Abraham Castellanos
email: abraham.castellanos-silva@gipsa-lab.grenoble-inp.fr

Doré Landau

Petros Ioannou
email: ioannou@usc.edu

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

Robust Direct Adaptive Regulation of Unknown Disturbances in the Vicinity of Low Damped Complex Zeros -Application to Active Vibration Control

Keywords: Adaptive Regulation, Active Vibration Control, Inertial Actuators, Multiple Narrow Band Disturbances, Youla-Kučera Parametrization, Internal Model Principle

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

The basic problem in active vibration control (AVC) and active noise control (ANC) is the strong attenuation of multiple narrow band disturbances 1 with unknown and varying frequencies. The problem has been approached both in continuoustime ( [START_REF] Marino | Robust adaptive compensation of biased sinusoidal disturbances with unknown frequency[END_REF], [START_REF] Marino | Output feedback stabilization of linear systems with unknown additive output sinusoidal disturbances[END_REF] among other references) and in discrete-time ( [START_REF] Valentinotti | Adaptive Rejection of Unstable Disturbances: Application to a Fed-Batch Fermentation[END_REF], [START_REF] Landau | Adaptive narrow band disturbance rejection applied to an active suspension -an internal model principle approach[END_REF] among other references). This paper consider the discrete-time case.

The disturbance model is assumed to be either a function equal to the sum of sinusoidal with unknown frequencies, amplitudes and phases or equivalently, a transfer function with unknown complex poles on the unit circle with white noise or a Dirac pulse as an input. When information about such model is available, it is possible to design an appropriate controller. If the model of a disturbance is considered, one has to address two issues: 1) its structure (complexity, order of the parametric model) and 2) the values of the parameters of the model. In general, one can assess from data the structure for such model of disturbance (using spectral analysis or order estimation techniques). However, the parameters of the model are unknown and may be time varying. This will require the use of an adaptive feedback approach in order to adapt to changes in parameters. An adaptive feedback approach, called adaptive regulation (known plant model and unknown/time varying disturbance model) is now widely accepted as the most effective approach for solving this class of problems. It is also assumed that the plant model is stable and this property could be the result of a robust control design that is already incorporated in the system under consideration. The problem of disturbance rejection and adaptive regulation as defined above has been previously addressed in a number of papers ( [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] 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], [START_REF] Landau | Adaptive regulation-rejection of unknown multiple narrow band disturbances (a review on algorithms and applications)[END_REF], [START_REF] Kinney | Robust estimation and automatic controller tuning in vibration control of time varying harmonic disturbances[END_REF], [START_REF] Chen | A minimum parameter adaptive approach for rejecting multiple narrow-band disturbances with application to hard disk drives[END_REF], [START_REF] Emedi | Fixed-order lpv controller design for rejection of a sinusoidal disturbance with time-varying frequency[END_REF], [START_REF] Bobtsov | An iterative algorithm of adaptive output control with complete compensation for unknown sinusoidal disturbance[END_REF], [START_REF] Celani | Output regulation for the tora benchmark via rotational position feedback[END_REF]) among others. More recent references on the subject are [START_REF] Marino | Disturbance cancellation for linear systems by adaptive internal models[END_REF], [START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF].

The Internal Model Principle implemented through a Youla-Kučera parametrization -also known as Q-parametrization -arises as a very attractive and efficient solution, since it allows the incorporation of the model of the disturbance in the controller without modifying the desired closed loop poles, defined by the designer ( [START_REF] Valentinotti | Adaptive Rejection of Unstable Disturbances: Application to a Fed-Batch Fermentation[END_REF], [START_REF] Landau | Adaptive narrow band disturbance rejection applied to an active suspension -an internal model principle approach[END_REF], [START_REF] Silva | Direct adaptive rejection of unknown time-varying narrow band disturbances applied to a benchmark problem[END_REF]) when a finite impulse response (FIR) filter structure is considered for the Q-filter. This parametrization allows one to have a two-stage controller: 1) a central controller for the stabilization and broadband disturbance rejection and 2) the adaptive part which deals with the rejection of the narrow band disturbance effects. The number of parameters to adapt is defined by the complexity of the assumed disturbance model. An international benchmark on adaptive rejection of narrow band disturbances has been organized and the results are published in a special issue of the European Journal of Control [START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF].

As indicated earlier the disturbance is considered to be periodic, i.e. the poles of the disturbance models are on the unit circle. All the stability proofs for the adaptation algorithms make the assumption that the plant zeros are different from the poles of the disturbance model in order to achieve disturbance compensation. However in practice the situation is more intricate since it is not clear what happens if the plant has very low damped complex zeros (often encountered in mechanical structures) and the frequency of the disturbance is close to the anti-resonance frequency (the resonance frequency of the plant zeros). Obviously even in the linear case with known parameters the design of the controller in this situation is difficult for robustness reasons. Finding a good control solution for this situation in an adaptive context is very challenging. In the international benchmark dedicated to adaptive regulation in the presence of unknown time varying disturbances [START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF] such a situation has been explicitly considered. Several solutions have been proposed and the most successful has been based on the appropriate choice of the desired closed loop poles to be achieved by the Youla-Kučera central controller and by using an adjustable Q-FIR filter with the minimum number of parameters [START_REF] Silva | Direct adaptive rejection of unknown time-varying narrow band disturbances applied to a benchmark problem[END_REF]. Recently in [START_REF] Jafari | Robust stability and performance of adaptive jitter supression in laser beam pointing[END_REF] it was suggested that over parametrization of the Q (FIR) filter can enhance the robustness of the linear and adaptive scheme in the vicinity of plant complex zeros 2 .

The aim of the article is to present simulation and realtime comparisons of these two approaches on a relevant example (The EJC Benchmark for adaptive regulation benchmark test bed [START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF]) and to evaluate also the combination of the two approaches. The paper is organized as follows: Section II-A presents the active vibration control system used for the real-time experiments. Section II-B introduces the plant/disturbance representation and the controller structure used in this work. The influence of low damped complex zeros for disturbance rejection is discussed in Section III. In this section the two approaches for improving the behavior of the system in the presence of periodic disturbances located near low damped complex zeros are also briefly presented. The direct adaptive feedback regulation scheme, based on a Youla-Kučera parametrization is recalled briefly in Section IV. The comparison between the two approaches presented in [START_REF] Silva | Direct adaptive rejection of unknown time-varying narrow band disturbances applied to a benchmark problem[END_REF] and in [START_REF] Jafari | Robust stability and performance of adaptive jitter supression in laser beam pointing[END_REF] and their combination is done in Section V through simulation and real-time results. Some concluding remarks are presented in Section VI.

II. PLANT DESCRIPTION AND CONTROLLER STRUCTURE A. System structure

The photo of the active vibration control experimental setup used in this study is presented in Fig. 1. The shaker acts as a disturbance source by introducing vibration forces and the inertial actuator can be used to counteract them by introducing vibrational forces in the opposite direction (inertial actuators use a similar principle as loudspeakers). This test bed was used as international benchmark in adaptive regulation. A detailed description together with the results were published in [START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF]. For sake of completeness some features are recalled here. The test bed consists of a passive damper, an inertial actuator, a mechanical structure, a transducer for the measurement of the residual force, a controller, a power amplifier and a shaker. The mechanical structure is such that the vibrations produced by the shaker, fixed to the ground, are transmitted to the upper side, on top of the passive damper. The inertial actuator is fixed to the chassis where the vibrations should be attenuated. The equivalent control scheme is shown in Fig. 2. The system input, u(t) is the position of the mobile part (magnet) of the inertial actuator, the output y(t) is the residual force measured by a force sensor. The plant transfer function (G = q -d B A ) between the input of the inertial actuator, u(t), and the residual force is called secondary path.

