Matěj Kuře

Jaroslav Bušek

Islam Boussaada
email: islam.boussaada@l2s.centralesupelec.fr

Wim Michiels
email: wim.michiels@cs.kuleuven.be

Silviu-Iulian Niculescu
email: silviu.niculescu@l2s.centralesupelec.fr

Tomáš Vyhlídal
email: tomas.vyhlidal@cvut.cz

Robust delayed resonator with acceleration feedback -design by double root assignment and experimental validation

Keywords: Double root assignment, active vibration control, robust control, acceleration feedback, spectral method, delayed resonator

A combination of analytical and numerical time-delay-system spectrum-shaping tools are applied to the design of the robust delayed resonator with an acceleration feedback. First, the delayed resonator model is turned into a dimension-less form with the objective to generalize the derived results. The main theoretical result is then provided as a complete parameterization of the proposed resonator feedback with two delay terms to assign a pair of roots with multiplicity two on the imaginary axis. In the frequency-domain, the double roots are projected to widening the stop-band in the active absorber frequency response, which increases its robustness in vibration suppression. On the other hand, they have a destabilizing effect on the overall system dynamics. The stabilization is subsequently performed by an additional controller via spectral optimization. The design is thoroughly validated by both simulations and experiments where the results are compared with the classical delayed resonator with lumped delay acceleration feedback.

Introduction

Passive as well as actively tuned vibration absorbers have become established tools in the vibration suppression task. The passive absorbers are easy to apply, but their applicability is limited to a narrow range close to the absorber natural frequency [START_REF] Preumont | Vibration control of active structures: an introduction[END_REF]. Besides, due to inherent damping and friction forces in their flexible links, the vibration suppression cannot be ideal. The applicable frequency range can be extended and vibration absorption quality can be enhanced by active tuning of the absorber properties. It can be done mechanically by adjusting the absorber mass, stiffness or damping, see e.g. [START_REF] Rana | Parametric study and simplified design of tuned mass dampers[END_REF][START_REF] Lin | Vibration control performance of tuned mass dampers with resettable variable stiffness[END_REF][START_REF] Gao | Design of the frequency tuning scheme for a semi-active vibration absorber[END_REF]. However, considerably better performance can be achieved if the absorber is actively actuated [START_REF] Preumont | Vibration control of active structures: an introduction[END_REF]. The delayed resonator (DR) concept proposed in 1990s by N. Olgac and his co-workers is a typical example of actively tuned vibration absorber. Next to becoming an established tool for vibration suppression, it has become one of the typical examples of benefits of time delays in vibration suppression applications. The time-delay feedback is applied to modify the active absorber properties resulting in the entire suppression of undesirable oscillations. The absorber is tuned to an ideal resonator, marginally stable, where the DR acts as a perfect resonator and absorbs the vibrations entirely.

The first concepts of DR used delayed feedback from absolute position of the absorber [START_REF] Olgac | A novel active vibration absorption technique: delayed resonator[END_REF], [START_REF] Olgac | Tunable active vibration absorber: the delayed resonator[END_REF]. Consequently a relative position of the absorber with respect to the primary structure was applied [START_REF] Olgac | Active vibration absorption using delayed resonator with relative position measurement[END_REF]. A damped centrifugal DR with angular position feedback with variable gain and time delay was considered in [START_REF] Hosek | A tunable torsional vibration absorber: the centrifugal delayed resonator[END_REF]. After these first results, including also extension to delayed velocity feedback [START_REF] Filipovic | Torsional delayed resonator with velocity feedback[END_REF], [START_REF] Filipović | Delayed resonator with speed feedback-design and performance analysis[END_REF], the research mainstream of the DR analysis and design switched to delayed feedback from acceleration sensor mounted on the absorber. This had a practical motivation as the accelerometers are easy to apply and of relatively low cost. To the best of the authors' knowledge, the DR with acceleration feedback was first introduced in [START_REF] Olgac | Active vibration control of distributed systems using delayed resonator with acceleration feedback[END_REF], where the DR based absorption methods were implemented on a distributed parameter structure under high frequency excitation. Research in DR with acceleration feedback then followed in many directions. An algorithm for multi-degree of freedom (MDOF) mechanical structures with multiple DRs is described in [START_REF] Jalili | Multiple delayed resonator vibration absorbers for multi-degreeof-freedom mechanical structures[END_REF]. In [START_REF] Renzulli | Robust control of the delayed resonator vibration absorber[END_REF], an automatic tuning method of the DR was proposed to handle parametric uncertainties. An auto-tuning algorithm to enhance the robustness against uncertainties was proposed in [START_REF] Hosek | A single-step automatic tuning algorithm for the delayed resonator vibration absorber[END_REF], see also [START_REF] Jalili | Identification and retuning of optimum delayed feedback vibration absorber[END_REF] for parameter identification and re-tuning the delayed feedback, and [START_REF] Jalili | A sensitivity study on optimum delayed feedback vibration absorber[END_REF] for sensitivity analysis and parameter optimization.

In the past decade, delayed resonator feedback was studied by a broader authors' team of this paper. A complete dynamics analysis of a DR with acceleration feedback was performed in [START_REF] Vyhlídal | Delayed resonator with acceleration feedbackcomplete stability analysis by spectral methods and vibration absorber design[END_REF] revealing an unfavorable neutral1 character of spectral properties of the DR. In order to mitigate this undesirable effect, an alternative distributed delay resonator was proposed and analyzed in [START_REF] Pilbauer | Delayed resonator with distributed delay in acceleration feedback -design and experimental verification[END_REF], resulting in retarded spectral properties. By applying the method of Cluster Treatment of Characteristic Roots [START_REF] Olgac | The cluster treatment of characteristic roots and the neutral type time-delayed systems[END_REF], it was also shown in [START_REF] Vyhlídal | Delayed resonator with acceleration feedbackcomplete stability analysis by spectral methods and vibration absorber design[END_REF] and [START_REF] Pilbauer | Delayed resonator with distributed delay in acceleration feedback -design and experimental verification[END_REF] that the operable frequency range is limited regardless the DR feedback type. From below, it is limited by the stability boundary, while the delay implementation aspects limit the range from the above -due to the exponential decay of the delay length with respect to growing frequency. A methodology for further extension of the operable frequency range of the DR with acceleration feedback was proposed in [START_REF] Kučera | Extended delayed resonators-design and experimental verification[END_REF], see also recent work [START_REF] Gao | Complete stability analysis and optimization of the extended delayed resonator with virtual natural frequency adjustment[END_REF]. It is based on extending the feedback by a delay free factor virtually modifying the mass of the absorber and thus its natural frequency. Besides, implementation aspects of the resulting, so-called extended delay resonators have been discussed.

The analysis performed in [START_REF] Vyhlídal | Delayed resonator with acceleration feedbackcomplete stability analysis by spectral methods and vibration absorber design[END_REF] and [START_REF] Pilbauer | Delayed resonator with distributed delay in acceleration feedback -design and experimental verification[END_REF] for DR with the acceleration feedback was extended to DR with position or velocity feedback in [START_REF] Vyhlídal | Analysis and design aspects of delayed resonator absorber with position, velocity or acceleration feedback[END_REF]. As pointed out, if not tuned properly, the DR can considerably lower the stability margin or even destabilize the system. Besides, the needed force and power of the DR for entire vibration of the primary were derived, which are independent of the DR type. As one of the main results, it was also demonstrated that the existing DR solutions suffer from extreme fragility in vibration suppression taking into account mismatch between the design and true excitation frequencies. Following from the frequency domain analysis, the entire vibration suppression only takes place if the resonant frequency adjusted by the resonator feedback is equal to the true excitation frequency. The performance of the resonator decays considerably even for very small differences in these frequencies.

Recent works of DR with acceleration feedback include also the work reported in [START_REF] Kammer | Electromechanical delayed resonator implementation using piezoelectric networks[END_REF], where the mechanical absorber structure is replaced by an electrical circuit that resonates. In [START_REF] Olgac | Actively tuned noncollocated vibration absorption: An unexplored venue in vibration science and a benchmark problem[END_REF], [START_REF] Jenkins | Real-time tuning of delayed resonator-based absorbers for spectral and spatial variations[END_REF], a challenging problem of non-collocated vibration absorption by DR was opened and analysed. The work [START_REF] Liu | A robust delayed resonator construction using amplifying mechanism[END_REF] supplements the DR with an amplifying mechanism. An interesting study on generalizing the DR towards fractional derivative feedback was recently presented in [START_REF] Cai | Spectrum-based stability analysis for fractionalorder delayed resonator with order scheduling[END_REF]. Let us also mention the approach of [START_REF] Eris | A new delayed resonator design approach for extended operable frequency range[END_REF] combining position and velocity feedback, the work reported in [START_REF] Valášek | Real-time tunable single-degree of freedom, multiple-frequency vibration absorber[END_REF] targeting multiple-frequencies by multi-parameter delayed position feedback and [START_REF] Vyhlídal | Analysis and optimized design of an actively controlled two-dimensional delayed resonator[END_REF], [START_REF] Šika | Two-dimensional delayed resonator for entire vibration absorption[END_REF] extending the DR concept to twodimensional vibration absorption using delayed position feedback at the three actuators of the 2D absorber. Motivated by DR applications, in [START_REF] Hernández-Villa | On stability of sdof systems with delayed position and velocity feedback[END_REF], stability analysis of systems with delayed position and velocity was performed. Regions in the feedback gain parameters were determined to assure delay-dependent exponential stability.

