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

published or not. The documents may come

Spectral design of robust delayed resonator by double-root assignment

Matěj Kuře, Tomas Vyhlidal, Wim Michiels,

Spectral design of robust delayed resonator by double-root assignment

Matej Kure * , * * Tomas Vyhlidal * , * * Wim Michiels * * * Islam Boussaada * * * *

INTRODUCTION

The delayed resonator (DR) concept proposed in 1990s by N. Olgac and his co-workers [START_REF] Olgac | A novel active vibration absorption technique: delayed resonator[END_REF]; [START_REF] Olgac | Delayed resonators as active dynamic absorbers[END_REF], has become an established tool for vibration suppression and one of the typical examples of benefits of time delay in the feedback loop. The timedelay feedback is applied to modify the active absorber properties resulting in 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 resonator feedback can be implemented using position, velocity or acceleration measurements, depending on the type of sensor selected for a particular vibration control application at hand. In this paper, the acceleration feedback is considered as it allows closing the feedback directly from the accelerometer sensors, which are easy to apply and of relatively low cost.

Modifications of the resonator concept include a singlemass dual frequency absorber [START_REF] Olgac | Single mass dual frequency fixed delayed resonator[END_REF], relative position feedback absorber in [START_REF] Olgac | Active vibration absorption using delayed resonator with relative position measurement[END_REF], a torsional vibrational absorber [START_REF] Hosek | A tunable torsional vibration absorber: the centrifugal delayed resonator[END_REF], to name a few. An auto-tuning algorithm enhancing robustness against uncertainties in the absorber parameters was presented in [START_REF] Hosek | A single-step automatic tuning algorithm for the delayed resonator vibration absorber[END_REF]. An algorithm for multidegree of freedom (MDOF) mechanical structures with multiple delayed resonators is described in [START_REF] Jalili | Multiple delayed resonator vibration absorbers for multi-degree-of-freedom mechanical structures[END_REF], where multiple absorbers are used to suppress different harmonics. Stability of the vibration absorber using acceleration and displacement feedback was derived in [START_REF] Alujević | Tuneable vibration absorber using acceleration and displacement feedback[END_REF] utilizing the Nyquist criterion. In [START_REF] Rivaz | An active dynamic vibration absorber for a hand-held vibro-elastography probe[END_REF] a delay free PI alternative of the resonator has been proposed for the acceleration feedback. 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 the time delay case.

Recently, a complete dynamics analysis of a DR with acceleration feedback was performed in Vyhlídal et al. (2014) revealing unfavorable neutral character of spectral properties of the DR, which is also transferred to the overall set-up. In order to mitigate this undesirable effect, an alternative distributed delay resonator (DDR) 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, which are more convenient. The second benefit of this novel resonator scheme is that the distributed delay provides filtering of the measured noise. 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 shown in Vyhlídal et al. (2014) and [START_REF] Pilbauer | Delayed resonator with distributed delay in acceleration feedback -design and experimental verification[END_REF] that for both the DR and DDR, their operable frequency range is limited. From below, it is limited by the stability boundary of the DR, 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 was recently proposed in [START_REF] Kučera | Extended delayed resonators-design and experimental verification[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 in [START_REF] Kučera | Extended delayed resonators-design and experimental verification[END_REF].

As it results from the frequency domain analysis performed in application sections of [START_REF] Pilbauer | Delayed resonator with distributed delay in acceleration feedback -design and experimental verification[END_REF]; [START_REF] Kučera | Extended delayed resonators-design and experimental verification[END_REF], the entire vibration suppression takes place only if the resonant frequency adjusted by the resonator feedback is equivalent with the true excitation frequency. The performance of the resonator decays considerably even for very small differences in these frequencies. The main objective of this paper is to propose a robust alternative to the classical delayed resonator, which will widen the frequency stop-band of the resonator. Note that in this aspect, the paper is analogous to the recent work [START_REF] Pilbauer | Optimized design of robust resonator with distributed timedelay[END_REF], where the optimization approach is applied directly on shaping the frequency characteristic of the DR. In this paper, we propose a more straightforward and easy to apply approach, which is based on double root assignment.

A fundamental feature of the resonator concept is that the parameters of the primary structure do not play a role in the resonator tuning. The purpose of the design is only to suppress the vibrations, while stability aspects of the overall set-up are left aside. As discussed in Vyhlídal et al. (2014); [START_REF] Pilbauer | Delayed resonator with distributed delay in acceleration feedback -design and experimental verification[END_REF], in fact, the resonator may decrease the stability margin or even destabilize the overall set-up. Thus, the second objective of the presented work is to handle this stability aspect by including a stabilizing controller, similarly as it was done in [START_REF] Fenzi | A probabilistic approach towards robust stability optimization, with application to vibration control[END_REF].

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 two analytic rules for its parametrizations. Next, design of the additional finite order controller is adressed together with its parametrization by minimizing the spectral abscissa. The validation of the proposed concept is done in section 4, where it is also compared with the classical delayed resonator concept. In section 5, brief conclusions are given.

PRELIMINARIES AND PROBLEM STATEMENT

The dynamic model of the absorber, depicted in Fig. 1, with the physical parameters m a , c a , k a denoting the mass, the damping and the stiffness coefficients, can be described by m a ẍa (ϑ) + c a ẋa (ϑ) + k a x a (ϑ) = ūa (ϑ) (1) with x a being the displacement of the absorber and u a a (scaled) external input. The damping ratio and the natural frequency of the absorber are given by ζ = ca 2 √ maka and Ω = ka ma , respectively. The objective of the absorber is to suppress vibrations of the primary structure being excited by a harmonic force f (t) at the frequency ωf .

The absorber equation can be turned to ẍa (ϑ) + 2ζΩ ẋa (ϑ) + Ω 2 x a (ϑ) = ūa (ϑ).

(2) Scaling the time ϑ with respect to the frequency Ω, i.e. by introducing dimensionless time t = Ωϑ, we obtain an absorber form ẍa (t) + 2ζ ẋa (t) + x a (t) = u a (t).

(3) with the only parameter ζ, assuming u a = 1 maΩ 2 ūa . Note that scaled excitation frequency is given by ω f = ωf Ω . Considering the scaled parameters with respect to m a and Ω, the overal set-up in Fig. 1 is given by ẍa 4)

(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) , (5)
where x p (t) is the position, u p (t) is the control input for positioning the primary structure and m p , c p , k p are the scaled mass, the damping and the stiffness parameters of the primary structure.

The absorber input u a is to be used for enhancing the absorber parameters so that the vibrations at the frequency ω f are entirely suppressed. Due to problem generalization purpose, we consider the feedback in the form U a (s) = P (s) X a (s) , (6) where P (s) is the feedback transfer function.

Compared to the classical delayed resonator concept, where (6) is the only active feedback at the set-up, we assume that also the primary structure is equipped with an active feedback

U p (s) = K (s) X p (s) , (7)
with controller K (s), which can be used for platform positioning or just for improving the overall dynamical properties in vibration suppression.