The control objective is to reject the effect of unknown narrow band disturbances on the output of the system (residual 2 This idea has not been explored by the participants to the benchmark. Note that the over parametrization of the Q filter for robustness with respect to uncertainties in the plant model has been proposed in [START_REF] Valentinotti | Adaptive Rejection of Unstable Disturbances: Application to a Fed-Batch Fermentation[END_REF], [START_REF] Jafari | Robust stability and performance of adaptive jitter supression in laser beam pointing[END_REF], however, here the objective of over parametrization is different. force), i.e. to attenuate the vibrations transmitted from the machine to the chassis. This requires that the compensator system (the secondary path) has enough gain in the frequency range where the narrow band disturbances are located [START_REF] Landau | Adaptive suppression of multiple time-varying unknown vibrations using an inertial actuator[END_REF].

The physical parameters of the system are not available.

The system has to be considered as a black box and the corresponding models for control design should be identified. The sampling frequency is F s = 800 Hz. Fig. 3 gives the frequency characteristics of the identified parametric model for the secondary path (the excitation signal was a PRBS). The system itself in the absence of the disturbances features a number of low damped vibration modes as well as low damped complex zeros (anti-resonance). This makes the design of the controller difficult for rejecting disturbances close to the location of low damped complex zeros. The most significant are those located at 46.45, 100.50 and 111.55 Hz (see the zoom of the frequency characteristics of the secondary path in Fig. 3). Note that the design of a linear controller for rejecting a disturbance at 50 or 95 Hz (as required by the benchmark) is difficult since each one of these frequencies is close to a pair of low damped complex zeros (damping around 0.005). The parametric model of the secondary path has a high order, n A = 22 and n B = 25. The system has a double differentiator behavior.

B. Plant/disturbance representation and controller structure

The structure of the linear time invariant discrete time model of the plant -the secondary path -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
)
where d is equal to the plant integer time delay (number of sampling periods) and

A(z -1 ) = 1 + a 1 z -1 + • • • + a n A z -n A ; B(z -1 ) = b 1 z -1 + • • • + b n B z -n B = z -1 B * (z -1 ) ; B * (z -1 ) = b 1 + • • • + b n B z -n B +1 ,
are polynomials in the complex variable z -1 and n A , n B and n B -1 represent their orders 3 . 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] Landau | A methodology for the design of feedback active vibration control systems[END_REF], [START_REF] Landau | Adaptive suppression of multiple time-varying unknown vibrations using an inertial actuator[END_REF].

Since the control objective is focused on regulation, the controller to be designed corresponds to a RS polynomial digital controller, ( [START_REF] Landau | Adaptive control[END_REF], [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF] -see also Fig. 2). The controller has the form K = R S , where R(z -1 ) and S(z -1 ), are polynomials in z -1 . Under the YK-parametrization (when a Q-FIR filter is considered) they have the following expressions:

R(z -1 ) = R 0 + AH S 0 H R 0 Q, ( 2
)
S(z -1 ) = S 0 -z -d BH S 0 H R 0 Q. ( 3
)
The YK-parametrization used here is depicted in Fig. 2, where both fixed and adaptive parts are shown. In this paper a YKparametrization using an equation-error disturbance observer is implemented along with a FIR filter representation of the optimal Q filter

Q(z -1 ) = q 0 + q 1 z -1 + • • • + q n Q z -n Q . ( 4
)
It can be shown [START_REF] Landau | Adaptive narrow band disturbance rejection applied to an active suspension -an internal model principle approach[END_REF] that for any arbitrary FIR Q(z -1 ) filter, the closed loop poles remain unchanged. They are defined by the stable polynomial

P(z 1 ) = A(z -1 )S 0 (z -1 ) + z -d B(z -1 )R 0 (z -1 ), (5)
whose order is

n P ≤ n A + n B + n H R 0 + n H S 0 + d -1
and where R 0 and S 0 are minimal degree solutions to the previous Bezout equation, whose structure is defined by the following equations:

R 0 (z -1 ) = r 0 0 + r 0 1 z -1 + . . . + r 0 n R 0 z -n R 0 = R ′ 0 • H R 0 ; (6) S 0 (z -1 ) = 1 + s 0 1 z -1 + . . . + s 0 n S 0 z -n S 0 = S ′ 0 • H S 0 , (7)
where H R 0 and H S 0 are fixed parts used to open the loop at some frequency or incorporate the model of the disturbance, respectively. Eqs. ( 6) and ( 7) and Fig. 2 describe the structure of the central controller. Pole Placement or other design technique can be used to place the poles.

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(t) ; ( 8
)
u(t) = - R(q -1 ) S(q -1 ) • y(t) , (9)
where p(t) is the resulting additive disturbance on the output of the system. We define the following sensitivity functions4 :

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

S yp (z -1 ) = AS P = A S 0 -z -d BH S 0 H R 0 Q P ; (10)
• Input sensitivity function (the transfer function between the disturbance p(t) and the input of the system u(t)):

S up (z -1 ) = - AR P = - A R 0 + AH S 0 H R 0 Q P , (11)
Using equations ( 8) and ( 9), one can write the output of the system as:

y(t) = A(q -1 )S(q -1 ) P(q -1 ) • p(t) = S yp (q -1 ) • p(t) . (12)
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(t) is a deterministic disturbance, so it can be written as

p(t) = N p (q -1 ) D p (q -1 ) • δ (t) , (13)
where δ (t) is a Dirac pulse 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 persistent (stationary) disturbances the roots of D p (z -1 ) are on the unit circle and they are complex and non repeated. 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 N p can be neglected.

It follows from eq. ( 12) that a complete rejection at a certain frequency ω k can be achieved when S yp (e -jω k ) = 0, which is possible if S(e -jω k ) = 0, i.e. the roots of D p (z -1 ) are contained in S(z -1 ). This is known as the internal model principle [START_REF] Francis | The internal model principle of control theory[END_REF]. In this case the modulus of the input sensitivity function (eq. ( 11)) becomes equal to the inverse of the plant gain which implies that if B(z -1 ) has a pair of low damped complex zeros close to ω k , the plant input will reach very high values, putting an important stress on the actuator. Another implication is that the modulus of the input sensitivity function will become very large and therefore the tolerance with respect to additive uncertainties will become very low. Therefore, the cancellation (or in general an important attenuation) of disturbance effect on the output should be done only in frequency regions where the system gain is large enough.

It is also well known that the use of the internal model for total rejection (asymptotically) of the disturbance raises the maximum value of the modulus of the output sensitivity function [START_REF] Silva | Direct adaptive rejection of unknown time-varying narrow band disturbances applied to a benchmark problem[END_REF], also known as "waterbed" effect. This may lead to unacceptable values for the modulus margin 5 -defined as ∆M = S yp (e -jω ) -1 max -and the delay margin if the controller design is not appropriately done [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF].

To illustrate the above, first consider the case when a double narrow band disturbance placed at 50 and 70 Hz must be canceled, (in this case D p (z -1 ) is known). In order that the polynomial Q(z -1 ) introduces the internal model of D p (z -1 ) in S(z -1 ), the following Diophantine equation must be satisfied

D p S ′ + z -d BH S 0 H R 0 Q = S 0 ( 14
)
where D p , H S 0 , H R 0 , d, B and S 0 are known, and S ′ and Q are unknown. Eq. ( 14) has a unique and minimal degree solution for S ′ and

Q with n S 0 ≤ n D p + n B + n H S 0 + n H R 0 + d -1, n S ′ = n B + d + n H S 0 + n H R 0 -1 and n Q = n D p -1.
Remark: It is assumed that D p and B do not have common factors but nothing is said of the feasibility of the solution if some roots of D p are very close to some of the roots of B. In such case, the solution of ( 14) will be computationally challenging due to ill conditioning. Furthermore, if n Q > n D P -1 then the solution is not unique leading to overparametrization. The central controller (R 0 , S 0 ) has been computed by Poles Placement and the desired poles have been selected using "Internal model control" [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF]. The system has n A poles. The closed loop will have

n P ≤ n A + n B + n H R 0 + n H S 0 + d -1 poles.
They are assigned as follows:

• The plant is stable and all plant poles [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF] are assigned as desired closed loop poles. • Among the remaining poles, 10 real poles are assigned at 0.3 for robustness reasons (robustifying poles called also robustifying filter) [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF], the other being at 0. • The loop is open at 0F s and 0.5F s by choosing H R 0 (z -1 ) = 1z -2 (controller gain equal to zero in steady state and at the Nyquist frequency). 5 The modulus margin is the minimum distance between the open loop transfer function hodograph and the Nyquist point [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF]. • No fixed parts are considered for S 0 (z -1 ), i.e. H S 0 (z -1 ) = 1. Using this central controller the resulting modulus margin will have unacceptable low values (the maximum of the modulus of the sensitivity functions is too high) as shown in Fig. 4(a). For the two sinusoidal disturbances at 50 Hz and 70 Hz one gets ∆M = 0.127 (17.9 dB). These values are far from the recommended value of ∆M = 0.5 -0.4. One observes also an important increase in the modulus of the input sensitivity function outside the attenuation band (50 to 95 Hz) in Fig. 4(b). This is why in [START_REF] Silva | Direct adaptive rejection of unknown time-varying narrow band disturbances applied to a benchmark problem[END_REF], the central controller design was considered as a major problem.

A. Improving the central controller design by poles selection [4]

In [START_REF] Silva | Direct adaptive rejection of unknown time-varying narrow band disturbances applied to a benchmark problem[END_REF] it was shown that using the plant model information (frequency characteristics), it is possible to keep the modulus of S yp (z -1 ) under an imposed maximum value by choosing appropriately some of the desired closed loop poles in P(z -1 ). In this approach the minimal degree for the polynomial Q(z -1 ) is preserved and an "equation error" Youla-Kučera observer is used. Basically it consist to place in addition of the plant poles, two pairs of complex poles close to the frequency region limits (50 and 95 HZ), and two other pairs of complex poles at 65 Hz and 80 Hz (these last values are not critical). The idea behind is that this will create "band stop filters " on the sensitivity functions around these frequencies which will allow to reduce significantly the waterbed effect (for more details on the effect of band stop filters see [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF]). The damping of these poles is chosen in relation with the attenuation imposed at the corresponding frequencies. In addition, the shaping of S up (z -1 ) outside the attenuation zone was considered in order to lower down its magnitude. This was done again by the technique of band stop filters. The sensitivity functions obtained with this design are also shown in Fig. 4 (dashed line). [START_REF] Jafari | Robust stability and performance of adaptive jitter supression in laser beam pointing[END_REF] In ( 14) the order of Q can be n Q > n D p -1. In this case the solution of ( 14) for Q, S ′ is not unique and an infinite number of polynomials with order n Q > n D p -1 satisfy [START_REF] Jafari | Robustness and performance of adaptive suppresion of unknown periodic disturbances[END_REF]. The approach of [START_REF] Jafari | Robust stability and performance of adaptive jitter supression in laser beam pointing[END_REF] is to consider the Q that minimizes S yp subject to the constraint of ( 14). The rationale behind this approach is that in the presence of noise or other uncertainties, denoted by v(t), we have

B. Improving robustness by increasing the number of parameters in the Q filter

y(t) = S yp (q -1 )p(t) + S yp (q -1 )v(t), (15)
since S yp contains the internal model D p of p(t), S yp (q -1 )p(t) will converge to zero asymptotically with time and y(t) will be driven by the term S yp (q -1 )v(t). The additional objective is to keep the modulus of S yp as low as possible. This gives rise to the following optimization problem min

Q J(Q) = H S 0 H R 0 q -d BQ S 0 ∞ ( 16
)
subject to [START_REF] Jafari | Robustness and performance of adaptive suppresion of unknown periodic disturbances[END_REF] and n Q > n D p -1. However another significant advantage of n Q > n D p -1 is that the ill conditioning that arises when the zeros of D p (z -1 ) are close to the zeros of B(z -1 ) is reduced as the order of Q increases. The details of the solution of the optimization problem in ( 16) are presented in [START_REF] Jafari | Robustness and performance of adaptive suppresion of unknown periodic disturbances[END_REF]. We should also note that the solution of the optimization problem shows the existence of an optimum Q, but in order to solve [START_REF] Landau | Adaptive regulation-rejection of unknown multiple narrow band disturbances (a review on algorithms and applications)[END_REF] we need to know the internal model D p (z -1 ).

The adaptive law that estimates the coefficients of Q search for the optimum Q given that it exists and does not require the a priori knowledge of the disturbances. The unknown optimal value is considered to be the desired one in the analysis of the adaptive scheme.

Note that the controller considered in [START_REF] Jafari | Robust stability and performance of adaptive jitter supression in laser beam pointing[END_REF] uses an "output error" type Youla-Kučera disturbance observer, however the methodology of [START_REF] Jafari | Robust stability and performance of adaptive jitter supression in laser beam pointing[END_REF] as indicated above extends to other types of disturbance observers without any significant difference 6 . The sensitivity functions obtained with this design are also shown in Fig. 4 (solid line).

C. Combination of the two approaches

It is obvious that the two approaches can be combined, i.e; one can consider to use the approach presented in III-A by using the improved controller design and augmenting the order of the Q filter.

IV. ADAPTIVE FEEDBACK REGULATION

In this section the direct adaptive algorithm -based on the YK-paremetrized controller -is recalled and used as basis for both comparison and combination of the two approaches to be evaluated. It can be used for the case of the minimal order of the polynomial Q (when n Q = n D p -1) or for the augmented case (when n Q > n D p -1).

As pointed out in [START_REF] Landau | Adaptive narrow band disturbance rejection applied to an active suspension -an internal model principle approach[END_REF], [START_REF] Landau | Adaptive suppression of multiple time-varying unknown vibrations using an inertial actuator[END_REF], the Internal Model Principle along with the YK-parametrization can be used to develop an algorithm to incorporate into the controller the assumed model of the disturbance (13) without changing the desired closed loop poles. From Fig. 2 we have that w(t) = A(q -1 )y(t)q -d B * (q -1 )u(t -1) = A(q -1 )p(t) [START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF] and replacing (3) in ( 12), the output of the closed loop system can be express as

y(t) = S 0 -q -d BH S 0 H R 0 Q P • w(t). ( 18
)
From the previous equation it is possible to derive the direct adaptive algorithm from [START_REF] Landau | Adaptive narrow band disturbance rejection applied to an active suspension -an internal model principle approach[END_REF], which uses the following a posteriori adaptation error

ε(t + 1) = θ T -θ T (t + 1) • φ (t) + v(t + 1), (19)
with the following definitions

θ = q 0 , q 1 , • • • , q n Q T , (20)
θ (t + 1) = q0 (t + 1), q1 (t + 1), • • • , qn Q (t + 1) T , (21)
φ (t) = [w 2 (t), w 2 (t -1), • • • , w 2 (t -n Q )] T , ( 22
)
w 2 (t) = q -d H R 0 H S 0 B * P • w(t), ( 23
)
v(t + 1) = S ′ D p P •w(t + 1) = S ′ AN p P • δ (t + 1), (24)
where v(t + 1) tends asymptotically towards zero (an asymptotically stable system excited by a Dirac pulse). The associated standard parameter adaptation algorithm (PAA) used is [START_REF] Landau | Adaptive suppression of multiple time-varying unknown vibrations using an inertial actuator[END_REF], [START_REF] Landau | Adaptive control[END_REF]:

θ (t + 1) = θ (t) + F(t)φ (t)ε(t + 1) (25)
ε(t + 1) = ε 0 (t + 1) 1 + φ T (t)F(t)φ (t) (26)
ε 0 (t + 1) = w 1 (t + 1) -θ T (t)φ (t) (27) w 1 (t + 1) = S 0 P • w(t + 1) (28)
F(t + 1) -1 = λ 1 (t)F(t) -1 + λ 2 (t)φ (t)φ T (t) (29) 0 < λ 1 (t) ≤ 1, 0 ≤ λ 2 (t) < 2, F(0) > 0 ( 30
)
where λ 1 (t), λ 2 (t) allow to obtain various profiles for the evolution of the adaptation gain F(t) (for details see [22][18]).