In [START_REF] Rivaz | An active dynamic vibration absorber for a hand-held vibroelastography probe[END_REF] a delay-free PI acceleration feedback of the absorber was proposed. However, due to risky noise integration phenomenon, the feedback needs to be supplemented with high-pass filters, which makes the overall feedback more complex compared to time delay feedback. The stability of the vibration absorber using acceleration and displacement feedback was analyzed in [START_REF] Alujević | Tuneable vibration absorber using acceleration and displacement feedback[END_REF] utilizing the Nyquist criterion. The work [START_REF] Mohanty | Linear and nonlinear analysis of piezoelectric based vibration absorber with acceleration feedback[END_REF] provides analysis of a primary structure and piezoelectric based dynamic vibration absorber. The active feedback is taken from the acceleration of the primary mass.

As mentioned above, and addressed thoroughly in [START_REF] Vyhlídal | Analysis and design aspects of delayed resonator absorber with position, velocity or acceleration feedback[END_REF], the main drawbacks of the classical DR applications are: i) extreme fragility in vibration suppression with respect to mismatch between the design and true excitation frequency, and ii) destabilizing effect of the DR deployment to the primary structure. Thus, the main objective of this paper is to propose a method for a straightforward design of a robust DR with acceleration feedback. The first work in this direction is [START_REF] Pilbauer | Optimized design of robust resonator with distributed time-delay[END_REF], where the DR feedback design is formulated as an optimization problem over the parameter set formed by the polynomial coefficients. The objective was to minimize the sensitivity of the resonator performance with respect to variations of the excitation frequency considering a system stability constraint. The arising non-convex and non-smooth optimization problem is widely discussed and a penalty method is developed, where the unconstrained problem is solved using available software tools for optimization and spectral design of time delay systems. In this paper, we utilize recent results on admissible multiplicity of fixed structure delay systems as in [START_REF] Boussaada | Characterizing the codimension of zero singularities for time-delay systems[END_REF][START_REF] Mazanti | Multiplicity-induced-dominancy for delaydifferential equations of retarded type[END_REF][START_REF] Boussaada | Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: A vandermonde-based approach[END_REF], which leads to much simpler design rules and feedback structure of the DR compared to [START_REF] Pilbauer | Optimized design of robust resonator with distributed time-delay[END_REF]. Analogously to [START_REF] Pilbauer | Optimized design of robust resonator with distributed time-delay[END_REF], the stabilization of the overall system is addressed, here by a supplementary finite order controller stabilizing the platform with a robust feedback through an additional actuator. For the controller tuning, the procedure synthesized in [START_REF] Michiels | Spectrum-based stability analysis and stabilisation of systems described by delay differential algebraic equations[END_REF] and implemented in the software package TDS-CONTROL [START_REF] Appeltans | Tds-control: a matlab package for the analysis and controller-design of time-delay systems[END_REF] is applied. As a preliminary step, the stability posture of the robust DR is analysed. To demonstrate the practical applicability of the method, an experimental validation of the overall control scheme is performed on a mechatronic system actuated by linear voicecoils and governed by an industrial controller.

The paper extends preliminary results presented in a conference publication [START_REF] Kuře | Spectral design of robust delayed resonator by double-root assignment[END_REF]. Compared to a single parameter set leading to a double root derived in [START_REF] Kuře | Spectral design of robust delayed resonator by double-root assignment[END_REF], an infinite number of parameter sets leading to a double root is presented. Additionally, an enhanced attention is paid to the stability aspects and experimental validation. The paper is organized as follows. In Section 2, the concept of vibration absorber and functioning of the overall set-up including a stabilizing controller is outlined. Section 3 then provides the main result with introducing the robust delayed resonator concept and providing analytic rules for double root assignment. Next, the design of the additional finite order controller is addressed together with its parametrization by minimizing the spectral abscissa. Both simulation and experimental validation of the proposed concept are done in Section 4, where the robust DR performance is compared with the performance of classical delayed resonator concept. In Section 5, the conclusions are given.

Problem statement and preliminaries

Consider the mechanical configuration of the platform P and active absorber A as shown in Fig. 1. The position of the primary structure and the position of the absorber are denoted by xp and xa , respectively. The physical parameters of the setup are the masses of the platform and absorber bodies mp , ma , the damping cp , ca , and the stiffness kp , ka of the links. The platform is excited by a harmonic force

f (ϑ) = fa cos(ω f (1 + δ)ϑ), ( 1
)
where ϑ is time, fa is the amplitude, ωf is the nominal excitation frequency and δ denotes its deviation in relative sense (satisfying |δ| < 1). Thus, the perturbed excitation frequency is given as ω = ωf (1 + δ). The control objective is to introduce an active actuation force ūa in order to fully suppress the vibration of the primary structure (x p ) in a robust way, i.e. with low sensitivity to the frequency deviation δ. In order to stabilize and enhance the robustness of the overall system, additional control force ūp is available.

Dimensionless model

For the generalization purposes, the model of the setup is derived in the dimensionless form. Considering xp = 0, the isolated absorber can be described by

ẍa (ϑ) + 2ζΩ ẋa (ϑ) + Ω 2 xa (ϑ) = 1 ma ūa (ϑ), (2)
where Ω = is the damping of the passive absorber. Scaling the time ϑ with respect to the frequency Ω, i.e., by introducing dimensionless time t = Ωϑ, and setting

x a (t) = xa (ϑ), u a (t) = 1 ma Ω 2 ūa (ϑ), with ϑ = t Ω ,
we obtain an isolated active absorber form

ẍa (t) + 2ζ ẋa (t) + x a (t) = u a (t), (3)
with the only parameter ζ.

Applying the same scaling of variables and time to the overall set-up according to Fig. 1 andsetting x

p (t) = xp (ϑ), we obtain { ẍa (t) + 2ζ ẋa (t) + x a (t) -2ζ ẋp (t) -x p (t) = u a (t), m p ẍp (t) + (2ζ + c p ) ẋp (t) + (1 + k p ) x p (t) -2ζ ẋa (t) -x a (t) = -u a (t) + u p (t) + f (t) , ( 4
) where m p = mp ma , c p = cp maΩ , k p = kp maΩ 2 are the scaled mass, the damping and the stiffness parameters of the primary structure, u p is defined as u p (t) = 1 maΩ 2 ūp (ϑ), with ϑ = tΩ, while the scaled excitation force is given by

f (t) = 1 ma Ω 2 f (ϑ).
(5)

Unifying excitation frequency with absorber natural frequency

Without loss of generality, we assume that the excitation frequency ωf in ( 1) is unified with the natural frequency of the absorber Ω, i.e. ωf = Ω. In this way, the force term in (4) becomes f (t) = fa maΩ 2 cos((1+δ)t). The unification of these frequencies can be achieved by modifying the physical parameters of the absorber. In particular, a modification of the mass is an easy task to do leading to

ma (ω f ) = ka ω2 f . ( 6
)
Alternatively, as shown in [START_REF] Kučera | Extended delayed resonators-design and experimental verification[END_REF], extending the resonator feedback by a delay-free acceleration term ūa (ϑ) = -h ẍa (ϑ) + va (ϑ),

with a parameter h and a new input va , allows virtual modification of the absorber mass from ma to ma + h. The unification of the excitation and natural frequencies implies

h(ω f ) = ka ω2 f -ma . ( 8
)
For practical aspects and limitations of this approach we refer the reader to [START_REF] Kučera | Extended delayed resonators-design and experimental verification[END_REF].

Active feedback to achieve ideal uni-frequency absorber

The structural analysis with the aim to achieve complete vibration suppression and stability posture is performed in the Laplace transform forms of the model and the controllers. A generalized absorber active feedback assumed in the form

U a (s) = P (s) X a (s) , (9)
with transfer function P (s) is applied to turn the physical absorber to an ideal absorber which entirely absorbs the vibration at excitation frequency ω f = 1. For stabilizing purposes, the primary structure is equipped with an active feedback

U p (s) = K (s) X p (s) , (10)
with controller K (s), which can be used for platform positioning or just for improving the overall dynamical properties in vibration suppression. The model ( 4)

is transformed into { R(s)X a (s) -H(s)X p (s) = U a (s), Z(s)X p (s) -H(s)X a (s) = -U a (s) + U p (s) + F (s), (11)
with

R (s) = s 2 + 2ζs + 1, Z (s) = m p s 2 + (2ζ + c p ) s + (1 + k p ), and
H (s) = 2ζs + 1.
Substituting ( 9) and ( 10) into [START_REF] Olgac | Active vibration control of distributed systems using delayed resonator with acceleration feedback[END_REF], the transfer function between the excitation force f and the position of the primary structure x p is then given by

G f,xp (s) = X p (s) F (s) = R(s) -P (s) (R(s) -P (s))(Z(s) -K(s)) + (P (s) -H(s))H(s) . ( 12
)
If the transfer function P (s) is parameterized so that the characteristic equation of the resonator composed of the absorber (3) and the feedback [START_REF] Filipovic | Torsional delayed resonator with velocity feedback[END_REF], given by

M (s) = R (s) -P (s) = 0, (13)
has a root couple s 1,2 = ±jω f , composing a pole couple of the resonator, then

G f,xp (jω f ) = 0, (14)
indicating that no vibration at the given frequency ω = ω f is transferred in this f to x p channel and the vibration is ideally suppressed. Note however that including the resonator feedback affects the dynamical properties of the entire system, which are determined by the roots of the characteristic equation

(R (s) -P (s)) (Z (s) -K (s)) + (P (s) -H (s)) H (s) = 0. (15)
Assuming the presence of the delay terms in P (s), equation ( 15) has in general infinitely many roots. For the stability implications, all of them need to be located safely in the left half of the complex plane.

