step change of the frequency of the disturbance occurs from 20Hz to 32 Hz respectively. One notes that the convergence of the output requires less than 0.25s This corresponds roughly to 6 periods for 32Hz. Same duration of the adaptation transient are obtained for the other frequencies step changes. These results have to be compared with the transients results given in [START_REF] Bodson | Adaptive algorithms for the rejection of sinusosidal disturbances with unknown frequency[END_REF], [START_REF] Marino | Robust adaptive compensation of biased sinusoidal disturbances with unknown frequency[END_REF], [START_REF] Ben Amara | Adaptive sinusoidal disturbance rejection in linear discrete-time systems -Part I: Theory[END_REF], [START_REF] Ben Amara | Adaptive sinusoidal disturbance rejection in linear discrete-time systems -Part II: Experiments[END_REF].

VII. APPLICATION 2 -ADAPTIVE REJECTION OF MULTIPLE NARROW BAND DISTURBANCES ON AN ACTIVE VIBRATION CONTROL SYSTEM USING AN INERTIAL ACTUATOR

A. The inertial actuator

In this application a different technological approach is used for suppressing the effect of vibrational disturbances. Instead of using an active suspension, one uses an inertial actuator which will create vibrational forces to counteract the effect of vibrational distrubances (inertial actuators use a similar principle as loudspeakers). The structure of the system is described in figure14. It consists on a standard passive damper and an inertial actuator fixed to the chassis where the vibrations should be attenuated. The testing setting is exactly the same as for the active suspension (see figure 4). The controller will act on the inertial actuator (through a power amplifier) in order to reduce the residual force. The equivalent control scheme is shown in figure 5. The system input is the position of the mobile part of the actuator. Like for the active suspension, the secondary path has a double differentiator behavior. The system has to be considered as a "black box" and the control objectives are similar to those for the active suspension; The sampling frequency is 800Hz.

B. Results obtained with the inertial actuator

The performance of the system for rejecting multiple unknown time varying narrow band disturbances will be illustrated using the direct adaptive control scheme presented in section IV. Since two simultaneous time varying frequency sinusoids will be considered as disturbances , one should take n D p = 4 and n Q = n D p -1 = 3 Same procedure for system identification in open and closed loop, as for the active suspension, has been used. The frequency characteristics of the primary path (identification in open loop) and of the secondary path (identification in closed loop)are shown in Figure 15. The secondary path has the following complexity: n B = 12, n A = 10, d = 0. The identification has been done using as excitation a PRBS ( with frequency divider p = 2 and N = 9). There exist several low damped vibration modes in the secondary path, the first vibration mode is at 51.58Hz with a damping of 0.023 and the second at 100.27Hz with a damping of 0.057. Only the "adaptive "operation regime has been considered for the subsequent results. Figure 16shows the spectral densities of the residual force obtained in open loop and in closed loop using the direct adaptation scheme (after the adaptation algorithm has converged). The results are given for the simultaneous applications of two sinusoidal disturbances (70Hz and 100Hz). One can remark a strong attenuation of the disturbances (larger than 45dB). Time domain results obtained with direct adaptation scheme in "adaptive" operation regime are shown in Figure17. The disturbances are applied at 1s (the loop has already been closed) and step changes of their frequencies occur every 3s. Figure 18 shows the corresponding evolution of the parameters of the polynomial Q. The convergence of the output requires less than 0.4s in the worst case. The results obtained in real time on active vibration control (using an active suspension or an inertial actuator) lead us to conclude that the direct adaptive control scheme provides better performance and is simpler than the indirect adaptive control scheme.

A similar approach has been used successfully on a chemical reactor and for noise cancellation in ducts. Extensions to the multivariable case have been recently done [START_REF] Ficocelli | Adaptive regulation of mimo linear systems against unknown sinusoidal exogenous inputs[END_REF]

Fig. 1 .

 1 Fig. 1.

Fig. 3 .

 3 Fig. 3. Active suspension system (scheme)

Fig. 4 . 1 Fig. 5 .

 415 Fig. 4. Active suspension system (photo)

Fig. 6 .

 6 Fig. 6. Frequency characteristics of the primary and secondary paths

Fig. 7 .Fig. 8 .

 78 Fig. 7. Time domain results with the direct adaptation method in self-tuning operation

Fig. 9 .

 9 Fig. 9. Spectral densities of the residual force in open and in closed loop, with the direct adaptation method in self-tuning operation

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Time domain results with the direct adaptation method in the adaptive case (trace = 3 • 10 -9 )

Fig. 12 .Fig. 13 .

 1213 Fig. 12. Real-time results obtained with the direct adaptive method and a chirp disturbance: (a) Open loop; (b) Closed loop

Fig. 14 .

 14 Fig. 14. Active vibration control using an inertial actuator (scheme)

Fig. 15 .Fig. 16 .Fig. 17 .

 151617 Fig. 15. Frequency characteristics of the primary and secondary paths (inertial actuator)

Fig. 18 .

 18 Fig. 18. Evolution of the parameters of the polynomial Q during adaptation

Throughout the paper it is assumed that the order of the disturbance model is known but the parameters of the model are unknown (the order can be estimated from data if necessary).

The complex variable z -1 will be used for characterizing the system's behavior in the frequency domain and the delay operator q -1 will be used for describing the system's behavior in the time domain.

Of course it is assumed that D p and B do not have common factors.