For a stability proof under the hypothesis model = plant, see

V. COMPARISON OF THE TWO APPROACHES AND THEIR COMBINATION -SIMULATION AND REAL TIME RESULTS

As seen in Section III, the standard central controller design using "internal model control" does not achieve a robust modulus margin, either for a single or double narrow band disturbances in the vicinity of low damped complex zeros, when the minimal degree for Q (n Q = n D p -1) is used. An improved central controller design has been proposed for this case. See III-A(this will be called "Case 1"). Alternatively using the standard central controller design the augmentation of the order of the Q filter has been considered (this will be called "Case 2"). Finally augmenting the order of the Q polynomial for the improved central controller design has also been considered (this will be called "Case 3").

Due to space constraints, only some of the results will be presented. To determine a relevant value of the degree of polynomial Q for the augmented solutions, simulations were carried out.The performance specifications ans constraints as well as the measurements procedures defined [START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF] are considered. Nevertheless, not all the criteria will be presented. Special attention is given to: global attenuation (GA) of the disturbance, disturbance attenuation (DA) and the maximum amplification (MA) along with the transient evaluation (TE). For the case of over parametrization a normalized parameter adaptation algorithm should be used [START_REF] Landau | Adaptive control[END_REF] and [START_REF] Petros | Robust adaptive control[END_REF]. However the normalization slow down significantly the adaptation transients and therefore for the purpose of comparison with the minimal parametrization approach the unnormalized adaptation algorithm has been used for all the cases.

Two different narrow band disturbances are tested on the simulator and experimental setup (Section II-A). The first disturbance corresponds to a double narrow band disturbance located at 50 and 70 Hz and the second is when the disturbance frequency corresponds to 75 and 95 Hz. This two set of frequencies were chosen due to the proximity that they have to the plant low damped complex zeros. For all simulations ans experiments, the parameter adaptation algorithm given in Section IV has been used. λ 1 and λ 2 have been updated in order to obtain a time decreasing adaptation gain combined with a constant trace adaptation gain (residual gain assuring the adaptive behavior). The value of the constant trace was 0.004xN where N is the number of parameters to be adapted. The corresponding updating formula can be found in [START_REF] Landau | Adaptive control[END_REF].

A. Performance criteria for simulation and real-time experiments

Simulations (SIM) and real-time (RT) experiments were conducted using both approaches and their combinations. The results were classified in frequency and in time domain. In frequency domain the objective is to strongly attenuate the disturbance with a limited amplification of the other frequencies. To evaluate the performance three indicators have been defined together with three target values according to [START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF]: Disturbance Attenuation (DA) (min = 40 dB), Global Attenuation (GA) (min = 30 dB), and Maximum Amplification (MA) (max = 7 dB) 8 .

In time domain the Transient Evaluation (TE in %) is considered. The transient evaluation criterion requires that the transient duration when a disturbance is applied, has to be smaller than 2 s. A percentage was established for the fulfillment of this criterion. TE = 0% indicates a transient duration of 4 s and TE = 100% a transient duration smaller or equal to 2 s. The percentage is assigned using the ratio (α) of the truncated two norm (N 2 T ) of the residual force evaluated in two periods of time. This means that 2 s after application of a disturbance the N 2 T of the output has to be equal or smaller than 1.21 of the steady state value of the N 2 T of the residual force. The N 2 T is evaluated over an interval of 3 s both for transient and steady state, taking into account that the disturbance is applied at t = 5 and the steady state is evaluated between 27 and 30 s. The performance variables are calculated using the following equations:

N 2 T (i : m) = m ∑ i y(i) 2 (31) α = N 2 T (7 : 10) N 2 T (27 : 30) (32) δ Trans = α -1.21 if α > 1.21 (33) δ Trans = 0 if α ≤ 1.21 (34)

B. Results in simulation and real-time experiments

Table I summarizes the results in frequency domain when a double disturbance at 50 and 70 Hz is introduced into the system. The results correspond to simulations and real-time experiments. For the over parametrized solutions the persistent excitation condition is not preserved.

As is seen from the results, the solution used in Case 1 shows its effectiveness by achieving very good results for GA and DA and being slightly over the limit in MA (for the RT). Since the objective is to compare the three cases, the number of parameters for Case 2 was augmented until the results were close to the ones of the Case 1. For this disturbance, 22 parameters ( nb. of parameters = n Q + 1) were necessary to achieve almost the same level of effectiveness. Nevertheless, the MA shows that keeping the modulus of S yp under the settled limit is difficult. It was found that even though the number of parameters could be increased, this have a negative impact either over the maximum amplification or the disturbance attenuation. So 22 parameters seems to be a good compromise. Finally, Case 3 shows that augmenting from 4 to 5 parameters, better GA (in SIM) and MA (in SIM) are achieved. However, the Disturbance attenuation (DA) in real time (RT) is slightly less good. In Fig. 5 the achieved modulus margin (∆M) is shown for all the cases. Since in all the cases the DA and GA were achieved, the figure aims to show the amplifications due to the waterbed effect. Case 1 and 3 handle in a better way this effect, while the Case 2 achieves almost the same profile but not for the amplifications at other frequencies different from 50 and 70 Hz.

Table II shows the results when the disturbance is located at 75 and 95 Hz. As in the previous test, the Case 1 shows simulation results close to the benchmark specifications. The RT results shows some differences, the most important in DA and MA. Taking these results as a reference, it results that one need 27 parameters for the Case 2 in order to achieve or improve the performance obtained in Case 1 (simulation results with 22 parameters shows less good results). As for the previous test, higher orders for Q(z -1 ) allows to achieve better DA for 95 Hz but not for 75 Hz. Also, MA achieves its minimum using 27 parameters but if more parameters are used, this result is degraded. Although simulations results where better, the real-time results were quite close to the ones obtained for the Case 1. Regarding the real-time results, the Case 3 achieves the best ones, showing clearly that the combination of both approaches allows to improve the result from a minimal degree solution (Case 1), without increasing significantly the number of parameters (Case 2). Fig. 6 displays the estimated output sensitivity function for the three cases, using the estimated parameters of Q(z -1 ). The figure is zoomed in order to show the amplifications due to the waterbed effect. It is clear that the Case 3 shows the lowest amplifications (higher ∆M).

The transient evaluation results are shown in Table III 94.2% and 99.7% in simulation and real-time respectively. These results are close to the imposed specifications.

The complexity of the various solutions are evaluated in Table IV by comparing the Computation Time (CT). As expected, the Case 1 shows the lowest CT since it uses the minimum number of parameters to be adapted. By augmenting the number of parameters, as in Case 3, the CT is increased. The most significant increases occur for the Case 2 for [50,70] Hz and and for [75,95] Hz. The computation time increases by a factor over 12 with respect to the Case 1.

VI. CONCLUDING REMARKS

Two approaches and their combination for handling the problem of attenuation of unknown narrow band disturbances in the vicinity of low damped plant complex zeros have thoroughly evaluated. One approach emphasizes the need for a careful selection of some of the desired closed poles in order to use a minimal number of parameters to adapt. The other approach overcome the problem of a careful design of the central controller by a significant over parametrization of the compensator filter to be adapted. The price to pay is a significant increase in the computer load. Combination of the two approaches can be considered. A small increase of the size of the minimal order of the filter to be adapted combined with an improved central controller design can further slightly improve the performance. It is however important to recall that strictly speaking the over parametrization requires to use a normalized parameter adaptation algorithm and this will augment significantly the duration of the adaptation transients.

Fig. 1 .

 1 Fig. 1. Active vibration control using an inertial actuator (photo).

Fig. 2 .