For the overall set-up (4)-( 7), the transfer function between the excitation force f and the position of the platform with feedback ( 6) is given by

G xp,f (s) = X p (s) F (s) = R(s) -P (s) (R(s) -P (s))(V (s) -K(s)) + (P (s) -Q(s))Q(s) (8) with R (s) = s 2 + 2ζs + 1 (9)
being the characteristic function of the absorber (1), and

V (s) = m p s 2 + (2ζ + c p ) s + (1 + k p ) , Q (s) = 2ζs + 1. ( 10
)
If the transfer function P (s) is parameterized so that the characteristic equation of the resonator composed of the absorber (1) and the feedback (6), given by M (s) = R (s) -P (s) = 0, (11) has a root couple s 1,2 = ±jω f , composing a pole couple of the resonator, then G xp,f (jω f ) = 0, (12) indicating that no vibrations at the given frequency ω = ω f are transferred in this f to x p channel and the vibrations are ideally suppressed.

Note however, that including the resonator feedback affects the dynamical properties of the entire system, which is given by the roots of the characteristic equation (R (s

) -P (s)) (V (s) -K(s)) + (P (s) -Q (s)) Q (s) = 0
(13) Assuming presence of the delay terms in P (s), the equation ( 13) has infinitely many roots. For the stability implications, all of them need to be located safely in the left half of the complex plane.

ROBUST RESONATOR FEEDBACK

The resonator feedback is considered in the following form

u(ϑ) = -hẍ a (ϑ) + ϑ 0 p(θ)x a (ϑ -θ)dθ, (14)
where p(ϑ) is the weighting function (impulse response) corresponding to the resonator transfer function P (s) , which is to be transformed to P (s) in the dimensionless form. Motivated by [START_REF] Kučera | Extended delayed resonators-design and experimental verification[END_REF], the term with h is included in ( 14) to modify virtually the mass of the absorber so that the natural frequency of the absorber Ω = ka ma+h is identical with the excitation frequency ωf , i.e. ω f = 1, which results to

h(ω f ) = k a ω2 f -m a . (15)
Note that if h = 0, for the design purposes, the absorber (1) as well as the primary structure needs to be scaled with respect to m+h, i.e. with respect to the virtually modified mass.

Coupling the absorber (1) with the feedback (14), and scaling the time with respect to Ω = ω f , the delayed resonator is given in the form

ẍa (t) + 2ζ ẋa (t) + x a (t) = t 0 p(θ)x a (t -θ)dθ. ( 16
)
where p(θ) is the scaled impulse response p(θ).

The transfer function is considered in the form

P (s) = L{p(t)} = X a (s) U a (s) = 2 i=0 a i e -sτi s s 2 , (17)
i.e. of (effectively) integral character with delays τ 0 = 0, τ 2 > τ 1 > τ 0 and parameters a 0 , a 1 , a 2 to be tuned. Note that the term s 2 corresponds to the acceleration measurements ẍa . When considering the measured output as y a = ẍa , the transfer function turns to the mentioned integral form Ya(s) Ua(s) = 2 i=0 aie -sτ i s .

As will be shown later, selection of this specific form of the transfer function ( 17) is advantageous due to easy parametrization of the resonator. However, on the other hand, it should be noted that from the practical point of view, the integral character of the transfer function can be risky when the measurements in the feedback path are noisy [START_REF] Kučera | Extended delayed resonators-design and experimental verification[END_REF]. As has been shown in [START_REF] Rivaz | An active dynamic vibration absorber for a hand-held vibro-elastography probe[END_REF], this problem can be solved by supplying the feedback path by a high-pass filter. Another option is to impose the finite impulse response of P (s), which can be achieved by satisfying the condition 2 i=0 a i = 0.

(18)

DR parametrization by assigning a double root

The objective in the robust resonator design is widening the frequency stop-band of |G xp,f (jω)|, where G xp,f (s) is given by ( 8), in the vicinity of ω f = 1. This can be done by assigning a double root s 1,2 = j to the characteristic equation of the delayed resonator ( 11), which appears in the numerator of (8). To perform the double root assignment, the root needs to be assigned not only to (11) given by

M (s) = s 2 + 2ζ - 2 i=0 a i e -sτi s + 1 = 0, (19)
but also to its first derivative

dM (s) ds = 2s + 2ζ - 2 i=0 a i e -sτi (1 -τ i s) = 0. ( 20
)
This will naturally lead to widening the range where |M (jω)| is close to zero in the vicinity of ω f = 1, due to its locally parabolic shape. This property is transferred to |G xp,f (jω)| supposing that the poles of G xp,f (s) are not located close to the zeros at j, which will be imposed by the stability requirement and will be targeted in the subsequent step.

Assigning the complex root s 1 = j to (20) and splitting the equation to real and imaginary parts, we obtain

(M (s)| s=j ) = - 2 i=0 a i sin τ i = 0, ( 21
) (M (s)| s=j ) = 2ζ - 2 i=0 a i cos τ i = 0. ( 22
)
Analogously, assigning the root to (20), we obtain the following two equations

dM (s) ds | s=j = 2ζ - 2 i=0 a i (cos τ i -τ i sin τ i ) = 0, (23) dM (s) ds | s=j = 2 + 2 i=0 a i (sin τ i + τ i cos τ i ) = 0. ( 24
)
Taking into account the equality ( 22), the equation ( 23) reduces to

Quasi-polynomial spectrum

a i τ i cos τ i = 0. (26)
Thus, we have four nonlinear equations ( 21), ( 22), ( 25) and ( 26), to determine the parameter set a i , i = 0, 1, 2, τ 1 , τ 2 (recall that the first delay was preselected as τ 0 = 0). The nonlinearity of the problem may be removed by preselecting also the other delay values. Equations ( 21) and ( 25) are satisfied independently of the gain values a i if the delays are selected as τ 1 = π and τ 2 = 2π (and possibly its 2k + 1 multiples, which are not considered here). Then, ( 22) and (26) form the linear set 2ζ -a 0 + a 1 -a 2 = 0, (27)

2 -πa 1 + 2πa 2 = 0. ( 28
)
Still the set of two equations with three variables have infinitely many solutions. We provide two reasonable options to solve the parametrization task.

Lemma 1: Consider the preselected delay values τ 0 = 0, τ 1 = π and τ 2 = 2π, then a universal parametrization of the resonator feedback ( 17) is given by

a 0 = 2ζ, a 1 = a 2 = - π 2 .
Proof: By pre-selecting a 0 = 2ζ, the other two parameters a 1 and a 2 result directly from ( 27) and ( 28). The universality lies in the independence of the resulting characteristic equation

M (s) = s 2 + π 2 e -sπ + e -s2π s + 1 = 0. ( 29
)
on the resonator parameters.