Main result -Robust resonator feedback design

The resonator feedback is considered in the form

u a (t) = t ∫ 0 α 0 ẍa (θ) + α 1 ẍa (θ -τ 1 ) + α 2 ẍa (θ -τ 2 ) dθ, (16)
with delays τ 2 > τ 1 > 0 and parameters α 0 , α 1 , α 2 satisfying

2 ∑ i=0 α i = 0, (17)
Figure 2: Impulse response of the distributed delay used in the resonator feedback [START_REF] Jalili | A sensitivity study on optimum delayed feedback vibration absorber[END_REF], considering the measured acceleration ẍa as the input while satisfying [START_REF] Vyhlídal | Delayed resonator with acceleration feedbackcomplete stability analysis by spectral methods and vibration absorber design[END_REF].

which imposes a finite length of the impulse response from ẍa to u a . The role of the gains and delays in ( 16) is shown in Fig. 2, where an impulse response of the distributed delay is shown, taking ẍa as the input. As can be seen, τ 1 determines the length of the first pulse, while the overall length is determined by τ 2 . The gains α i , i = 0..2 then determine the magnitude distribution of the pulses. Taking into consideration the feedback transfer function form

P (s) = α 0 + α 1 e -sτ 1 + α 2 e -sτ 2 s s 2 , ( 18
)
the characteristic equation ( 13) of the delayed resonator composed of the absorber (3) and active feedback ( 16) is then given

M (s) = s 2 + ( 2ζ -α 0 -α 1 e -sτ 1 -α 2 e -sτ 2 ) s + 1. (19)
Notice that despite the involved acceleration feedback, due to the distributed delay nature, the characteristic equation structure indicates the presence of a point spectrum (retarded case), [START_REF] Pilbauer | Delayed resonator with distributed delay in acceleration feedback -design and experimental verification[END_REF].

DR parameterization by assigning a double root

The objective in the design of the robust resonator is to widen the frequency stopband of |G f,xp (jω)| in the vicinity of ω = 1. This can be done by assigning a double root s 1,2 = j to the characteristic equation of the delayed resonator [START_REF] Michiels | Stability, Control, and Computation for Time-Delay Systems: An Eigenvalue-Based Approach[END_REF], which appears in the numerator of [START_REF] Jalili | Multiple delayed resonator vibration absorbers for multi-degreeof-freedom mechanical structures[END_REF]. Theorem 1. For a given damping factor ζ ∈ [0, 1], under the design condition [START_REF] Vyhlídal | Delayed resonator with acceleration feedbackcomplete stability analysis by spectral methods and vibration absorber design[END_REF], the multiplicity of s 1,2 = j as a root of (19) cannot exceed 2. The required multiplicity 2 is reached if and only if, for arbitrary positive integers k 1 and k 2 satisfying k 1 ̸ = k 2 and at least one of them being odd, the parameters of the resonator's feedback are given by

                           τ 1 = k 1 π, τ 2 = k 2 π, α 0 = - 2 π (1 + πk 2 ζ)(-1) k 1 +k 2 -πk 1 ζ -1 k 1 + (k 2 -k 1 -k 2 (-1) k 1 )(-1) k 2 , α 1 = 2 π (-1) k 1 ((-1) k 2 + π(-1) k 2 k 2 ζ -1) k 1 + (k 2 -k 1 -k 2 (-1) k 1 )(-1) k 2 , α 2 = - 2 π πk 1 ζ -(-1) k 1 + 1 k 1 + (k 2 -k 1 -k 2 (-1) k 1 )(-1) k 2 . ( 20
)
Proof. Substituting s = j in the quasipolynomial function [START_REF] Michiels | Stability, Control, and Computation for Time-Delay Systems: An Eigenvalue-Based Approach[END_REF] as well as its first derivative and separating the real and imaginary part, we get the following system of four trigonometric polynomials

         -α 1 sin(τ 1 ) -α 2 sin(τ 2 ) = 0, (2 ζ -α 0 ) -α 1 cos(τ 1 ) -α 2 cos(τ 2 ) = 0, α 1 τ 1 sin(τ 1 ) + α 2 τ 2 sin(τ 2 ) + (2ζ -α 0 ) -α 1 cos(τ 1 ) -α 2 cos(τ 2 ) = 0, 2 + α 1 τ 1 cos(τ 1 ) + α 2 τ 2 cos(τ 2 ) + α 1 sin(τ 1 ) + α 2 sin(τ 2 ) = 0. (21)
From the first two equations we eliminate the trigonometric functions in

τ 1        sin (τ 1 ) = - α 2 sin (τ 2 ) α 1 , cos (τ 1 ) = - -2ζ + α 0 + α 2 cos (τ 2 ) α 1 , ( 22
)
which after being substituted in the remaining two equations of [START_REF] Olgac | The cluster treatment of characteristic roots and the neutral type time-delayed systems[END_REF] gives

   sin (τ 2 ) = 0, cos (τ 2 ) = τ 1 (2ζ -α 0 ) + 2 α 2 (τ 1 -τ 2 ) . ( 23
)
Substituting again [START_REF] Gao | Complete stability analysis and optimization of the extended delayed resonator with virtual natural frequency adjustment[END_REF] in [START_REF] Kučera | Extended delayed resonators-design and experimental verification[END_REF] allows us to prove that

τ 1 = k 1 π and τ 2 = k 2 π, ( 24
)
where k 1 and k 2 are positive integers.

Substituting the obtained values of τ 1 and τ 2 and s = j in the expression of M (s) as well as in the expression of M ′ (s) one obtains:

   -j ( -2 ζ + α 0 + α 1 (-1) -k 1 + (-1) k 2 α 2 ) = 0, 2 ζ -α 0 -α 1 (-1) -k 1 -(-1) k 2 α 2 + j ( 2 + α 1 k 1 π (-1) -k 1 + (-1) k 2 α 2 k 2 π ) = 0, (25)
which gives

α 1 = - 2 + (2ζ -α 0 ) k 2 π (-1) -k 1 π (k 1 -k 2 ) , α 2 = 2 + (2ζ -α 0 ) k 1 π (-1) k 2 π (k 1 -k 2 ) . ( 26
)
Solving the set of equations ( 17) and ( 26) gives [START_REF] Pilbauer | Delayed resonator with distributed delay in acceleration feedback -design and experimental verification[END_REF]. To prove that the maximal multiplicity of s = j as a root of ( 19) is two, one can substitute the obtained parameter values in equation M ′′ (j) = 0, leading to an inconsistency. □ Remark 2. As stated in the theorem, the only possible control laws of the form (16) feature delays τ 1 = k 1 π and τ 2 = k 2 π with either (k 1 , k 2 ) both odd, or one of them even and the other one odd. One may wonder about a physical interpretation for the exclusion of possibilities with k 1 and k 2 both even. If both were even, the controller's transfer function would satisfy in view of [START_REF] Vyhlídal | Delayed resonator with acceleration feedbackcomplete stability analysis by spectral methods and vibration absorber design[END_REF], i.e., a harmonic function of frequency equal to one would be filtered out by the control. Hence, the damping of the absorber, having (modified) natural frequency one, cannot be compensated. Note that control law ( 16) can be interpreted as a generalization of Pyragas type feedback (also called time-delayed feedback). This type of feedback has originally been developed for nonlinear systems, where the filtering property induced by time-shifted differences is employed to preserve the shape of a periodic orbit with given period or an equilibrium's position to be stabilized [START_REF] Pyragas | Continuous control of chaos by self-controlling feedback[END_REF]. In [START_REF] Huijberts | Stabilisability via time-delayed feedback: an eigenvalue optimisation approach[END_REF]Figure 2.3] the impact of the filtering property on the stabilizabilty of linear second-order systems is highlighted. □ Remark 3. Notice that the multiplicity of a given root of a quasipolynomial is bounded by the generic Polya and Szegö bound denoted P S B , see for instance [START_REF] Pólya | Problems and Theorems in Analysis II: Theory of Functions[END_REF][START_REF] Boussaada | Characterizing the codimension of zero singularities for time-delay systems[END_REF]. Such a bound is defined in [START_REF] Boussaada | Characterizing the codimension of zero singularities for time-delay systems[END_REF] as the degree of the quasipolynomial and corresponds to the sum of the involved polynomials' degrees plus the number of delays. In [START_REF] Boussaada | Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: A vandermonde-based approach[END_REF], it is shown that the multiplicity of purely imaginary roots (with a non-vanishing frequency) of a generic quasipolynomials never reaches P S B and a sharper bound for its admissible multiplicities is established. Accordingly, the degree of the quasipolynomial M (s) defined by (19) is 6. In particular, a characteristic root located on the real (or imaginary) axis may have the maximal multiplicity equal to 6 (or 3). However, in our case, as stated in the Theorem above, the particular structure of the characteristic function does not allow an imaginary root with a multiplicity larger than 2. For a deeper discussion on generic maximal multiplicity of a real characteristic root and its properties, the reader is referred to [START_REF] Mazanti | Multiplicity-induced-dominancy for delaydifferential equations of retarded type[END_REF]. □