 2 Fig. 2. Direct adaptive scheme using a YK-parametrization of the controller. Dashed line: fixed part, dotted line: adaptive part, solid line: active vibration control set up.

Fig. 3 .

 3 Fig. 3. Magnitude of the frequency response for secondary path model (top). Zoom at the low damped complex poles and zeros (bottom).

Fig. 4 .

 4 Fig. 4. Modulus of the sensitivity functions for different cases. Dash-dot line: basic design, dashed line: design from [4], solid line: overparametrized solution.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Output sensitivity function comparison, for rejecting a double disturbance located at [50,70] Hz.

  . All the cases except Case 2 with n Q = 26 for a disturbance located at [75,95] Hz pass the transient evaluation. For Case 2 with n Q = 26 for a disturbance located at [75,95] Hz on achieves

The complex variable z -1 will be used for characterizing the system's behavior in the frequency domain and the delay (shift) operator q -1 (x(t) = q -1 x(t + 1)) will be used for describing the system's behavior in the time domain.

The arguments (z -1 ) and (q -1 ) will be omitted in some of the following equations to make them more compact.

For a definition of the various types of Youla-Kučera disturbance observers see[START_REF] Landau | Benchmark on adaptive regulation -rejection of unknown/time-varying multiple narrow band disturbances[END_REF].

Convergence of the adaptation error for any initial conditions, boundedness of all the variables, convergence of the estimated parameters toward the optimal values for n Q = n Dp -1

GA and MA give indication about the quality of the control which is supposed to introduce a very limited amplification at frequencies different from the frequency of the disturbance.

The CT for an open loop test is 12.9 µs is subtracted from the total computation time


I D Landau
email: ioan-dore.landau@gipsa-lab.grenoble-inp.fr

Robust and Adaptive Active Vibration Control Using an Inertial Actuator

Tudor-Bogdan Airimitoaie and Ioan Doré Landau

Abstract-An active vibration control system using an inertial actuator will be used to assure the vibration isolation of a chassis in the presence of multiple unknown time varying tonal vibrations located in two distinct regions in the frequency domain. The objective is to minimize the residual force by applying an appropriate control effort through the inertial actuator. The system does not use any additional sensor for getting in realtime information upon the disturbances. A hierarchical feedback approach will be used for the control of the system. At the first level a robust linear controller will be designed taking advantage of the knowledge of the domains of variation of the frequencies of the vibrations. To further improve the performance, a direct adaptive control algorithm will be added. Its design takes into account the internal model principle for disturbance rejection and is conveniently implemented through the Youla-Kučera parametrization of the controller. Experimental results obtained on a relevant test bench will illustrate the methodology.

Index Terms-Adaptive regulation, system identification, active vibration control, robust control, Youla-Kučera parametrization, internal model principle.

I. Introduction

A CTIVE VIBRATION CONTROL (AVC) is a current challenge in todays technology development for smart vibration isolation systems. Fundamentally in AVC a compensator system is introduced with the aim to generate a secondary "source." This compensator conveniently driven will interfere destructively with the vibrations coming from the original source (in general non accessible).

The technology for the implementation of the AVC in mechanical structures has evolved towards using inertial (electrodynamic) actuators [START_REF] Marcos | The straight attraction[END_REF]. In this paper we will consider the case of AVC systems using inertial actuators in a co-located configuration "actuator-sensor."

Vibrations in terms of control terminology are "disturbances" which have to be compensated (see also [START_REF] Li | Multi-modal vibration control using amended disturbance observer compensation[END_REF], [START_REF] Li | Composite multi-modal vibration control for a stiffened plate using non-collocated acceleration sensor and piezoelectric actuator[END_REF]). They can be conveniently described in the frequency domain. One encounters either "tonal" (sinusoidal) disturbances with unknown and time varying frequency or "narrow band disturbances" located in a certain frequency region.

Initial solutions for this problem have been based on "feedforward" compensation techniques, provided that an "image" (called also "reference") of the disturbance can be obtained by using an additional sensor [START_REF] Elliott | Performance of feedforward and feedback systems for active control[END_REF]. This approach has a number of disadvantages: (i) it requires the use of an additional transducer, (ii) difficult choice for the location of this additional transducer, (iii) requires adaptation of many parameters, and (iv) presence of a physical internal "positive" feedback coupling (a challenging stability issue). In fact, in AVC, in order to attenuate narrow band disturbances and tonal disturbances, a feedback control structure can be used.

In managing the vibration attenuation by feedback, the shape of the modulus of the "output sensitivity function" (the transfer function between the disturbance and the residual acceleration/force) is fundamental both from performance and robustness considerations. Three basic concepts are to be considered: the Bode Integral, the Modulus margin and the Internal Model Principle (IMP). The Bode Integral just says that the area of attenuation in the frequency domain (attenuation × frequency range) leads to an amplification at other frequencies such that the integral of the modulus of the sensitivity function over the frequency domain is equal to zero. The modulus margin, which is the inverse of the maximum of the modulus of the output sensitivity function, gives the minimum distance with respect the the Nyquist instability point and as such is a major indicator both for robustness and performance. The IMP says that in order to cancel completely a disturbance the controller should contain the model of the disturbance1 (see also [START_REF] Francis | The internal model principle of control theory[END_REF]).

Several problems have been considered in the field of AVC. The case of full rejection of single or multiple tonal disturbances (up to 3) located quite distantly in the frequency domain with unknown and time varying frequencies over a significant frequency range has been extensively covered in the literature and compared experimentally [START_REF] Landau | Benchmark on adaptive regulation: rejection of unknown/timevarying multiple narrow band disturbances[END_REF]. 2 An adaptive feedback approach taking advantage of the IMP as well as of a special parametrization of the controller (the Youla-Kučera parametrization [START_REF] Anderson | From Youla-Kucera to identification, adaptive and nonlinear control[END_REF]) has been considered. Note that Youla-Kučera parametrization introduces explicitely a distrubance observer. A rapprochement with DOB control method [START_REF] Li | Piezoelectric vibration control for all-clamped panel using dob-based optimal control[END_REF] can be established.

In this paper, a different pertinent problem encountered in practice will be examined. One considers multiple unknown and time varying sinusoidal disturbances located within two distinct relatively small frequency ranges. To be specific, two cases will be considered: (i) the case of two time varying tonal disturbances located in two distinct frequency regions and (ii) the case of four simultaneous tonal disturbances, two located in one limited frequency range and the other two in another frequency range.

In this context, a very important problem is to be able to counteract the very low frequency oscillations which are generated when the two frequencies are very close. This phenomenon is considered for example in [START_REF] Li | Piezoelectric Multimode Vibration Control for Stiffened Plate Using ADRC-Based Acceleration Compensation[END_REF]. It also occurs often on ships with two thrusters which can not be perfectly synchronized. A typical image of the phenomenon is shown in Figure 1. Since these disturbances are located within two relatively small frequency ranges, it is possible to consider a robust linear control design which will shape the output sensitivity function in such a way that a sufficient attenuation is introduced in these two frequency regions but avoiding significant amplification at other frequencies (both for performance and robustness reason). This problem in the context of active noise control has been considered in [START_REF] Carmona | Active noise control of a duct using robust control theory[END_REF] and the shaping of the output sensitivity function has been achieved using the convex optimization procedure introduced in [START_REF] Langer | Combined pole placement/sensitivity function shaping method using convex optimization criteria[END_REF]. It will be shown in this paper that an elementary procedure for shaping appropriately the modulus of the sensitivity functions can be implemented using stop band filters as shaping tools. For a basic reference on this approach see [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF].

To further improve the performance, an algorithm for direct adaptive rejection of the disturbances will be added [START_REF] Landau | Adaptive narrow band disturbance rejection applied to an active suspension -an internal model principle approach[END_REF].