Applying the QPmR algorithm Vyhlídal and Zítek (2014), the rightmost spectrum of the retarded characteristic equation ( 29) is shown in Fig. 2. Next to the spectrum values denoted as black dots, the iso-lines (M (s)) = 0 and (M (s)) = 0 are shown. The root multiplicity two at s = j is confirmed by that two couples of (M (s)) = 0 and (M (s)) = 0 isolines intersect at this point, see [START_REF] Vyhlídal | Qpmr -quasipolynomial root-finder: Algorithm update and examples[END_REF]. As can be seen from the figure, the given root s 1,2 is the rightmost root.

From the physical and univerality points of view, this setting may be considered as a reasonable option. However, a drawback of this setting is that as the equation ( 18) is not satisfied, the impulse response of P (s) is not of finite length, which can have undesirable consequences for the control loop performance, taking into account accumulation of the measurement noise due to the integral character of P (s) s 2 . This problem can be overcome by the setting given in the following Lemma obtained directly by solving the set of equations ( 18), ( 25) and ( 26).

Lemma 2: Consider the preselected delay values τ 0 = 0, τ 1 = π and τ 2 = 2π, then parametrization of the resonator feedback ( 17) for which the impulse response is of finite length is given

a 0 = 3 2 ζ + 1 π , a 1 = -ζ, a 2 = - 1 2 ζ - 1 π .

Stabilizing the overall set-up

The absorber and primary structure coupled together as given in ( 4) and ( 5) 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)
(30) where the output y a is the measured acceleration y a = ẍa , and matrices A, B 1 , B 2 , B 3 correspond to a linear model of a mechanical system (4)-( 5) translated into state space representation 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 , i.e. the position of the primary structure.

Concerning the resonator feedback implementation, which in Laplace form is given as

U a (s) = N i=0 a i e -sτi s Y a (s), ( 31
)
it can be turned to

ua (t) = N i=0 a i y a (t -τ i ). ( 32
)
It should be mentioned that this transformation results in additional dynamics, characterized by the introduction of two eigenvalues at zero [START_REF] Pilbauer | Distributed delay input shaper design by optimizing smooth kernel functions[END_REF].

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 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), (33)
with k denoting order of the controller.

The system (30) and the feedback (32) create together System of Delay Differential Algebraic Equations (DDAE)

     ẋ(t) = Ax(t) + B 1 u a (t) + B 2 u p (t) + B 3 f (t), ua (t) = 2 i=0 a i y a (t -τ i ), y a (t) = C a Ax(t) + C a B 1 u a (t) + C a B 2 u p (t) + CB 3 f (t), y p (t) = C p x(t)
(34) which can be rewritten in a compact form of DDAEs [START_REF] Michiels | Stability, Control, and Computation for Time-Delay Systems: An Eigenvalue-Based Approach[END_REF] as

E ξ(t) = A 0 ξ(t) + 2 i=1 A i ξ(t -τ i ) + B 0 f (t), (35)
where ξ = [x u a y a y p ] T is a new state vector.

For the resulting system, it is required that the spectrum of (35) lies safely in the left half of the complex plain. Defining the spectral abscissa of the system as α(K) := sup( (s) :

s ∈ Σ), (36) with
Σ := s ∈ C : det sE -A 0 - 2 i=1 A i e -sτi = 0 , (37)
the requirement on the stability is given as α < 0. Note that if α is very close to the imaginary axis, next to the undesirably small stability margin, it may result in undesirably long transients due to the low damping or slow modes brought by the rightmost roots. Therefore, it will be required so that the spectral abscissa is located sufficiently far to the left of the imaginary axis. This motivates a controller design based on minimizing the spectral abscissa over the free parameters, the elements of matrices (A K , B K , C K , D K ). For this task 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].

It is important to note that the function K → α(K) is in general not everywhere differentiable, even not everywhere Lipschitz continuous, even though the dependence of the system matrices of the closed-loop system on the controller parameters is smooth. To tackle these challenges, the core optimization tasks in the method of [START_REF] Michiels | Spectrum-based stability analysis and stabilisation of systems described by delay differential algebraic equations[END_REF] are done using the software HANSO (Hybrid Algorithm for Nonsmooth Optimization) [START_REF] Overton | Hanso: a hybrid algorithm for nonsmooth optimization[END_REF].

CASE STUDY EXAMPLE

Let's consider coupled absorber with the platform as shown in Fig. 1 with following parameters m a = 0.223 kg, c a = 1.273 kgs -1 , k a = 350 Nm -1 on the absorber's side and mp = 1.52 kg, cp = 10.11 kgs -1 , kp = 1960 Nm -1 on the primary structure. These parameters were also considered in [START_REF] Kučera | Extended delayed resonators-design and experimental verification[END_REF].

To describe this set-up by ( 4)-( 5) we use normalized parameters. Absorber has natural frequency Ω = 39.627 s -1 and damping ratio ζ = 0.072. For simplicity, we assume that the excitation frequency ωf = Ω, which implies h = 0 in (14). Then, the primary structure has normalized weight m p = 6.816, damping c p = 1.144 and stiffness k p = 5.6. System is excited by the external harmonic force 16)-( 17) alone, drDR -system interconnection (34) without controller (unstable), and with controller (33) (stable).

with amplitude ∆ f = 10 N, which implies the normalized amplitude ∆f = 0.029.

For parametrization of the robust resonator, we apply Lemma 1. In Fig. 3 the spectrum of the double root delayed resonator (drDR) ( 16)-( 17) as well as the spectrum of its coupling with the primary structure (34) is shown. As can be seen, the spectrum of the interconnected system is unstable, as two root couples are located to the right of the stability boundary. Thus, the controller (33) of the order k = 3 is tuned by minimizing the spectral abscissa, as described above. After spectral abscissa optimization, defining the controller matrices (38) the drDR with the primary structure is stable (as shown in Figure 3).

A K = -1
In the next step, the performance validation is done via simulations. Next to the nominal excitation frequency ω = 1 case, two cases with 3% mismatch between the design and true excitation frequency (ω = 0.97 and ω = 1.03) are considered to study the robustness. For comparison results by the classical delayed resonator with lumped delay feedback

u a (t) = g cDR ẍa (t -τ cDR ) (39)
are shown, with gain g cDR = 0.1441 and time delay τ cDR = 1.5708, parametrized by assigning a single root s 1 = j to the delayed resonator [START_REF] Olgac | A novel active vibration absorption technique: delayed resonator[END_REF], [START_REF] Olgac | Delayed resonators as active dynamic absorbers[END_REF]. From the results shown in Figure 4 it can be seen that for nominal frequency ω = 1 both DRs performs well. However, for the cases with the 3% frequency mismatch, the drDR performs considerably better with approximately ten times smaller residual vibration amplitudes. 16)-( 17) and the stabilizing controller (33) (red line), the results by the classical scheme with cDR with feedback (39) (blue line) are shown.