P (j) = j (α 0 + α 1 + α 2 ) = 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 -0.3 -0.2 -0.1 0 0.1 0.2 . k 1 = 1; k 2 = 2 k 1 = 1; k 2 = 3 k 1 = 2; k 2 = 3 k 1 = 3; k 2 = 4 k 1 = 1; k 2 = 4
In Fig. 3, the real part of the rightmost root of ( 19), obtained after elimination of the assigned double root and denoted by γ, is shown with respect to ζ and selected k 1 , k 2 . As can be seen, the double-root DR (drDR) may become unstable (γ > 0) for some of the settings. This does not happen e.g. for the smallest possible delays

k 1 = 1, k 2 = 2.
Remark 4. For the physical implementation, the dimensionless form of the whole resonator feedback needs to be turned to the dimensional form. Considering [START_REF] Olgac | Active vibration absorption using delayed resonator with relative position measurement[END_REF] and [START_REF] Jalili | A sensitivity study on optimum delayed feedback vibration absorber[END_REF], it reads as

ūa (ϑ) = -h ẍa (ϑ) + ∫ ϑ 0 ᾱ0 ẍa (η) + ᾱ1 ẍa (η -τ1 ) + ᾱ2 ẍa (η -τ2 ) dη (27)
Resonator Primary Structure where

(2ζs + 1 -P (s))(R(s) -P (s)) -1 (2ζs + 1)Z(s) -1 W a W p Z a Z p
τi = 1 Ω τ i , i = 1, 2, ( 28
)
and ᾱj = ( ma + h)Ωα j , j = 0..2. ( 29
)
□

Instability of the DR due to a double root assignment

We give some intuition why the inclusion of a (robust) delayed resonator might negatively affect the stability of the overall set-up, which motivates us to include the additional active control [START_REF] Filipović | Delayed resonator with speed feedback-design and performance analysis[END_REF] of the primary structure.

With F ≡ 0 and U p ≡ 0, the combined system can be interpreted as a feedback interconnection of system { (R(s) -P (s)) X a (s) = W a (s) Z a (s) = (2ζs + 1 -P (s))X a (s) [START_REF] Eris | A new delayed resonator design approach for extended operable frequency range[END_REF] and system { Z(s) X p (s) = W p (s) Z p (s) = (2ζs + 1)X p (s),

where W a (s) and W p (s) represent virtual input forces, while Z a (s) and Z p (s) represent virtual measurements. The coupling between both (sub)systems, which are well posed, is described by W a (s) = Z p (s), W p (s) = Z a (s), see Fig. 4. It should be noted that this decomposition is quite natural. For W a (s) ≡ 0 the state equation of system [START_REF] Eris | A new delayed resonator design approach for extended operable frequency range[END_REF] corresponds to the dynamics of the resonator when the primary structure is fixed (X p (s) ≡ 0). Its characteristic equation is given by

det (R(s) -P (s)) = 0, (32)
showing the influence of the active control. Similarly, assuming zero input the state equation of system (31) describes the dynamics of the primary structure when the absorber's position is fixed. Hence, the characteristic equation det (Z(s)) = 0 is in accordance with the presence of two spring-damper combinations. For P (s) ≡ 0 both subsystems depicted in Fig. 4 are exponentially stable provided ζ > 0 and c p > 0. When the active resonator feedback is determined to assign simple zeros ±j to the transfer function from f to x p , these zeros appear as solutions of [START_REF] Vyhlídal | Analysis and optimized design of an actively controlled two-dimensional delayed resonator[END_REF]. Hence, the upper subsystem is not anymore exponentially stable, but still stable in the sense of Lyapunov. If the overall system is exponentially stable, the lower subsystem can be seen as taking the role of a stabilizing controller for the upper subsystem. Thus, the degree of instability induced by the active feedback of the delayed resonator is compensated by the stability of the primary structure and/or the coupling.

When double zeros are assigned by the resonator feedback, these appear as double non-semisimple eigenvalues of the upper subsystem in Fig. 4, which becomes polynomially unstable (modes of the form c 1 t cos(t) + c 2 t sin(t)). In such a situation, the stabilization task and the subsequent task of having the closed-loop spectrum sufficiently bounded away from the imaginary axis are much more difficult. This argument, as well as numerical experiments demonstrating a potential loss of stability, lead us to supplement the scheme with active feedback of the primary structure.

Stabilizing the overall set-up

The absorber and primary structure coupled together, as given by (4), can be described as

       ẋ(t) = Ax(t) + B 1 u a (t) + B 2 u p (t) + B 3 f (t), y a (t) = C a ẋ(t) = C a Ax(t) + C a B 1 u a (t) + C a B 2 u p (t) + C a B 3 f (t), y p (t) = C p x(t), ( 33
)
where the output y a is the measured acceleration y a = ẍa , and matrices I, A, B 1 , B 2 , B 3 correspond to a linear model of a mechanical system (4) with a state vector x = [x a ẋa x p ẋp ] T . Matrix C a = [0 1 0 0] defines the measured acceleration of the absorber, and C p = [0 0 1 0] determines the control system output y p = x p . Concerning the resonator feedback ( 16), it can be turned to

ua (t) = α 0 y a (t) + α 1 y a (t -τ 1 ) + α 2 y a (t -τ 2 ). (34)
System [START_REF] Šika | Two-dimensional delayed resonator for entire vibration absorption[END_REF] and resonator feedback [START_REF] Hernández-Villa | On stability of sdof systems with delayed position and velocity feedback[END_REF] can be recast into a Delay Differential Algebraic Equations (DDAE) description of the form

E ξ(t) = A 0 ξ(t) + 2 ∑ i=1 A i ξ(t -τ i ) + B 0 f (t) + B 1 u p (t), Y(t) = Cξ(t), (35)
where ξ = [x T u a y a y p ] T is the (pseudo)state vector, E = diag(I, 1, 0, 0), and Y(t) = y p (t).

In order to stabilize and optimize the spectrum of the resulting infinite dimensional system, following the methodology proposed in [START_REF] Michiels | Spectrum-based stability analysis and stabilisation of systems described by delay differential algebraic equations[END_REF], we consider a fixed k th -order dynamic feedback controller in the form

K : { ẋK (t) = A K x K (t) + B K y p (t), u p (t) = C K x K (t) + D K y p (t). ( 36
)
To assure the system is stable with sufficiently large margin and to avoid undesirably long transients, we design the controller based on minimizing the spectral abscissa over the free parameters, i.e. the elements of matrices (A K , B K , C K , D K ). This objective function is defined as

γ(K) := sup (ℜ(s) : s ∈ Σ(K)) , ( 37
)
where Σ(K) denotes the spectrum of the closed loop system ( 35)- [START_REF] Alujević | Tuneable vibration absorber using acceleration and displacement feedback[END_REF]. For its optimization we use the algorithm and software described in [START_REF] Michiels | Spectrum-based stability analysis and stabilisation of systems described by delay differential algebraic equations[END_REF].

Let us remark that even in the ideal case where both the DR and main structure are stable, there is no guarantee that their interconnection is stable. Thus application of a stabilizing controller proposed above may be needed also when a classical (quasi-stable) DR is deployed.

Remark 5. For the physical implementation, the dimensional form of (36) turns into

K : { ẋK (ϑ) = A K x K (ϑ) + B K y p (ϑ), u p (ϑ) = C K x K (ϑ) + D K y p (ϑ), ( 38
)
where

A K = A K Ω, B K = B K Ω, C K = C K m a Ω 2
, and D K = D K m a Ω 2 are the corresponding matrices of the controller. □ Remark 6. For the comparison, we present the alternative robust resonator feedback form proposed in [START_REF] Pilbauer | Optimized design of robust resonator with distributed time-delay[END_REF] and outline its optimization based design procedure. The resonator feedback is considered in the form