Another important point of the methodology for designing AVC systems is the fact that one uses for design discrete-time models of the system directly estimated from data (both the orders of the model and the parameters). The motivation is two fold: on one hand it is practically impossible to obtain an accurate model of the plant from physical modeling based on continuous time equations and on the other hand very performant methods for estimating orders and parameters of discrete time dynamical models directly from data are available. Furthermore designing the control law based on the identified discrete time model will directly provide the control algorithm to be implemented on the computer controlling the system.

The paper is organized as follows. In section II, the test bench and its characteristics will be presented. Section III presents briefly the equations describing the system model and the controller. Section IV presents the performances specification to be achieved and the design of the robust controller. Section V presents the adaptive control algorithm. Experimental results for both robust linear control and adap-tive control are given in Section VI. Concluding remarks are given in Section VII. The AVC system that will be used for the experiments is shown in Fig. 2. The same system has been used for an international benchmark on adaptive disturbance regulation (see [START_REF] Landau | Benchmark on adaptive regulation: rejection of unknown/timevarying multiple narrow band disturbances[END_REF]).The system has been designed in order to test the performance of active control by reducing to a minimum the "passive vibration attenuation". It consists of a shaker (fixed to the ground), a passive damper, an inertial actuator, a mechanical structure, and a transducer for the residual force. For control purposes, 2 desktop computers are used, one with a Microsoft Windows operating system and Matlab/Simulink environment (for controller design, implementation and simulation) and the second with the real time operating system Matlab xPC Target (for real-time operation of the AVC system) -see also Fig. 3 for more details. The latter is connected through a power amplifier to the inertial actuator.

II. System Presentation

The mechanical construction is such that the vibrations produced by the shaker are transmitted to the upper part of the mechanical structure, on top of the passive damper. The inertial actuator is fixed to the chassis where the vibrations should be attenuated. The controller, through the power amplifier, generates current in the coil which produces motion in order to reduce the residual force. In Fig. 3, the control signal u(t) represents the position of the magnet inside the inertial actuator. The measured output of the system (residual force) is y(t) which enters the xPC Target dedicated computer. Finally, for testing purposes, the disturbance p(t) induced by the shaker on the residual force is operated from the computer through the disturbance input u p (t). The transfer function between the disturbance input u p (t) and the measured output y(t) is called primary path. The transfer function between the control input u(t) and the measured output is called secondary path. Note that the system has a double differentiator behavior (input=position, output=force).

The physical parameters of the system being unknown, black-box discrete time linear model identification has to be done in order to obtain a dynamical model of the primary and secondary paths. 3 The sampling period is T s = 0.00125 sec ( f s = 800 Hz). Fig. 4 gives the frequency characteristics of the identified model for the primary and secondary paths. As it can be seen, there is an important number of very low damped complex poles (resonances) and complex zeros (anti-resonances). The primary path model is used only for simulation purposes. The region for active vibration control is 50 to 95 Hz.

III. System Description

The linear time invariant (LTI) discrete time model of the secondary path, used for controller design is

G(z -1 ) = z -d B(z -1 ) A(z -1 ) = z -d 1 + a 1 z -1 + • • • + a n A z -n A b 1 z -1 + • • • + b n B z -n B ( 1
)
and d is the plant pure time delay in number of sampling periods. 4The output of the plant y(t) and the input u(t) may be written as (see Fig. 5):

y(t) = q -d B(q -1 ) A(q -1 ) • u(t) + p(t), (2)
S 0 (q -1 ) • u(t) = -R 0 (q -1 ) • y(t). (3)
In (2), p(t) is the effect of the disturbances on the measured output 5 and R 0 (z -1 ), S 0 (z -1 ) are polynomials in z -1 having the following expressions:6

S 0 = 1 + s 0 1 z -1 + . . . + s 0 n S z -n S = S 0 • H S 0 , (4)
R 0 = r 0 0 + r 0 1 z -1 + . . . + r 0 n R z -n R = R 0 • H R 0 , (5)
where H S 0 (z -1 ) and H R 0 (z -1 ) represent 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) and S 0 (z -1 ) and R 0 (z -1 ) are the solutions of the Bezout equation

P 0 = A • H S 0 • S 0 + z -d B • H R 0 • R 0 . (6)
In the last equation, P 0 (z -1 ) represents the characteristic polynomial, which specifies the desired closed loop poles of the system. The transfer functions between the disturbance p(t) and the output of the system y(t) and from disturbance to the control input u(t), denoted respectively output sensitivity function and input sensitivity function, are given by

S yp (z -1 ) = A(z -1 )S 0 (z -1 ) P 0 (z -1 ) (7)
and

S up (z -1 ) = - A(z -1 )R 0 (z -1 ) P 0 (z -1 ) . ( 8
)
It is important to remark that one should only reject disturbances located in frequency regions where the plant model has enough gain. This rule results from [START_REF] Anderson | From Youla-Kucera to identification, adaptive and nonlinear control[END_REF] and noticing that perfect rejection at a certain frequency ω 0 is obtained iff S 0 (e -jω 0 ) = 0. At this frequency, under perfect rejection of disturbances, one gets

S up (e -jω 0 ) = - AR 0 0 + e -d jω 0 BR 0 = - A e -d jω 0 B = 1 G(e -jω 0 ) . (9)
Eq. ( 9) corresponds to the inverse of the gain of the system to be controlled. Its implication 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 [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF]. Eq. ( 9) 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.

IV. Robust Control Design

Before presenting the objectives for regulation and robustness, a few notions about feedback disturbance attenuation should be reminded. In the case of a feedback controlled system, the Bode integral constraint leads to a waterbed effect on the output sensitivity function (transfer function from disturbance p(t) to output y(t) in closed loop, see Section III). In other words, forcing the magnitude of the output sensitivity function at certain frequencies below 0 dB (in order to attenuate disturbances) has an inverse effect at neighboring frequencies, where an amplification will be observed. Recalling from [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF] that the minimal distance between the Nyquist plot of the open loop transfer function and the critical point -1+0i (also called modulus margin) corresponds to the inverse of the maximum of the output sensitivity function, it can be concluded that "too much" attenuation at some frequencies can have a bad effect on the robust stability of the closed loop system.

The secondary path has enough gain in the frequency region of operation (see also Fig. 4).

For the design of the linear robust digital controller the following specifications are considered:

• up to 4 sines disturbances are supposed to affect the output of the system • their frequencies are varying within a ±2.5 Hz frequency band around 60 Hz and 80 Hz.

• the controller should attenuate the disturbances by a minimum of 14 dB. • the maximum allowed amplification of the output sensitivity function is of 8dB.

• the effect of disturbances on the control input should be attenuated above 100 Hz in order to improve robustness with respect to unmodeled dynamics and nonlinear phenomena (S up (e -jω ) < -20dB, ∀ω ∈ [100 Hz, 400 Hz]) • the gain of the controller should be zero at at 0 Hz and 0.5 f s (400 Hz) (the system has no gain at 0 Hz -double differentiator -and very low gain at 0.5 f s )

It is shown in [12, Property 7, Section 3.6.1] that very accurate shaping of the output or the input sensitivity function can be obtained by the use of band-stop filters (BSF). These are IIR filters obtained from the discretization of continuoustime filters of the form

F(s) = s 2 + 2ζ num ω 0 s + ω 2 0 s 2 + 2ζ den ω 0 s + ω 2 0 ( 10

2

. Depending on whether the filter is designed for shaping the output or the input sensitivity function, the numerator of the discretized filter is included in the fixed part of the controller denominator H S 0 or numerator H R 0 , respectively. The filter denominator is always included in the closed loop characteristic polynomial. As such, the filter denominator influences the design of the controller indirectly in the computation of S 0 and R 0 as solutions of the Bezout equation ( 6). They will be used for a fine shaping of both the output and input sensitivity functions.