Finally, the enhancement of the robustness is studied via the frequency responses. Next to the magnitude |G xp,f (jω)| of the transfer function ( 8) between the excitation force f and the position of the primary x p , also the normalized transmissibility function

T (ω) = R(jω) -P (jω) Q(jω) (40)
in the vicinity of ω = 1 is studied. Notice that this measure, which was introduced in Pilbauer et al. ( 2017), is fully determined by the absorber parameters in R(s) and Q(s) defined by ( 9) and ( 10) and by the parameters of active feedback P (s). The denominator Q(s) normalizes the function (40) in order to achieve lim ω→0 T (ω) = 1. Note that by coupling equation ( 4) with the resonator feedback, the transmissibility can be interpreted as T

(ω) = 1 |Gx a ,xp (jω)| , where G xa,xp (s) = Xa(s) Xp(s) .
If the cDR (39) is applied, then T (ω) goes to zero for the nominal frequency ω = 1 as shown in Fig. 5. 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. As can be seen in Fig. 5, drDR transmissibility has parabolic shape in the vicinity of ω = 1 with considerably smaller increase in its neighborhood, compared to the cDR. This broadening of the effective frequency stop-band is transformed to G f,xp (jω) shown in the Fig. 6. Another positive aspect is considerably lower H ∞ norm (maximum of the frequency response) compared to the case with cDR, caused by applying the platform position controller K, which is not coupled for the cDR case.

CONCLUSION

A new type of delayed resonator with a double root assigned at nominal frequency was presented as the main given by (8): i) with robust drDR ( 16)-( 17) and the stabilizing controller (33), ii) with a classical cDR with feedback (39) result. For the parametrization of the proposed resonator feedback with time delays, two fully analytic formulas have been proposed. The overall design procedure gains from the fact that it was proposed for the dimensionless models, normalized with respect to the absorber mass and natural frequency. An important aspect is also application of delay free feedback part in the resonator feedback ( 14), by which the resonant frequency of the absorber is unified with the excitation frequency. This step makes the successive parametrization much easier, as already revealed in [START_REF] Kučera | Extended delayed resonators-design and experimental verification[END_REF]. It was also demonstrated that application of the additional controller of the platform position is beneficial from both the stability and robustness points of view.

Fig. 1 .

 1 Fig. 1. Primary Structure (P), with an active vibration absorber (A) to suppress displacement x p induced by harmonic disturbance force f (t)

Fig. 2 .

 2 Fig. 2. Rightmost spectrum of the double root DelayedResonator characteristic equation (29), parametrized by Lemma 1.2

Fig. 3 .

 3 Fig.3. Spectra of the drDR (16)-(17) alone, drDR -system interconnection (34) without controller (unstable), and with controller (33) (stable).

  [6.6243 -0.1113 -0.3087] , D K = [-4.2371] ,

Fig. 4 .

 4 Fig. 4. Simulation results of delayed resonator performance at three excitation frequencies i) ω = 0.97, ii) the nominal ω = 1 and iii) ω = 1.03. Next to the results of the scheme with robust drDR (16)-(17) and the stabilizing controller (33) (red line), the results by the classical scheme with cDR with feedback (39) (blue line) are shown.

Fig. 6 .

 6 Fig. 5. Transmisibility of the drDR (16)-(17) and classical DR with the feedback (39).

ACKNOWLEDGEMENT

The presented research has been supported by the Czech Science Foundation under the project 17-20943S.


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

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

On the Dominancy of Multiple Spectral Values for Time-delay Systems with Applications

Keywords: Time-delay systems, stability analysis, spectral abscissa, rightmost spectral values

A further extension of a result on maximal multiplicity induced-dominancy for spectral values is analytically derived for generic retarded second-order systems with a single delay in the parameter space. Several examples illustrate the applicative perspectives of the result, towards a rightmost spectral value assignment approach.

INTRODUCTION

The present study concerns a frequency domain approach in the stability analysis of linear time-invariant retarded Time-delay systems. The investigation of conditions on the equation parameters that guarantee the exponential stability of solutions is a question of ongoing interest. In the Laplace domain, where a number of effective methods have been proposed, the stability analysis amounts to studying the distribution of the characteristic quasipolynomial function's roots, see for instance [START_REF] Cooke | On zeroes of some transcendental equations[END_REF][START_REF] Walton | Direct method for tds stability analysis[END_REF][START_REF] Stépán | Retarded Dynamical Systems: Stability and Characteristic Functions[END_REF][START_REF] Michiels | Stability and stabilization of time-delay systems[END_REF][START_REF] Olgac | An exact method for the stability analysis of time delayed linear time-invariant (lti) systems[END_REF][START_REF] Sipahi | Stability and stabilization of systems with time delay[END_REF]. The delay effect in controllers design was first introduced in [START_REF] Suh | Proportional minus delay controller[END_REF] where is shown that delayed proportional controller performs an averaged derivative action and thus can replace the proportional-derivative controller, see also [START_REF] Atay | Balancing the inverted pendulum using position feedback[END_REF]. Also, under appropriate conditions the Time-delay may have a stabilizing effect in the control design, see for instance [START_REF] Niculescu | Delay Effects on Output Feedback Control of Dynamical Systems[END_REF]. Indeed, that the closed-loop stability is guaranteed precisely by the existence of the delay. In the context of mechanical engineering problems, the effect of Time-delay was emphasized in [START_REF] Stépán | Retarded Dynamical Systems: Stability and Characteristic Functions[END_REF] where concrete applications are studied, such as, the machine tool vibrations and robotic systems.

In recent works, it is shown that the multiple spectral values for Time-delay systems can be characterized using a Birkhoff/Vandermonde-based approach; see for instance (Boussaada andNiculescu, 2016a,b, 2014;Boussaada et al., 2016). More precisely, in (Boussaada and Niculescu, 2016b), it is shown that the admissible multiplicity of the zero spectral value is bounded by the generic

The authors are partially financially supported by a grant from Hubert Curien (PHC) BRANCUSI 2017, project number 38390ZL and a grant from PHC BALATON 2018, project number 40502NM.