ūa (ϑ) = -h ẍa (ϑ) + ∫ τ 0 g(η) ẍa (ϑ -η) dη, ( 39
)
where the delay distribution function is in the polynomial form

g(η) = N ∑ i=0 a i η i e -λη , ( 40
)
with parameters a i and delay length τ . Note that the function e -λη with λ > 0 is included due to implementation of the delay. The required widening of the frequency stop band is achieved in [START_REF] Pilbauer | Optimized design of robust resonator with distributed time-delay[END_REF] by minimizing the squared transmissibility function in the vicinity of the target frequency. Prefixing the delay length τ , it leads to solving the Quadratic optimisation problem over N + 1 parameter set p = [a 0 , a 1 , ..a N ] T . However, due to destabilizing effect of the robustified DR, the optimization problem was extended by including the stability constraint γ(p) < 0, where γ(p) denotes the spectral abscissa of the whole system. Applying the penalty method, the constrained quadratic optimization problem is turned to an unconstrained nonlinear optimization problem. In the case study section of [START_REF] Pilbauer | Optimized design of robust resonator with distributed time-delay[END_REF], the problem was successfully solved by HANSO (Hybrid Algorithm for Nonsmooth Optimization) [START_REF] Overton | Hanso: a hybrid algorithm for nonsmooth optimization[END_REF]. Note that HANSO is also involved in the software tool [START_REF] Michiels | Spectrum-based stability analysis and stabilisation of systems described by delay differential algebraic equations[END_REF] used to design the controller [START_REF] Alujević | Tuneable vibration absorber using acceleration and displacement feedback[END_REF]. □

Case study validation

The validation is performed both by simulations and experiments. For the purpose of the latter, an experimental set-up has been built, which is shown in Fig. 5 together with implementation of the control scheme. The set-up built according to the scheme in Fig. 1 consists of two carts. The absorber cart ( ma ) is interconnected with the primary structure cart ( mp ) by a pair of springs. Another pair of springs is used to fix the primary structure cart to the left and right base elements. Both the carts slide on rails -the absorber-rails are fixed to the primary structure while the primary-rails are fixed to the base. The carts are actuated by two voice-coil linear motors generating forces ūa and ūp . The damping in the connecting links is mainly due to viscous friction between the bearings and the rails. The position xp of the primary structure is measured by an incremental position sensor while acceleration of the absorber ẍa is measured by an accelerometer. The proposed control scheme consists of the resonator feedback ( 16) and the stabilizing controller [START_REF] Alujević | Tuneable vibration absorber using acceleration and displacement feedback[END_REF].

Concerning the hardware implementation visualized in Fig. 5, the excitation of the primary structure is performed via the voice coil linear motor (AVM40-20-0.5, with continuous force 9.93 N) connecting the primary with the base. The absorber is actuated by a second voice coil (LVCM-032-076-02, with continuous force 8 N), which is placed between the primary structure and the absorber. The control algorithms [START_REF] Jenkins | Real-time tuning of delayed resonator-based absorbers for spectral and spatial variations[END_REF] and [START_REF] Pilbauer | Optimized design of robust resonator with distributed time-delay[END_REF] are implemented in LabVIEW ™ and performed using the CompactRIO controller with 1 kHz sampling. Note that Euler explicit method was used for solving both ( 27) and [START_REF] Pilbauer | Optimized design of robust resonator with distributed time-delay[END_REF]. FPGA module of CompactRIO is responsible for fast sensor measurement, particularly from the incremental position sensor requiring fast quadrature encoder signal. Additionally, the CompactRIO controller is equipped with the three additional modules. The first, NI-9870 module communicates by RS-232 bus with instrument control unit (ICU) which translates commands into electric current forcing both voice coils to move (with current/torque control loop). The second, NI-9401 module, detects pulses of the quadrature signal generated by the incremental position sensor AS5304 during its motion. The AS5304 is a sensor with integrated Hall elements for measuring linear motion using a multi-pole magnetic strip placed on the base alongside the main rail. The resolution of the position measurement is 25 µm. The last one is the NI-9230 module, which is designed to measure signals from both the integrated electronic piezoelectric (IEPE) type sensors and non IEPE type sensors. A miniature piezoelectric accelerometer KS95B.100 with high resonant frequency is connected to the module and measures acceleration of the absorber.

The parameters of the system according to Fig. 1 identified on the set-up are given as follows: ma = 0.52 kg, ca = 1.45 N s m -1 , ka = 418.50 N m -1 , mp = 1.157 kg, cp = 3.92 N s m -1 , kp = 1503.30 N m -1 . The absorber damping and natural frequency are then given as Ω = 28.37 s -1 and ζ = 0.0491. The excitation frequency of the force f acting on the primary structure is selected to be identical with Ω, i.e. ωf = 28.37 s -1 . Thus, the parameters in the dimensionless model [START_REF] Gao | Design of the frequency tuning scheme for a semi-active vibration absorber[END_REF] with ω f = 1 are given as ζ = 0.0491, m p = 2.225, c p = 0.2657, k p = 3.592. Let us mention that when the excitation frequency need be shifted, the absorber mass can be easily adjusted according to (6) by selecting appropriate number and weight of the steel plates seen in the top of the absorber in Fig. 5. Alternatively, the virtual mass adjustment can be performed by including delay free acceleration feedback [START_REF] Olgac | Active vibration absorption using delayed resonator with relative position measurement[END_REF] with h determined by [START_REF] Hosek | A tunable torsional vibration absorber: the centrifugal delayed resonator[END_REF], as proposed in [START_REF] Kučera | Extended delayed resonators-design and experimental verification[END_REF].

For parametrization of the robust resonator with the delay distribution given by ( 18), Theorem 1 is applied. Selecting k 1 = 1, k 2 = 2, implies by (20) τ 1 = π, τ 2 = 2π, α 0 = 0.3920, α 1 = -0.0491, α 2 = -0.3429 (τ 1 = 0.1107 s, τ2 = 0.2215 s, ᾱ0 = 5.7832 kgs -1 , ᾱ1 = -0.7250 kgs -1 , ᾱ2 = -5.0582 kgs -1 ). Applying the QPmR algorithm [START_REF] Vyhlidal | Mapping based algorithm for large-scale computation of quasipolynomial zeros[END_REF], the rightmost spectrum of the retarded characteristic equation ( 19) is shown in Fig. 6. Next to the spectrum values denoted as black dots, the iso-lines ℜ(M (s)) = 0 (solid) and ℑ(M (s)) = 0 (dashed) are shown. The root multiplicity two at s = j is confirmed by two couples of ℜ(M (s)) = 0 and ℑ(M (s)) = 0 iso-lines intersecting at this point, see [START_REF] Vyhlidal | Mapping based algorithm for large-scale computation of quasipolynomial zeros[END_REF]. As can be seen from the figure, the given roots s 1,2,3,4 are the rightmost roots.

In Fig. 7, the spectrum of the double root delayed resonator (drDR) (3) and ( 16), as well as the spectrum of its coupling with the primary structure (4) and ( 16) are shown. As can be seen, the spectrum of the interconnected system is unstable. Thus, the controller (36) of the order k = 2 is tuned by minimising the system spectral abscissa [START_REF] Michiels | Spectrum-based stability analysis and stabilisation of systems described by delay differential algebraic equations[END_REF], leading to

A K = [ 0.1956 1.0532 -1.1881 -1.0502 ] , B K = [ 0.9371 0.4681 ] , C K = [ 1.2523 -0.0416 ] , D K = [ -0.1992 ] , (41)
which in the dimensional form reads as

A K = [ 5
respectively.

As shown in Fig. 7, the drDR with the controlled primary structure is stable with spectral abscissa γ = -0.1046 (γ = -2.9675 s -1 ). 16), alone, drDR -primary structure interconnection, (4) and ( 16), without the controller (unstable), and with the controller [START_REF] Alujević | Tuneable vibration absorber using acceleration and displacement feedback[END_REF] given by ( 12) with: i) drDR and the stabilizing controller, and ii) cDR (results shown for scaled parameters).

In the next step, the performance validation is done via simulations. Next to the nominal excitation frequency ω f = 1 case, the setting with 3% (δ = ±0.03) and 5% (δ = ±0.05) mismatch between the design and true excitation frequencies are considered to study the robustness. Considering ω = ωf (1 + δ) in the excitation force [START_REF] Preumont | Vibration control of active structures: an introduction[END_REF], the absorption robustness is studied for ω = 26.95 s -1 (δ = -0.05), ω = 27.52 s -1 (δ = -0.03), ω = 28.37 s -1 (δ = 0), ω = 29.22 s -1 (δ = 0.03) and ω = 29.79 s -1 (δ = 0.05). For comparison, results of a classical delayed resonator (cDR) with lumped delay feedback, in dimensionless form given by u a (t) = g cDR ẍa (t -τ cDR ), [START_REF] Appeltans | Tds-control: a matlab package for the analysis and controller-design of time-delay systems[END_REF] and in dimensional form by

u a (ϑ) = g cDR ẍa (ϑ -τ cDR ), (44)
respectively, are shown, with gain g cDR = 2ζ = 0.0982 and time delay τ cDR = π 2 (g cDR = ma g cDR = 0.0511 kg, τ cDR = 0.0554 s), parametrized by assigning a single root pair s 1,2 = ±j (s 1,2 = ±j28.37) to the delayed resonator [START_REF] Olgac | A novel active vibration absorption technique: delayed resonator[END_REF]. From the results shown in Fig. 8, it can be seen that for nominal frequency ω = 1 (ω = 28.37 s -1 ) both DRs perform well. However, for the cases with the 3% and 5% frequency mismatch, the drDR performs considerably better with substantially smaller residual vibration amplitudes.