The steps for the design of the linear controller are: 7 1) include all (stable) secondary path poles in the closed loop characteristic polynomial. 2) open the loop at 0 Hz and at 400 Hz by setting the fixed part of the controller numerator

H R = (1 + q -1 ) • (1 -q -1 )
3) 3 BSFs on S yp have been used around each of the frequencies where attenuation is desired in order to assure the desired attenuation within ±2.5 Hz (see Table I for specifications). 4) 1 BSF has been used on S up to reduce its magnitude above 100 Hz (see Table I for specifications). 5) to improve robustness 2 complex conjugate poles have been added to the characteristic polynomial, one at 55 Hz and the second at 95 Hz, both of them with 0.1 damping factor. The output and input sensitivity functions with this linear controller can be analyzed in Figs. 6 and7, respectively. From Fig. 6, it can be observed that the desired attenuation of 14 dB and the maximum amplification of 8 dB on S yp are achieved.

V. Adaptive Feedback Control

This section gives a brief presentation of the direct adaptive control scheme implemented on top of the central controller. The Youla-Kučera (YK) parametrization of controller is used (see for implementing the adaptive loop. In this context, the controller polynomials are parametrized using an finite impulse response (FIR) filter of the form

Q(z -1 ) = q 0 + q 1 z -1 + . . . + q n Q z -n Q , (11)
and the central controller polynomials given in eqs.( 4) and ( 5) obtained as solutions of the Bezout equation ( 6). As such, the controller polynomials become

R = R 0 + AQH S 0 H R 0 , (12)
S = S 0 -z -d BQH S 0 H R 0 . ( 13
)
The purpose of the central controller ( R 0 S 0 ) in the Youla-Kučera parametrization is that of verifying stability and robustspecifications. It should be observed that the characteristic polynomial of the closed loop remains unchanged

P = AS + z -d BR = AS 0 + z -d BR 0 , ( 14
)
in the presence of the Q(z -1 ) FIR filter. The role of the Q filter, which will be adjusted in real time using a parameter adaptation algorithm, is to assure the rejection of unknown time-varying disturbances. The schematic representation of the closed loop with the adaptive Youla-Kučera parametrized 7 The software iREG has been used for the design of this robust digital controller but the same results can be obtained using functions written in Matlab/Scilab languages (see http://www.gipsa-lab.grenoble-inp.fr/~ioandore. landau/identificationandcontrol/). controller of ( 12) and ( 13) is shown in Fig. 8. Note that in Fig. 8, w(t) = A • p(t) and as such the Youla-Kučera parametrization is a disturbance observer and w(t) is feedback through the filter Q (like in DOB control method [START_REF] Li | Piezoelectric vibration control for all-clamped panel using dob-based optimal control[END_REF]).

A key aspect of this methodology is the use of the IMP. It is supposed that p(t) is a deterministic disturbance given by

p(t) = N p (q -1 ) D p (q -1 ) • δ (t), (15)
where δ (t) is a Dirac impulse and N p , D p are coprime polynomials of degrees n N p and n D p , respectively 8 . In the case of stationary narrow-band disturbances, the roots of D p (z -1 ) are on the unit circle.

Internal Model Principle [5]:

The effect of the disturbance (15) upon the output y(t) = A(q -1 )S(q -1 ) P(q -1 )

• N p (q -1 ) D p (q -1 ) • δ (t), (16)
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 iff the polynomial S(z -1 ) in the RS controller has the form (based on eq. ( 4))

S(z -1 ) = D p (z -1 )H S 0 (z -1 )S (z -1 ). (17)
Thus, the pre-specified part of S(z -1 ) should be chosen as H S (z -1 ) = D p (z -1 )H S 0 (z -1 ) and the controller is computed solving

P = AD p H S 0 S + z -d BH R 0 R , (18)
where P, D p , A, B, H R 0 , H S 0 and d are given 9 . The Q polynomial allows the introduction in the controller of the model of the disturbance (i.e. if D p is the model of the disturbance, it exist a polynomial Q of order n D p -1 such that S given by ( 13) can be factorized as S D p -see [START_REF] Landau | Adaptive control[END_REF]).

Assuming that the structure of the disturbance is known, i.e. n D p , the order of the Q polynomial is fixed as

n Q = n D p -1.
Let define the estimate of Q at time t by 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(t + 1) = A • y(t + 1) -q -d B * • u(t), (19)
w 1 (t) = S 0 P • w(t), w 2 (t) = q -d B * H S 0 H R 0 P • w(t) (20)
and define the following observation vector

φ T (t) = [w 2 (t) w 2 (t -1) . . . w 2 (t -n Q )]. (21)
The a priori adaptation error can be defined as (see [START_REF] Landau | Adaptive narrow band disturbance rejection applied to an active suspension -an internal model principle approach[END_REF] for more details)

ε 0 (t + 1) = w 1 (t + 1) -θ T (t)φ (t). (22)
For the estimation of the parameters of Q(t, q -1 ) an 'Integral' Parameter Adaptation Algorithm (I-PAA) is used:

θ (t + 1) = θ (t) + F(t)φ (t)ε(t + 1), (23a)
ε(t + 1) = ε 0 (t + 1) 1 + φ T (t)F(t)φ (t) , (23b)
ε 0 (t + 1) = w 1 (t + 1) -θ T (t)φ (t), (23c)
F(t + 1) = 1 λ 1 (t)   F(t) - F(t)φ (t)φ T (t)F(t) λ 1 (t) λ 2 (t) + φ T (t)F(t)φ (t)   , 1 ≥ λ 1 (t) > 0, 0 ≤ λ 2 (t) < 2, (23d)
where λ 1 (t), λ 2 (t) allow to obtain various profiles for the evolution of the adaption gain F(t) (for more details see [START_REF] Landau | Adaptive control[END_REF]).

Another more general choice of adaptation algorithm is the 'Integral+Proportional' Parameter Adaptation Algorithm (IP-PAA) (see [START_REF] Landau | Adaptive control[END_REF], [START_REF] Airimiţoaie | Improving adaptive feedforward vibration compensation by using "Integral+Proportional" adaptation[END_REF] for more details).

At each sampling time the following procedure is applied for adaptive operation:

1) Get the measured output y(t + 1) and the applied control u(t) to compute w(t + 1) using (19). 2) Compute w 1 (t + 1) and w 2 (t) using (20) with P given by ( 14). 3) Estimate the Q-polynomial using the I-PAA of eqs. (23). 4) Compute and apply the control (see Fig. 8): S 0 (q -1 )u(t + 1) = -R 0 (q -1 )y(t + 1) -Q(t + 1, q -1 )w(t + 1).

VI. Experimental Results

The experimental results presented in this section are obtained using the identified model of the secondary path (see also Section II). Details on system identification and the model used throughout this section can be found on the benchmark web site at http://www.gipsa-lab.grenoble-inp.fr/~ioandore. landau/benchmark_adaptive_regulation/files/Simulator_2.zip. The secondary path model is given in the file model_sec2.mat. The orders of this system are: n A = 22, n B = 25, and d = 0.

A. Central Controller for Youla-Kučera Parametrization

The design of the central controller used in the Youla-Kučera parametrization is similar to the design of the robust linear controller with the exception that the BSFs on S yp have not been used and the resulting free roots to be assigned have been moved from 0 to 0.2. Remark that the order of the characteristic polynomial is given by n P = n A +n B +n H S +n H R +d -1 which in our case gives 22 + 25 + 0 + 4 + 0 -1 = 50. Given the roots already specified (28 as can be concluded from the design of the robust controller excepting roots given by BSFs for S yp ), it follows that 22 roots can be selected. These 22 auxiliary poles at 0.2 have the effect of reducing the magnitude of S up above 100 Hz. They were not used in the robust linear design.

B. Single-Mode Vibration Control

The results in this subsection are obtained by considering 2 sinusoidal disturbances with time varying frequencies on the system output. The time variations of the frequencies are obtained by using 2 independent pseudo random binary sequences (PRBS). The 2 sinusoidal disturbances vary around 60 and 80 Hz, respectively, remaining inside the ±2.5 Hz frequency intervals for which the robust linear controller introduces 14 dB of attenuation.