Polya and Szegö bound denoted P S B , which is nothing but the degree of the corresponding quasipolynomial, see for instance [START_REF] Pólya | Problems and Theorems in Analysis[END_REF]. In (Boussaada and Niculescu, 2016a), it is shown that a given CIR with non vanishing frequency never reaches P S B and a sharper bound for its admissible multiplicities is established. Moreover, in (Boussaada et al., 2016), the variety corresponding to a multiple root for scalar Time-delay equations defines a stable variety for the steady state. The multiplicity of a root itself is not important as such but its connection with the dominancy of this root is a meaningful tool for control synthesis. An example of a scalar retarded equation with two delays is studied in (Boussaada and Niculescu, 2016a) where it is shown that the multiplicity of real spectral values may reach the P S B . In addition, the corresponding system has some further interesting properties: (i) it is asymptotically stable, (ii) its spectral abscissa (rightmost root) corresponds to this maximal allowable multiple root located on the imaginary axis. Such observations enhance the outlook of further exhibiting the existing links between the maximal allowable multiplicity of some negative spectral value reaching the quasipolynomial degree (i.e the number of the involved polynomials plus their degree minus one) and the stability of the trivial solution of the corresponding dynamical system. This property induced from multiplicity appears also in optimization problems since such a multiple spectral value is nothing but the rightmost root, see also [START_REF] Vanbiervliet | A nonsmooth optimisation approach for the stabilisation of time-delay systems[END_REF][START_REF] Michiels | Continuous pole placement for delay equations[END_REF]. Also notice that the property was already observed in [START_REF] Ramirez | Design of proportional-integral-retarded (pir) controllers for second-order lti systems[END_REF], where a tuning strategy is proposed for the design of a delayed Proportional-Integral controller by placing a triple real dominant root for the closed-loop system. However, the dominancy is only checked using a Mikhailov curve and QPmR toolbox, see for instance [START_REF] Vyhlídal | Mapping based algorithm for large-scale computation of quasi-polynomial zeros[END_REF]. To the best of our knowledge, the first time an analytical proof of the dominancy of a spectral value for the scalar equation with a single delay was presented in [START_REF] Hayes | Roots of the transcendental equation associated with a certain difference-differential equation[END_REF]. The dominancy property is further explored and analytically shown in the case of second-order systems and a rightmost root assignment based design using delayed state-feedback is proposed in [START_REF] Boussaada | On the coalescence of spectral values and its effect on the stability of time-delay systems: Application to active vibration control[END_REF](Boussaada et al., , 2018) ) where its applicability in damping active vibrations for a piezo-actuated beam is proved.

The multiplicity of a root itself is not important as such but its connection with the dominancy of this root is a meaningful tool for control synthesis. This work, further explores such a connexion and gives an analytical proof for the dominancy of the spectral value with maximal multiplicity for second-order systems controlled via a delayed proportional-derivative controller.

PREREQUISITES AND MOTIVATING EXAMPLES

Second-order linear systems capture the dynamic behavior of many natural phenomena, and have found wide applications in a variety of fields, such as vibration and structural analysis. In the sequel we recall some hints, recent results and examples motivating the use of delay in controller design for stabilizing the steady state solution corresponding to such a class of systems. Consider the generic second-order system with a single time delay:

ẋ = A 0 x(t) + A 1 x(t -τ ) (1)
with the state-vector x = (x 1 , x 2 ) ∈ R 2 , under appropriate initial conditions belonging to the Banach space of con-

tinuous functions C([-τ N , 0], R 2 ).
Here τ is a positive constant delay and the matrices A j ∈ M 2 (R) for j = 0 . . . 1. It is well known that the asymptotic behavior of the solutions of ( 1) is determined from the spectrum ℵ designating the set of the roots of the associated characteristic function (denoted in the sequel ∆(s, τ )). Namely, the characteristic function corresponding to system (1) is a quasipolynomial ∆ :

C × R + → C of the form: ∆(s, τ ) = det s I -A 0 -A 1 e -τ s . (2)
To start with, let us recall a generic result on the location of spectral values corresponding to (2). The proof of the proposition below can be found in [START_REF] Michiels | Stability and stabilization of time-delay systems[END_REF]. Proposition 1. If s is a characteristic root of the system (1), then it satisfies

|s| ≤ ||A 0 + A 1 e -τ s || 2 . ( 3
)
The above proposition provides a generic envelope curve around the characteristic roots corresponding to (1).

In particular, the present work is focused on Time-delay systems characterized by the quasipolynomial function of the form ∆(s, τ ) = P 0 (s) + P 1 (s) e -τ s , (4) and we are concerned by the problem of the analytical characterization of the rightmost root. More precisely, equation ( 4) is written as:

∆(s, τ ) = s 2 + c 1 s + c 0 + (β 0 + sβ 1 ) e -τ s .
(5)

The case β 1 = 0, yields a rightmost root with maximal multiplicity as characterized in (Boussaada et al., 2018). The problem of stabilization of a chain of integrators is considered in [START_REF] Niculescu | Stabilizing a chain of integrators using multiple delays[END_REF]) where a single integrator can be stabilized by a single delay statefeedback. It is asserted that either 2 distinct delays or a proportional+delay are sufficient to stabilize a chain including integrators. In [START_REF] Kharitonov | Static output feedback stabilization: necessary conditions for multiple delay controllers[END_REF], a like assertion is shown to be also necessary to stabilize the double integrator. In conclusion, there exists at least a spectral value for (6) with positive real part. As a matter of fact, consider the following quasipolynomial function: ∆(s, τ ) = s 2 + α e -τ s .

(6) It can be checked that the maximal admissible multiplicity is 2 and it can be attained if, and only if,

α = -4 e -2 τ 2 , s = - 2 τ . ( 7
)
As a result, s 0 = -2 τ , while being a multiple root it cannot be dominant. Figure 1 illustrates the particular quasipolynomial ( 6)-( 7) with τ = 1, that is s+2) .

∆(s, 1) = s 2 -4e -(
(8) where the dominancy property of the multiple root is lost since s 1 ≈ 0.557 is a root of (8). This is structurally explained by the sparsity of the corresponding quasipolynomial.

Sunflower Equation

In (Boussaada et al., 2016) the well-known Sunflower model is considered. Namely, the helical movement of a growing plant is governed by the following delay equation:

ẍ + a τ ẋ + b τ sin(x(t -τ )) = 0 (9)
This model is known to reproduce the dynamics of the upper part of the stem of the plant, which performs a rotating movement. Here the state x(t) designates the angle of the plant with respect to the vertical line, the delay τ corresponds to a geotropic reaction time, and a and b some positive parameters. The corresponding linearized system with α = a/τ and β = b/τ is given by: ẍ + α ẋ + βx(t -τ ) = 0 (10)

In (Boussaada et al., 2016) it is shown that equation (10) admits a spectral value at z with multiplicity 2 if, and only if, (α, z) = (α + , z + ) or (α, z) = (α -, z -) where:

                         α -= -2 + 4 + τ 2 β 2 e 1/2 τ -β- 2+ √ 4+τ 2 β 2 τ τ -2 , z -= - β 2 - 2 + 4 + τ 2 β 2 2τ α + = -2 + 4 + τ 2 β 2 e 1/2 τ -β+ -2+ √ 4+τ 2 β 2 τ τ -2 , z + = - β 2 + -2 + 4 + τ 2 β 2 2τ (11) It is also proved that if (z, α) = (z + , α + ) (respectively (z -, α -)) and τ β > 2 √ 3 (τ β < 2 √ 3) then z + (respectively z -)
is the rightmost root and the corresponding steady state solution is asymptotically stable. The next section enunciates the main contribution, which extends the above results.