Subsequently, the enhancement of the robustness is studied via the magnitudes of the frequency responses |G f,xp (jω)| of the transfer function [START_REF] Jalili | Multiple delayed resonator vibration absorbers for multi-degreeof-freedom mechanical structures[END_REF] between the excitation force f and the position of the primary x p . If the cDR ( 43) is applied, then |G f,xp (jω)| goes to zero for the nominal frequency ω = 1 as shown in Fig. 9. For the frequencies in the vicinity of ω = 1, this function tends to rapidly increase and therefore, if there is an uncertainty in the excitation frequency, the quality of the vibration suppression is likely to decay. On the other hand, the |G f,xp (jω)| magnitude with drDR and stabilizing feedback has parabolic shape in the vicinity of ω = 1 with considerably smaller increase in its neighborhood, compared to the cDR. Another positive aspect of the given solution is considerably lower H ∞ norm (maximum of the frequency response magnitude) when the stabilizing controller is applied, compared to cDR case without the controller. Though, a positive impact of the controller is observed when it is used together with the cDR as it lowers the H ∞ norm substantially also for this case. For the demonstration purposes, in Fig. 9, we also show the amplitude of the drDR without the stabilizing feedback. As can be seen, the H ∞ norm increases substantially, although the characteristics close to the target frequency remained unchanged. This confirms the fact that the robustness in vibration suppression is truly imposed by the drDR.

Finally, the experimental validation is performed analogously to the simulation result set shown in Fig. 8. The experimental results are shown in Fig. 10 2 . As can be seen, considerably better robustness in vibration suppression is confirmed for the drDR. For the 3% and 5% mismatch between true and design excitation frequencies, considerable smaller amplitudes can be observed for the results by drDR compared to the results by cDR. Note that compared to the simulation results, certain residual vibration can be observed for the nominal frequency (δ = 0). The most likely reason is in the considerable measurement noise of the accelerometer. Besides, the unmodelled nonlinear dynamics (mainly due to the Coulomb friction) could negatively contribute to the physical drDR(cDR)'s performance decay. However, despite to this imperfection, it can be concluded that the main objective of the experimental validation of robustness enhancement due to drDR application has been fulfilled.

To conclude, let us discuss briefly the results against those achieved in simulation case study of [START_REF] Pilbauer | Optimized design of robust resonator with distributed time-delay[END_REF]. Let us note that in [START_REF] Pilbauer | Optimized design of robust resonator with distributed time-delay[END_REF] only a single actuator is considered governed by [START_REF] Boussaada | Characterizing the codimension of zero singularities for time-delay systems[END_REF], i.e. with ūp = 0. Applying the complex optimization based design, analogous results in vibration suppression robustness were achieved in [START_REF] Pilbauer | Optimized design of robust resonator with distributed time-delay[END_REF] as in the above case study with simple structure drDR feedback, though with slight residual vibration for the nominal frequency. However, including the stability condition did not have such a positive effect on overall H ∞ norm, which is observed in the current set-up with a separate stabilizing controller.

Conclusions

A robust version of the delayed resonator with acceleration feedback for vibration suppression is proposed as the main result. The resonator feedback is in a form of distributed delay which is composed of two segments. The length of the segments and the gains are determined utilizing recent analytical results on admissible multiplicity of fixed structure delay systems. An enhanced attention is paid to stability aspects of both the isolated resonator and the overall system, i.e. the resonator deployed at the primary structure to be silenced. It is shown that the double root placed at the imaginary axis destabilizes the resonator, independently of the location of its other roots in the infinite spectrum. This is likely to have negative consequences for the stability of the overall system. To handle this risky feature, the control scheme is supplemented by a stabilizing finite order controller acting at the primary structure. The controller parameters are determined by an optimization method with the objective to minimize the spectral abscissa of the overall control scheme formulated as an interconnected time delay system.

The proposed robust vibration absorption method is thoroughly validated on an extensive case study. First, the numerical and simulation based analysis is performed on a model of the physical laboratory set-up which is proposed and built for that purpose. The enhanced robustness in vibration absorption is then confirmed by both simulations and experiments. The case study analysis also revealed positive effect of the stabilizing feedback on the robustness of the overall control scheme. For the experimental validation the control algorithms were implemented in LabVIEW ™ and ran on the NI CompactRIO hardware.

Even though the presented results proved conceptual applicability of the proposed vibration suppression method with enhanced robustness against frequency mismatch, one has to take into account the assumptions that need to be fulfilled and technical limitations. First of all, for tuning the parameters of the absorber feedback, a very precise mathematical model of the absorber needs to be available. As a rule, determining the absorber mass and link stiffness are relatively easy tasks. However, as a rule, it is not the case for the damping. In our configuration of the set-up, for example, the damping forces are composed of inherent damping of the springs and the friction of the bearings, which is temperature dependent and is likely to vary in time. Another limitation comes from the well-known fact that even though the active feedback and/or mechanical adjustment of some of the physical parameters (mass in our example) can extend the applicable frequency range, it still has some relatively strict limits due to mechanical construction of the absorber, which remains fixed. Let us also mention that the proposed two step control feedback design where the vibration absorption and stabilization tasks are handled separately does not need to lead to an optimal solution, e.g. with respect to the overall energy needed for the control.

Motivated by the above outlined imperfections and limitations, in the subsequent research, methods to further enhance the vibration suppression quality, robustness in both vibration suppression and stability will be investigated. It will include an application of filtration techniques to handle the noise of the accelerometer sensors and algorithms to compensate for the inherent nonlinearities such as Coulomb friction. On the construction side of the mechatronic absorber, the current bearings will be substituted by low-friction bearings. In the control design research direction, the attention will be paid to simultaneous design of the active absorber and the resonator, targeting multiple frequencies and non-collocated vibration suppression.

Figure 1 :

 1 Figure 1: Primary Structure (P) with an active vibration absorber (A) to suppress displacement xp induced by harmonic disturbance force f (t).

√ ka ma is the natural frequency and ζ = ca 2 √

 2 ma ka

Figure 3 :

 3 Figure 3: Real part of the drDR non-assigned rightmost root with respect ζ and k 1 , k 2 .

Figure 4 :

 4 Figure 4: A feedback interconnection interpretation of the coupling between absorber and primary structure.

Figure 5 :

 5 Figure 5: Experimental setup built according to Fig. 1 controlled by NI CompactRIO, where the control algorithms (7) and (36) are implemented in LabVIEW ™ .

Figure 6 :

 6 Figure 6: Rightmost spectrum of characteristic equation (19), parameterized by Theorem 1 considering ζ = 0.0491, k 1 = 1, k 2 = 2: black dots -roots, solid iso-line -ℜ(M (s)) = 0, dashed iso-line -ℑ(M (s)) = 0 (results shown for scaled parameters).

Figure 7 :

 7 Figure 7: Spectra of the drDR, (3) and (16), alone, drDR -primary structure interconnection, (4) and (16), without the controller (unstable), and with the controller (36) (stable) (results shown for scaled parameters).

7 !

 7 = 29:79 rad s !1 ! = 1:05 cDR -sim drDR -sim

Figure 8 :

 8 Figure 8: Simulation results of DRs performance at five excitation frequencies, considering ω = ωf (1 + δ) in (1). The DR feedback and stabilization switched on at t = 3 s.

Figure 9 :

 9 Figure 9: Magnitude of the transfer function G f,xp (jω) given by (12) with: i) drDR and the stabilizing controller, and ii) cDR (results shown for scaled parameters).

7 !

 7 = 29:79 rad s !1 ! = 1:05 cDR drDR

Figure 10 :

 10 Figure 10: Experimental results of DRs performance at five excitation frequencies, considering ω = ωf (1 + δ) in (1). The DR feedback and stabilization switched on at t = 3 s.

  (stable) (results shown for scaled parameters).

		2				7 ! = 26:95 rad s !1 ! = 0:95
	[mm]	0
	7 x p
		-2
		34	35	36	37	38	39	40	41	42	43	44	45
		2				7 ! = 27:52 rad s !1 ! = 0:97
	[mm]	0
	7 x p
		-2
		34	35	36	37	38	39	40	41	42	43	44	45
		2				7 ! = 28:37 rad s !1 ! = 1:00
	[mm]	0
	7 x p
		-2
		34	35	36	37	38	39	40	41	42	43	44	45
		2				7 ! = 29:22 rad s !1 ! = 1:03
	[mm]	0
	7 x p
		-2
		34	35	36	37	38	39	40	41	42	43	44	45
		34	35	36	37	38	39	40	41	42	43	44	45
							# [s]

For an appropriate definition of the neutral functional differential equations and their properties, we refer to[START_REF] Hale | Introduction to functional differential equations[END_REF][START_REF] Michiels | Stability, Control, and Computation for Time-Delay Systems: An Eigenvalue-Based Approach[END_REF] and the references therein.

Video record of the experiments is shown at https://control.fs.cvut.cz/en/aclab/experiments/robustdr

Acknowledgment

The presented research was supported by the Czech Science Foundation under the project 21-00871S, by the collaborative project CELSA/20/013 funded by KU Leuven and the Czech Technical University in Prague in the framework of the CELSA alliance, by the project C14/22/092 of the Internal Funds KU Leuven, by the project G092721N of the Research Foundation-Flanders (FWO -Vlaanderen), and by a public grant overseen by the French National Research Agency (ANR) as part of the "Investissement d'Avenir" program, through the iCODE project funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-02. The first author also acknowledges the support of the Grant Agency of the Czech Technical University in Prague, grant No. SGS23/157/OHK2/3T/12.