Note that all subsequent experiments start at 10 seconds. This period has been introduced in order to give enough time to activate the electronic boards for real time experimentation. Also, the system operates in open loop for 5 seconds (from 10 to 15 sec). 10 Finally, 5 seconds before the end of the experiments, the system is switched back to open loop and the system input and the disturbances are removed so that the residual forces can be compared to the system noise.

In Fig. 9, time domain experimental results are shown for the open loop, the closed loop with the linear controller and for the closed loop with the adaptive controller.

For adaptive regulation, the I-PAA is used. The order of the Q polynomial has been chosen equal to 3 (4 parameters to be adapted). The initial diagonal adaptation gain matrix used is F(0) = α • I, with α = 0.2 and I the identity matrix (initial trace of 0.8). A constant trace adaptation algorithm is used with constant trace of 0.8 (see [START_REF] Landau | Adaptive control[END_REF] for further details on the choice of the adaptation gain). The evolution of the parameters of the Q polynomial can be viewed in Fig. 10. As it can be observed, the vector of the estimated Q parameters, θ is initialized at zero. Once the loop is closed, the adaptive algorithm starts to adjust the parameters in order to reduce the residual force. The parameters of the Youla-Kučera filter evolve continuously during the experiments in order to adjust to the changing frequencies of the disturbances.

To compare the attenuation results of the robust linear controller with the adaptive regulator, the global attenuation is computed over the last 3 seconds of each closed loop experimentation. For the robust linear controller the global attenuation is 25.70 dB,while in the adaptive case it is 39.68 dB. A small additional improvement can be obtained by using the IP-PAA. 11 Finally, experimental results for frequencies variations in ±5 Hz intervals around 60 and 80 Hz are shown in Fig. 11. As expected the linear controller results are not good (we are outside of the domain considered for design). The last 5 sec without disturbance and in open loop operation are also plotted as reference. The levels of the control signal for the linear and adaptive controllers, are approximately the same: ±0.14 V for the linear case and ±0.13 V for the adaptive case.

C. Two-Mode Vibrational Interference Control

This subsection deals with the AVC of two-mode sinusoidal vibrations (vibration interference). The phenomena occurs when two tonal vibrations have very close frequencies. This phenomena is shown in the upper plot of Fig. 12 where 2 10 To avoid large transients when switching on the controllers, a bumpless transfer scheme from open to closed loop has been used (see also [START_REF] Landau | Digital Control Systems -Design, Identification and Implementation[END_REF]Chapter 8]). 11 Global attenuation in decibels (dB) is computed as -20 log 10 pairs of neighboring sinusoidal disturbances are introduced, one pair around 60 Hz (at 59.9 and 60.1 Hz) and the second around 80 Hz (at 79.9 and 80.1 Hz). The same robust linear controller as described earlier can be used as its attenuation frequency band is large enough to accommodate the neighboring disturbances. For adaptive regulation, the I-PAA has been used with an initial diagonal adaptation gain matrix F(0) = α • I, with α = 0.2 and I the identity matrix (initial trace of 0.8), and a decreasing gain followed by constant trace adaptation. The constant trace is chosen equal to 0.02. The number of parameters for the Q polynomial is also equal to 4 (order equal to 3). Augmenting the order of the polynomial Q to 7 (8 parameters -two for each sinusoidal disturbance) does not improve the performance (probably because the frequencies of the pair of sines are too close). Time domain results are shown in Figs. 12. The global attenuation for the robust linear controller is 27.50 dB and for the adaptive controller is 45.59 dB.

Power spectral densities (PSD) estimates of the two control schemes are given in Fig. 13. It can be observed that the attenuation introduced by the robust linear controller in the desired frequency zone is equal to 14 dB which is coherent with the design done in Section IV. The adaptive regulator has better attenuation of disturbances and also does not amplify at other frequencies more than the linear controller. Adaptation capabilities are tested and the results are compared to the linear robust controller in Fig. 14. In this figure, all 4 sinusoidal disturbances are modified at 35 seconds by adding 5 Hz to their frequencies. The constant trace is increased to 0.1. As such the new disturbance frequencies are centered around 65 Hz (64.9 and 65.1 Hz) and 85 Hz (84.9 and 85.1 Hz). As expected the linear robust controller fails to provide an acceptable attenuation. The adaptation transient is about 1.5 sec. The input signal for both linear control and adaptive control is shown in Figure 15. Their level is comparable in the nominal situation but when the frequencies changes the level of the control signal for the adaptive controller is lower.

VII. Conclusion

It was shown in this paper that provided that the region of variation of the unknown multiple tonal vibrations is limited, efficient robust active compensation can be achieved with a properly designed linear controller. However, adding an adaptive loop enhances drastically the performance. The use of the adaptive approach allows to achieve the same level of performance even if the variations of the frequencies of the vibrations go beyond the initial specified frequency region. In this new context the performance of the robust controller are unsatisfactory.

Figure 1 .

 1 Figure 1. Vibrational interference caused by two sinusoidal disturbances with close frequencies.

Figure 2 .Figure 3 .

 23 Figure 2. The active vibration control system (photo).

Figure 4 .

 4 Figure 4. Frequency characteristics of the secondary and the primary paths.

Figure 5 .

 5 Figure 5. Feedback regulation scheme for rejection of disturbances.

) using the bilinear transform s = 2 T s 1 -z - 1 1+z - 1 .

 2111 The use of BSFs introduces an attenuation M = 20 log ζ num ζ den at the normalized discretized frequency ω d = 2 • arctan ω 0 T S

Figure 6 .

 6 Figure 6. Output sensitivity function with the linear controller (upper figure) and zoom in the 50 Hz to 90 Hz frequency interval (lower figure).

Figure 7 .Figure 8 .

 78 Figure 7. Input sensitivity function with the linear controller.

  var(y CL ) var(y OL ) , where y OL denotes residual acceleration in open loop and y CL is the closed loop residual acceleration.

Figure 9 .Figure 10 .

 910 Figure 9. Residual force in closed loop with linear controller (upper plot) and with adaptive controller (lower plot). The experiments are started in open loop for 5 seconds. Range of frequency variation: ±2.5 Hz.

Figure 11 .

 11 Figure 11. Residual force in closed loop with linear controller (upper plot) and with adaptive controller (lower plot). The experiments are started in open loop for 5 seconds. Range of frequency variation: ±5 Hz.

Figure 12 .Figure 13 .

 1213 Figure 12. Residual force in closed loop with linear controller (upper plot) and with adaptive controller (lower plot). The loop is closed at t=15 sec.

Figure 14 .

 14 Figure 14. Vibrational interference. Residual force with step frequency changes (+5 Hz at 35 sec) in closed loop with linear controller (upper plot) and with adaptive controller (lower plot). The system is in open loop until t=15 sec.

Figure 15 .

 15 Figure 15. Control signals for step frequency changes (+5 Hz) experiment: linear controller (upper plot) and adaptive controller (lower plot).

Table I

 I Band-stop filters for output and input sensitivity functions.

		Frequency [Hz]	Amplification [dB]	Damping
			57.5			-17		0.1
			59.8			-25		0.5
	S yp		62 77.5			-15 -13		0.1 0.05
			79.8			-20		0.2
			82			-12		0.05
	S up		155			-16		0.5
	0	50	100	150	200	250	300	350	400

In the case of a single "tonal" disturbance (for a pure single frequency vibration), this corresponds to a zero damped second order tuned at this frequency. The gain of the output sensitivity function at this frequency will be 0 (-∞ dB).

More details can be found on the website http://www.gipsa-lab. grenoble-inp.fr/~ioandore.landau/benchmark_adaptive_regulation/

Both the orders and the parameters of the models have been estimated from data.

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

The disturbance passes through a so called primary path which is not represented in this figure, and p(t) is its output.

The argument (z -1 ) will be omitted in some of the following equations to make them more compact.

Throughout the paper, n X denotes the degree of the polynomial X.

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