MAIN RESULT

The following result generalizes the result from in (Boussaada et al., 2018) which is restricted to β 1 = 0. Theorem 2. Consider the quasipolynomial function ( 5):

∆(s, τ ) = s 2 + c 1 s + c 0 + (β 0 + sβ 1 ) e -τ s , The following assertions hold:

i) The multiplicity of any given root of the quasipolynomial function ( 5) is bounded by 4, it can be attained only on the real axis. ii) The quasipolynomial (5) admits a real spectral value at s = s ± with algebraic multiplicity 4 if, and only if, either

                 s + = -2 + √ -2 + c 0 τ 2 τ , β 0 = 2 e -2+ √ -2+c0τ 2 -5 ± √ -2 + c 0 τ 2 τ 2 , β 1 = -2 e -2+ √ -2+c0τ 2 τ , c 1 = -2 √ -2 + c 0 τ 2 τ (12) or                  s -= -2 - √ -2 + c 0 τ 2 τ , β 0 = 2 e -2- √ -2+c0τ 2 -5 - √ -2 + c 0 τ 2 τ 2 , β 1 = -2 e -2- √ -2+c0τ 2 τ , c 1 = 2 √ -2 + c 0 τ 2 τ ( 13
) where τ is arbitrarily chosen satisfying c 0 τ 2 ≥ 2. iii) If either (12) or ( 13) is satisfied then s = s ± is the rightmost root of (4).

A complete proof of the main result will be presented in an extended version of the paper. Its sketch is summarized below.

Sketch of the Proof:

The degree of the quasipolynomial function is equal to 4 as defined above. First, the van-ishing of the quasipolynomial ∆ yields the elimination of the exponential term as a rational function in s. The substitution of the obtained equality in the first three derivatives gives a system of algebraic equations. Solving them, one obtains the two solutions ( 12) and (13). Next using the argument principle one shows the dominancy of s ± ; see Figure 4. Further explanation can be found in the next section. For an effective implementation for complex integral computations we refer the reader to [START_REF] Xu | Delay-dependent stability analysis by using delay-independent integral evaluation[END_REF].

ILLUSTRATIVE EXAMPLE

Consider the damping-free oscillator controlled by a delayed proportional-derivative controller ξ(t) + γξ(t) = u(t),

u(t) = -βξ(t -τ ) -α ξ(t -τ ) ( 14
)
where γ a real parameter. The corresponding quasipolynomial function is given by ∆(s, τ ) = s 2 + γ + (β + α s) e -s τ (15) If γ = 0 then the control problem ( 14) reduces to the stabilization of the double integrator using delayed PD controller. Otherwise, using a linear transformation, it is sufficient to study the two cases γ = 1 and γ = -1 to get a complete picture of the effect of the parameter γ on the dominancy of admissible multiple roots.

Double integrator stabilized by delayed PD-Controller

A result from [START_REF] Niculescu | Stabilizing a chain of integrators using multiple delays[END_REF][START_REF] Kharitonov | Static output feedback stabilization: necessary conditions for multiple delay controllers[END_REF], mentioned in Section 2.1, asserts that a delayed proportional controller (with a single delay) is not able to stabilize a double integrator. Here we investigate the effect of the additional delayed derivative term and explore its stabilizing effect through the multiplicity induceddominancy property. Consider the quasipolynomial function ∆(s, τ ) = s 2 + (β + α s) e -s τ (16) where α = 0. Proposition 3. The following assertions hold for (16):

i) The multiplicity of any given root of the quasipolynomial function ( 16) is bounded by 3, it can be attained only on the real axis. ii) The quasipolynomial ( 16) admits a real spectral value at s = s ± with algebraic multiplicity 3 if, and only if, 16)

                 α = 2 -1 - √ 2 e -2- √ 2 τ , β = 2 e -2- √ 2 -7 -5 √ 2 τ 2 , s -= -2 - √ 2 τ (17) or                  α = 2 √ 2 -1 e -2+ √ 2 τ , β = 2 e -2+ √ 2 -7 + 5 √ 2 τ 2 , s + = -2 + √ 2 τ (18)
for τ = 1/3 and ( 18) is satisfied. (Down) Spectrum distribution corresponding to ( 16) for τ = 1/3 and ( 17) is satisfied, the dominancy property is lost.

iii) A spectral value of ( 16) with maximal multiplicity (multiplicity 3) is dominant if and only if s = s + .

Sketch of the Proof: The degree of the quasipolynomial function is equal to 4 as defined in Section 2. First, the vanishing of the quasipolynomial ∆ yields the elimination of the exponential term as a rational function in s. The substitution of the obtained equality in the first three derivatives gives a system of algebraic equations. Solving them, one shows that the solutions set is empty. Thus, the maximal multiplicity is less or equal to 3. Solving the two first derivatives yields to the solutions ( 17) and ( 18).

The dominancy of s + proof follows the same steps as that of Theorem 2. First, using Proposition 1, one establishes a generic supremum bound for the real and imaginary parts of roots of ( 21)-( 23). Then define an integration contour γ = ∪ 6 k=1 C k which is taken as a counterclockwise closed curve, then an integral over γ is defined as the sum of the integrals over the directed smooth curves that make γ up, as illustrated in Figures 4. Elementary calculations give parametrization of γ on each C k . Since ∆ is analytic then the argument principle asserts:

1 2iπ γ ∂ s ∆(s, τ ) ∆(s, τ ) ds = Z, (19)
where Z designates the number of the quasipolynomial roots enclosed by γ. Furthermore, the left-hand side of (24) gives:

γ ∂ s ∆(s, τ ) ∆(s, τ ) ds = lim →0 6 k=1,k =4 1 0 ṡk (t) ∂ s ∆(s k (t), τ ) ∆(s k (t), τ ) dt + lim →0 π 2 -π 2 ṡ4 (t) ∂ s ∆(s 4 (t), τ ) ∆(s 4 (t), τ ) dt
(20) where s k (t) designates the parametrization of s along C k for k ∈ {1, . . . , 6}. Some tedious but elementary computations lead to Z = 0 in case ( 18), but Z = 1 in case (17). Finally, Figure 2 22) is satisfied then s = -√ 2 is the rightmost root of (4).

if, α = - √ 2e -2 , β = -5 e -2 , τ = √ 2. (22) iii) If (
Sketch of the Proof: Following Theorem 2, s 0 = -√ 2 is root of (21) of multiplicity 4 with parameters values:

α 0 = - √ 2e -2 , β 0 = -5 e -2 , τ 0 = √ 2 (
23) Furthermore, s 0 is the rightmost root of ( 21)-( 23). The dominancy proof follows the same steps as that of Theorem 2. First, using Proposition 1, one establish a generic supremum bound for the real part as well as the imaginary part of roots of ( 21)-( 23). Then define an integration contour γ = ∪ 6 k=1 C k which is taken as a counterclockwise closed curve, then an integral over γ is defined as the sum of the integrals over the directed smooth curves that make γ up, as illustrated in Figures 4. Since ∆ is analytic function then the principle of argument asserts:

1 2iπ γ ∂ s ∆(s, τ ) ∆(s, τ ) ds = Z, (24)
where Z designates the number of the quasipolynomial roots enclosed by γ. Furthermore, the left-hand side of (24) gives:

γ ∂ s ∆(s, τ ) ∆(s, τ ) ds = lim →0 6 k=1,k =4 1 0 ṡk (t) ∂ s ∆(s k (t), τ ) ∆(s k (t), τ ) dt + lim →0 π 2 -π 2 ṡ4 (t) ∂ s ∆(s 4 (t), τ ) ∆(s 4 (t), τ ) dt (
                 s + = -2 + √ τ 2 + 2 τ , β = 2 e -2+ √ τ 2 +2 -7 + 5 √ τ 2 + 2 -τ 2 τ 2 , α = 2 -1 + √ τ 2 + 2 e -2+ √ τ 2 +2 τ (27)
or Throughout the above result, one can illustrate various scenarios. In the first one, the multiple root is dominant and stable, for instance when condition ( 27) is satisfied and τ = 1, see Figure 5 (Left). In the second, the multiple root is dominant and unstable which occurs when condition ( 27) is satisfied and τ = 2, see Figure 5 (Right). In the last, the multiple spectral value is not dominant which is illustrated in Figure 6.