Valentin Favier

Jérémy Charriot

Louis Crampette

Arnaud Bourdin
email: bourdin@chu-montpellier.fr

Engi Ahmed

What place will tezepelumab hold in the treatment paradigm in chronic rhinosinusitis?

Annualized Asthma Exacerbation Rate Arachidonic acid 15-lipoxygenase ALOX15, 15-lipoxygenase ATP adenosine triphosphate CD Cluster of differentiation CRS Chronic Rhinosinusitis CRSwNP Chronic Rhinosinusitis with nasal polyps CRSsNP Chronic Rhinosinusitis without nasal polyps DC dendritic cells FeNO Fractional exhaled nitric oxide MC mast cells IgE immunoglobulin ILC innate lymphoid cell IL interleukin IL-R interleukin-receptor mAb monoclonal antibodies Nitric oxide synthase NOS NP nasal polyp RORα RAR-related orphan

1.Introduction

Tezepelumab is the sixth biologic approved for severe asthma. Interestingly, the five already approved demonstrated clinical improvement in chronic rhinosinusitis with nasal polyps (CRSwNP), and by extension, in Chronic rhinosinusitis (CRS). What would be the benefit of adding a sixth in this rapidly changing landscape? Actually, the mechanism of action looks essential for determining the potential benefits of these drugs. Mechanistic overlap between these monoclonal antibodies (mAbs) can explain some of the overlapping reported evidence. Indeed, by targeting the inflammatory cascade upstream of the others, tezepelumab would at least in theory provide substantial additional benefits.

CRS and asthma are chronic inflammatory disorders that frequently coexist. Nose and bronchial epithelium have the same embryological origin and also share common molecular and cellular pathophysiology, leading to the concept of 'unified airway disease' [START_REF] Passalacqua | United airways disease: therapeutic aspects[END_REF]. There is a clear correlation between possibilities of achieving clinical control both of the upper and the lower airways in patients with coexisting CRS and asthma, and these patients are more difficult to treat [START_REF] Laidlaw | Chronic rhinosinusitis with nasal polyps and asthma[END_REF]. More than 80% of the patients with CRSwNP exhibit type 2 (T2) inflammation, characterized by eosinophilic tissue infiltration. Indeed, severe CRSwNP is typically characterized by T2 inflammation. Surgical intervention remains a mainstay of therapy for CRSwNP. Surgical treatment has a reported recurrence rate highly varying from 40% to 79%, depending on multiple factors including presence of type 2 inflammation. Tissue eosinophilic infiltration and the presence of galectin-10, the constitutive protein of Charcot Leyden crystals are other known prognostic factors for recurrence after surgery [START_REF] Chen | Predictive significance of Charcot-Leyden crystal structures for nasal polyp recurrence[END_REF]. Thus, patients who have recurrence despite surgery may ben-efit from a biologic agent. As such, biologics, which target T2 inflammation, represent promising therapeutic options for patients with severe CRSwNP.

The type 2 inflammatory pattern is characterized by eosinophils, dendritic cells (DCs), innate lymphoid cells type 2 (ILC2), mast cells, and T-helper 2 (Th2) cells. Eosinophils are found in up to 80% of surgically removed nasal polyps. Increased amounts of IL-4, IL-5, and IL-13 can be found in surgical tissue resection of nasal polyp. TSLP is increased in the epithelium from the upper airways in the polyps and, more interestingly, also in the non-polyp areas [START_REF] Nagarkar | Thymic stromal lymphopoietin activity is increased in nasal polyps of patients with chronic rhinosinusitis[END_REF]. TSLP expression was uniquely restricted to basal cells, especially in eosinophilic CRSwNP Figure 1.

There is increasing evidence of the heterogeneity of inflammation during CRS. Although type 2 inflammation is the hall-mark of most CRSwNP, a non-type 2 endo type with predominant neutrophilic inflammation has also been described, also known as type 1 and type 3 endotypes. Non T2 CRSwNP/CRS are less associated with asthma and are more prevalent in Asian population. Inflammatory mechanisms are poorly understood. In non-T2 inflammation, IL-8 promotes neutrophils recruitment. Type 1 inflammation is characterized by a Th1 polarization with high levels of IFN-γ and TGFβ, whereas type 3 is more likely to be driven by Th17 pathway, through IL-17-A. ALOX15 axis: ALOX15 (15-LO) produces a variety of proinflammatory lipid metabolites, notably catalyses the conversion of arachidonic acid to 15-hydroxyeicosatetraenoic acid (15-HETE), eotaxin-3, and eoxins. Eotaxin-3 plays a key role in the recruitment of eosinophils and is elevated in polyps. ALOX15 is mainly expressed in polyps, especially in basal cells and is inducible by IL-4 and IL-13. T2-polyps have been shown to be mainly composed of monotonous p63-expressing basal cells associated to a loss of the other classical epithelial subtypes notably secretory and ciliated cells [START_REF] Ordovas-Montanes | innate immune memory" and the memory of inflammation is not restricted to the hematopoietic lineage[END_REF]. Single cell RNA sequencing of NP highlighted an impairment in the differentiation ability of basal cells that accumulate at this early ontogenic state ('basal cell hyperplasia'). Basal cells in eosinophilic CRSwNP showed the highest expression of proinflammatory genes, especially those linked to type 2 inflammation, including the nitric oxide synthase (NOS2), that may be linked to high exhaled nitric oxide (FeNO) levels, TSLP, IL-4/13-induced genes, chemokines such as eotaxin-3, CLC (Charcot-Leyden crystal), and protease inhibitors such as cystatins. Eotaxin 3 is a potent chemoattractant for eosinophils through the binding to the cell surface chemokine receptor CCR3.Arachidonic acid 15lipoxygenase (ALOX15) encodes the enzyme 15-lipoxygenase (15-LO), which can oxidize polyunsaturated fatty acids and also promote the production of proteins such as eotaxin-3 and eoxins in a IL-13 dependent manner, which are associated with CRS and asthma [START_REF] Feltenmark | Eoxins are proin-flammatory arachidonic acid metabolites produced via the 15-lipoxygenase-1 pathway in human eosinophils and mast cells[END_REF][START_REF] Kristjansson | A loss-of-function variant in ALOX15 protects against nasal polyps and chronic rhinosinusitis[END_REF].

In the polyp tissue, the top genes expressed are related to cystatins proteins family, CST1, 2, & 4 [START_REF] Nocera | Cystatin SN is a potent upstream initiator of epithelial-derived type 2 inflammation in chronic rhinosinusitis[END_REF]. CST1&2 are cysteine protease inhibitors that are supposed to play a protective role against allergen, viral, and bacterial proteases. They are among the most highly overexpressed proteins in both nasal polyp epithelium and mucus-derived exosomes in CRSwNP and are correlated with CRSwNP disease severity [START_REF] Nocera | Cystatin SN is a potent upstream initiator of epithelial-derived type 2 inflammation in chronic rhinosinusitis[END_REF]. CST1 expression can be induced by TSLP and can also stimulate TSLP expression in the presence of IL-4 and double-stranded DNA, thereby forming a positive feedback loop. P-glycoprotein (permeability glycoprotein, P-gp, encoded by ABCB1a gene) is an ATP-dependent efflux pump that is capable of regulating cytokine transport and is expressed in the nasal and can also be transferred through exosomes [START_REF] Bleier | Regional expression of epithelial MDR1/P-glycoprotein in chronic rhinosinusitis with and without nasal polyposis[END_REF]. One hypothesis is that the structure of TSLP enables the alarmin to act as a P-gp pump substrates.

2.Tezepelumab for managing CRS with or without NP

The use of biologics in severe CRSwNP is now based on high levels of clinical evidence, making them the treatment of choice in severe CRS. Thus, the collaboration between Ear-Nose-Throat (ENT) specialists and pulmonologists is increasingly necessary to manage type-2 inflammatory diseases. To date, there is no data in the literature regarding biologic switching in CRSwNP, when a first option has not been effective enough; this comes in contrast with what happens in the management of severe asthma where switches are now commonly admitted and performed. When there is no associated asthma, the ENT must consider the clinical and biological characteristics of the patient to choose a second option. In the case of moderate to severe asthma associated with CRS comorbidity, collaboration between clinical disciplines is critical to enhance multidisciplinary approach (pulmonologist, allergy specialist dermatologist, pediatrician, gastroenterologist, ideally a pathologist and ENT physician) to define the most adequate therapeutic strategy, especially for the choice of biologic, that will be mainly driven by asthma phenotype. If no asthma disorder is present, ENT specialist will initiate the biologic. But the difficulty lies in cases of dissociated response, when asthma is well controlled, but rhinosinusitis is not. The ENT specialist could propose recurrent sinonasal surgeries to remove the polyps or a switch of biologics with the risk of altering the control of asthma. In a network meta-analysis that aimed to compare the efficacy of biologics on a composite outcome (7 criteria), dupilumab was the only one that achieved an improvement of all the items compared to other biologics (i.e. omalizumab, mepolizumab, benralizumab [START_REF] Oykhman | Comparative efficacy and safety of monoclonal antibodies and aspirin desensitization for chronic rhinosinusitis with nasal polyposis: a systematic review and network meta-analysis[END_REF]).