CONCLUSION

A new extension of a dominancy result based on the maximal multiplicity of spectral value is analytically shown for generic second-order systems with a single delay. The dominancy property is parametrically analyzed in the case of a damping-free oscillator. Also examples of multiple roots loss of dominancy are provided. Unlike methods based on finite spectrum assignment, the method proposed in this work does not render the closed loop system finite dimensional but consists in controlling its rightmost spectral value, see (Boussaada and Niculescu, 2018) for further applications of the approach.

Fig. 1 .

 1 Fig. 1. Sparsity-induced loss of dominancy for the multiple spectral value. Each intersection between the blue/red curves corresponds to a spectral value of (8). The roots' distribution is illustrated using QPmR toolbox from (Vyhlídal and Zitek, 2009) 2.1 Multiple spectral values for Time-delay systems are not necessarily dominant

Fig. 2 .

 2 Fig. 2. (Up) Spectrum distribution corresponding to (16)for τ = 1/3 and (18) is satisfied. (Down) Spectrum distribution corresponding to (16) for τ = 1/3 and (17) is satisfied, the dominancy property is lost.

  illustrate the result. 4.2 Harmonic oscillator stabilized by delayed PD-controller Consider the problem of stabilization of a classical harmonic oscillator using PD controller: ∆(s, τ ) = s 2 + 1 + (β + α s) e -s τ (21) Proposition 4. Consider the quasipolynomial function (21) for which the following assertions hold: i) The multiplicity of any given root of the quasipolynomial function (5) is bounded by 4, it can be attained only on the real axis. ii) The quasipolynomial (5) admits a real spectral value at s = -√ 2 with algebraic multiplicity 4 if, and only

  Figures 3 and 4 illustrate the distribution of the spectrum of (21)-(23).

Fig. 3 .

 3 Fig. 3. (Left) Spectrum distribution corresponding to (21)-(23). The black points correspond to the spectral values, the solid blue line is the zero-modulusmanifold, the solid brown line is the zero-real partmanifold, the solid red line is the zero-imaginary partmanifold where x and y represent respectively the real part and the imaginary part of the complex variable s. (Right) Zoom on the dominant non resonant spectral value corresponding to (21)-(23) located at λ = -√ 2. The solid blue line is the zero-modulusmanifold and the solid red line is the zero-imaginary part-manifold of (21)-(23).

Fig. 4 .

 4 Fig. 4. The used contour for applying the argument principle to investigate the dominancy of the multiple root in blue solid line. The dashed red line gives the generic spectrum envelope established from Proposition 1 . 4.3 Controlling unstable second-order system via a delayed Proportional-Derivative Controller Consider the problem of stabilization of the second-order sparse polynomial (having two real roots with different signs): ∆(s, τ ) = s 2 -1 + (β + α s) e -s τ . (26) Proposition 5. The following assertions hold for (26): i) The multiplicity of any given root of the quasipolynomial function (26) is bounded by 3, it can be attained only on the real axis. ii) The quasipolynomial (26) admits a real spectral value at s = s ± with algebraic multiplicity 3 if and only if either         

Fig. 5 .

 5 Fig. 5. (Up)The spectral distribution of (26) under the conditions (27) with τ = 1. The multiple spectral value is dominant and stable. (Down) The spectral distribution of (26) under the conditions (27) with τ = 2. The multiple spectral value is dominant but unstable.

Fig. 6 .

 6 Fig.6. The spectral distribution of (26) under the conditions (28) with τ = 1. The multiple spectral value is not dominant ACKNOWLEDGEMENTS

ACKNOWLEDGEMENTS

We thank Karim L. Trabelsi (IPSA Paris, France) for careful reading of the manuscript and for valuable discussions.


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

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

Karim Trabelsi
email: karim.trabelsi@ipsa.fr

Towards a Decay Rate Assignment Based Design for Time-Delay Systems with Multiple Spectral Values

Recent results on maximal multiplicity induceddominancy for spectral values in reduced-order Time-Delay Systems naturally apply in controllers design. As a matter of fact, the approach is merely a delayed-output-feedback where the candidates' delays and gains result from the manifold defining the maximal multiplicity of a real spectral value, then, the dominancy is shown using the argument principle. Various reduced order examples illustrate the applicative perspectives of the approach.

I. INTRODUCTION

The present study centers on stabilizing-controllers design for linear time-invariant retarded time-delay systems. The investigation of conditions on the equation parameters that guarantee the exponential stability of solutions is a question of ongoing interest. In particular, an efficient way to study a solution's stability is the frequency domain approach since in the Laplace domain, where a number of effective methods have been proposed, the stability analysis amounts to studying the distribution of the characteristic quasipolynomial function's roots, see for instance [START_REF] Bellman | Differential-difference equations[END_REF], [START_REF] Cooke | On zeroes of some transcendental equations[END_REF], [START_REF] Walton | Direct method for tds stability analysis[END_REF], [START_REF] Stépán | Retarded Dynamical Systems: Stability and Characteristic Functions[END_REF], [START_REF] Hale | Introduction to functional differential equations[END_REF], [START_REF] Michiels | Stability and stabilization of timedelay systems[END_REF], [START_REF] Olgac | An exact method for the stability analysis of time delayed linear time-invariant (lti) systems[END_REF], [START_REF] Sipahi | Stability and stabilization of systems with time delay[END_REF]. The idea of exploiting the delay effect in controllers design was first introduced in [START_REF] Suh | Proportional minus delay controller[END_REF] where it is shown that the conventional proportional controller equipped with an appropriate time-delay performs an averaged derivative action and thus can replace the proportional-derivative controller, see also [START_REF] Atay | Balancing the inverted pendulum using position feedback[END_REF]. Furthermore, it was stressed in [START_REF] Niculescu | Delay Effects on Output Feedback Control of Dynamical Systems[END_REF] that timedelay has a stabilizing effect in the control design. Indeed, the closed-loop stability is guaranteed precisely by the existence of the delay. In the context of mechanical engineering problems, the effect of time-delay was emphasized in [START_REF] Stépán | Retarded Dynamical Systems: Stability and Characteristic Functions[END_REF] where concrete applications are studied, such as the machine tool vibrations and robotic systems.