In case of CRSwNP control failure with dupilumab, while asthma is controlled, the clinical challenge will then be to achieve outcomes qualitatively and quantitatively within the range of those achieved by dupilumab. Whether non-eosinophilic patients with CRSwNP will also improve with tezepelumab as shown in asthma will be of great interest.

Recent cases in the literature report the combination of biologics in severe asthma, either to reach a better control of asthma in poor controlled severe asthma patients or insufficiently controlled associated comorbidities such as CRSwNP. The only randomized controlled clinical trial that evaluated the safety and efficacy of combination of monoclonal antibodies during severe asthma did not highlighted an increase of side effects nor efficacy in the combination mAbs arm (i.e. anti-IL-33 itekepimab and dupilumab. The safety and efficacy of such therapeutic strategies (i.e. switching and combination of bio-logics) need to be address in randomized clinical trials.

3.Indirect evidence of Tezepelumab efficacy from asthma clinical trials (Table 1)

Tezepelumab is currently FDA approved for the treatment of moderate-to-severe asthma. In the phase 2b PATHWAY and phase 3 NAVIGATOR studies, it has been shown to reduce exacerbation rates independently of baseline blood eosinophil count and improved the level of asthma control, quality of life, and even lung function [START_REF] Menzies-Gow | Tezepelumab in Adults and Adolescents with Severe, Uncontrolled Asthma[END_REF][START_REF] Corren | Tezepelumab in Adults with Uncontrolled Asthma[END_REF]. Next to that, there is some indirect evidence from clinical trials that it could also be effective in the treatment of chronic rhinosinusitis with nasal polyposis.

First, in the phase 2 placebo-controlled study CASCADE, Tezepelumab showed a reduction in the number of airway submucosal eosinophils by 85%, even in patients with a low submucosal eosinophil counts at baseline, suggesting that tezepelumab may act through other unknown mechanisms, expected in blocking an upstream epithelial cytokine. However, no impact was shown on other subtypes of inflammatory cells hypothetically due to a « less severe » profile of patients included in comparison with other clinical trials [START_REF] Diver | Effect of tezepelumab on airway inflammatory cells, remodelling, and hyperresponsiveness in patients with moderate-to-severe uncontrolled asthma (CASCADE): a double-blind, randomised, placebo-controlled, phase 2 trial[END_REF].

Next, considering the « allergic response pathway» which may maintain a high level of Type 2 inflammation in some patients with CRS, interesting insights have been brought by the CATNIP study [START_REF] Corren | Effects of combination treatment with tezepelumab and allergen immunotherapy on nasal responses to allergen: a randomized controlled trial[END_REF]. The objective was to assess the impact of combination treatment with tezepelumab and subcutaneous cat allergen immunotherapy (SCIT) on responses to Nasal Allergen Challenge (NAC). At week 52, the association of tezepelumab and SCIT was the most effective in reducing the NAC induced total nasal symptom scores (TNSS). More interestingly, one year after treatment cessation, a persistent clinical tolerance was observed that could be explained by a downregulation of a gene network related to type 2 inflammation, especially a reduced expression of a mast cell gene signature in nasal brushings.

From a more clinical perspective, in NAVIGATOR, tezepelumab impressively improved nasal symptoms (SNOT-22 score) in patients with comorbid nasal polyps [START_REF] Menzies-Gow | Tezepelumab efficacy in patients with severe, uncontrolled asthma and comorbid nasal polyps in NAVIGATOR[END_REF]; these patients had a greater response in terms of asthma as well.

4.Safety

Additional data from both real-life and NCT studies are needed to address the question of the long-term safety of tezepelumab. The DESTINATION study (NCT03706079) has shown that tezepelumab is well tolerated in patients with severe asthma over 2 years. Indeed, the roles of TSLP in different type of cancers high-lighted that TSLP can have pleiotropic effects and has both pro-tumoral and anti-tumoral effects, depending on the type of tumor. TSLP can also be protective regarding host defense against helminth or in S. aureus infections. However, the incidences of benign and malignant neoplasms and severe infections were similar in the tezepelumab and placebo groups in different NCT studies (NAVIGATOR, CASCADE, and PATHWAY studies). No hypersensitivity reactions have been reported. In the long-term DESTINATION study, significantly higher incidence of cardiac serious adverse events has been reported in patients receiving tezepelumab compared to placebo, without an increase of cardiovascular deaths.

5.Expert opinion

CRS is a heterogeneous disease divided into the two phenotypes: CRS with and without nasal polyps. Eosinophilic inflammation is present in 85% of CRSwNP. However, during Chronic Rhinosinusitis without nasal polyps (CRSsNP), inflammation is highly heterogeneous, with 5%, 45%, and 15% of type 1, type 2, and type 3 inflammation, respectively, classified on nasal pro-tein expression of IFN-γ, eosinophil, and IL-17-A. Multiple bio-logics are efficient for treating either lower and upper airways. The challenge is now to treat both sides and even all the manifestations falling within the T2 spectrum. Targeting T2 inflammation in CRSwNP, and especially TSLP, may reverse structural changes such as nasal polyps through the 'reprogramming' of structural cells and their interaction with the environment. TSLP is produced by the epithelia and is found in the airways, skin, and eye mucosa, sustaining the hypothesis of a universal molecular pathway to transduce a warning signal and supporting, beyond the concept of the unified airways, the idea that one biologic in a patient should target all the spectrum of T2 inflammation expressing in different organs. The ongoing study WAYPOINT (NCT04851964) might confirm those findings of tezepelumab for the treatment of CRSwNP.

Tezepelumab is actually considered as an available drug to cover the unmet need in T2 and low T2 severe asthma and acts upstream the inflammation cascade. Data are needed to evaluate its efficacy in non T2 CRS/CRSwNP

Figure 1 .

 1 Figure 1. Inflammatory T2 pathway and mechanisms of action of Tezepelumab during chronic rhinosinusitis (CRS) and during chronic rhinosinusitis with nasal polyps (CRSwNP). Note: Left side: Biological mechanisms involved during CRS in a non-polyp area and monoclonal thymic stromal lymphopoietin (TSLP) antibody effects. The production of TSLP can be triggered by endogenous or external signals such as viruses, fungi (i.e. aspergillus) through the Toll Like Receptor type 2 (TLR2), proteases (House Dust Mite, Alternaria), and particulate matter. TSLP promotes T2 inflammation through its receptor, TSLPR, expressed by DCs, ILC2, mast cells, eosinophils, and also fibroblasts and results in the induction of Th2 differentiation. IL-13 and IL-4 through the IL-4 Rα can act on airway epithelium and induce mucins production and activation of macrophages. Prostaglandin D2 (PGD2) causes further recruitment of ILC2s and TH2 cells by acting on CRTH2. IL-5 acts on eosinophils via IL-5 R and induce their recruitment and activation. Tezepelumab directly binds TSLP, avoiding the interaction of the alarmin with the target cells. Blocking TSLP decreased the interleukins (IL) 4, 5 and 13 release. Right side: Hypothetical mechanism of polyp growth: Reprogramming of susceptible basal cells by T2 environment during eosinophilic CRSwNP. Polyps are composed by basal cells. In polyp, TSLP expression is uniquely restricted to basal cells in airway epithelium. IL-4/IL-13 axis: Because epigenetic changes persist in dividing cells, T2 environment in susceptible basal cells cause stable alterations in the cell phenotype. Basal cell differentiation is intrinsically impaired through the chronic IL-4/IL-13 exposure.

Cystatins & P-glycoprotein axis: CST1 and CST2 are profoundly and focally overexpressed within nasal polyp epithelium, secreted in exosomes. Stimulation by TSLP induces CST1 in nasal epithelial cells and can reciprocally stimulate TSLP expression in combination with double-stranded DNA and IL-4, thereby forming a proinflammatory feedback loop. CST1 enhanced eosinophil activation and recruitment through induction of IL-5 by airway epithelial cells. TSLP stimulates degranulation of eosinophils, release of eosinophil eosinophilic cationic protein (ECP) and the formation of eosinophil extracellular traps (EETs) Table1.

Funding

This paper was not funded.

Declaration of interest

A Bourdin reports grants, personal fees, non-financial support, and/or other from AstraZeneca and Boehringer Ingelheim; Astra Zeneca, GSK, Novartis, Sanofi Regeneron, Boehringer Ingelheim and Chiesi. C Jeremy reports funding's by GlaxoSmithKline, Astra Zeneca, Sanofi. L Crampette reports speaker fees, non-financial support and/other from Medtronic, Novartis, Zambon, ALK, AstraZeneca, Sanofi Genzyme. A Engi reports personal fees, non-financial support from Sanofi, Astra Zeneca The authors have no other relevant affiliations or financial involvement with any organization or entity with a financial interest in or financial conflict with the subject matter or materials discussed in the manuscript apart from those disclosed.

Reviewer disclosures

Peer reviewers on this manuscript have no relevant financial or other relationships to disclose.ORCIDValentin Favier http://orcid.org/0000-0002-7999-951XJérémy Charriot http://orcid.org/0000-0002-1420-5774