In recent works, the characterization of multiple spectral values for time-delay systems of retarded type were established using a Birkhoff/Vandermonde-based approach; see for instance [START_REF] Boussaada | Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: A Vandermonde-based approach[END_REF], [START_REF]Characterizing the codimension of zero singularities for timedelay systems[END_REF], [START_REF]Computing the codimension of the singularity at the origin for delay systems: The missing link with Birkhoff incidence matrices[END_REF], [START_REF] Boussaada | Multiplicity and stable varieties of time-delay systems: A missing link[END_REF]. In particular, in [START_REF]Characterizing the codimension of zero singularities for timedelay systems[END_REF], it is shown that the admissible multiplicity of the zero spectral value is bounded by the generic Polya and Szegö bound denoted P S B , which is merely the degree of the corresponding quasipolynomial 1 , see for instance [START_REF] Pólya | Problems and Theorems in Analysis[END_REF]. 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 a given crossing imaginary root with a non vanishing frequency never reaches P S B and a sharper bound for its admissible multiplicities is established.

Moreover, in [START_REF] Boussaada | Multiplicity and stable varieties of time-delay systems: A missing link[END_REF], the manifold corresponding to a multiple root for scalar time-delay equations defines a stable manifold for the steady state. An example of a scalar retarded equation with two delays is studied in [START_REF] Boussaada | Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: A Vandermonde-based approach[END_REF] where it is shown that the multiplicity of real spectral values may reach the P S B . In addition, the corresponding system has some further interesting properties: (i) it is asymptotically stable, (ii) its spectral abscissa (rightmost root) corresponds to this maximal allowable multiple root located on the imaginary axis. Such observations enhance the outlook of further exhibiting the existing links between the maximal allowable multiplicity of some negative spectral value reaching the quasipolynomial degree and the stability of the trivial solution of the corresponding dynamical system. This interesting property induced by multiplicity appears also in optimization problems since such a multiple spectral value is indeed the rightmost root, see also [START_REF] Vanbiervliet | A nonsmooth optimisation approach for the stabilisation of time-delay systems[END_REF]. Also notice that the property was already observed in [START_REF] Ramirez | Design of proportional-integral-retarded (pir) controllers for second-order lti systems[END_REF], where a tuning strategy is proposed for the design of a delayed Proportional-Integral controller by placing a triple real dominant root for the closed-loop system. However, the dominancy is only checked using a Mikhailov curve and QPmR toolbox, see for instance [START_REF] Vyhlidal | Mapping based algorithm for large-scale computation of quasi-polynomial zeros[END_REF].

It is worth noting that the rightmost root for quasipolynomial function corresponding to stable time-delay systems is actually the exponential decay rate of its time-domain solution, see for instance [START_REF] Mori | On an estimate of the decay rate for stable linear delay systems[END_REF] for an estimate of the decay rate for stable linear delay systems. To the best of our knowledge, the first time an analytical proof of the dominancy of a spectral value for the scalar equation with a single delay was presented in [START_REF] Hayes | Roots of the transcendental equation associated with a certain difference-differential equation[END_REF]. The dominancy property is further explored and analytically shown in scalar delay equations in [START_REF] Boussaada | Multiplicity and stable varieties of time-delay systems: A missing link[END_REF], then in second-order systems controlled by a delayed proportional is proposed in [START_REF] Boussaada | Further remarks on the effect of multiple spectral values on the dynamics of time-delay systems. application to the control of a mechanical system[END_REF], [START_REF] Boussaada | On the coalescence of spectral values and its effect on the stability of timedelay systems: Application to active vibration control[END_REF] where its applicability in damping active vibrations for a piezo-actuated beam is proved. An extension to the delayed proportional-derivative controller case is studied in [START_REF] Boussaada | On the dominancy of multiple spectral values for time-delay systems with applications[END_REF] where the dominancy property is parametrically characterized. We emphasize that the idea of using roots assignment for controller-design for time-delay system is not new. As a matter of fact, an analytical/numerical stabilization method for retarded timedelay systems related to the classical pole-placement method for ordinary differential equations is proposed in [START_REF] Michiels | Continuous pole placement for delay equations[END_REF], see also [START_REF] Zitek | Dimensional analysis approach to dominant three-pole placement in delayed pid control loops[END_REF] for further insights on pole-placement methods for retarded time-delays systems with proportional-integralderivative controller-design.

This work provides an overview of those recent results on the dominancy criterion for scalar and second-order systems and it further explores the applicability of such a criterion in a third-order model describing the Mach number regulation in a wind tunnel. Roughly speaking, the Mach number regulation in a wind tunnel is based on Navier-Stokes equations for unsteady flow and contains control laws for temperature and pressure regulation. Here, the model we consider consists of a system of three state equations with a delay in one of the state variables.

II. PREREQUISITES

Consider the generic second-order system with a single time delay:

χ = A 0 χ(t) + A 1 χ(t -τ ), (1)
where χ = (χ 1 , . . . , χ n ) ∈ R n is the state-vector, under appropriate initial conditions belonging to the Banach space of continuous functions

C([-τ N , 0], R n ).
Here τ is a positive constant delay and A j ∈ M n (R) for j = 0 . . . 1 are real valued matrices. It is well known that the asymptotic behavior of the solutions of (1) is determined from the spectrum ℵ designating the set of the roots of the associated characteristic function (denoted ∆(s, τ ) in the sequel). Namely, the characteristic function corresponding to system (1) is a quasipolynomial ∆ : C × R + → C of the form:

∆(s, τ ) = det s I -A 0 -A 1 e -τ s . (2)
To start with, let us recall a generic result on the location of spectral values corresponding to [START_REF] Cooke | On zeroes of some transcendental equations[END_REF]. The proof of the proposition below can be found in [START_REF] Michiels | Stability and stabilization of timedelay systems[END_REF].

Proposition 1. If s is a characteristic root of system (1), then it satisfies

|s| ≤ ||A 0 + A 1 e -τ s || 2 . (3)
The above proposition combined with the triangular inequality provides a generic envelope curve around the characteristic roots corresponding to system [START_REF] Bellman | Differential-difference equations[END_REF].

In particular, the present work is focused on time-delay systems characterized by the quasipolynomial function of the form

∆(s, τ ) = P 0 (s) + P 1 (s) e -τ s , (4)
where deg(P 0 ) > deg(P 1 ). We shall consider the problem of the analytical characterization of its rightmost root. The starting point of this work in progress and the first analytical proof of the multiplicity-induced dominancy was proposed in [START_REF] Boussaada | Multiplicity and stable varieties of time-delay systems: A missing link[END_REF]. Indeed, a simple scalar differential equation with one delay representing a biological model describing the dynamics of a vector disease model was considered. In its linearized version, the infected host population ξ(t) is governed by:

ξ(t) + a 0 ξ(t) + a 1 ξ(t -τ ) = 0, (5)
where a 1 > 0 designates the contact rate between infected and uninfected populations assuming that the infection of the host recovery proceeds exponentially at a rate -a 0 > 0.

It was shown that for a given positive delay, equation ( 5) admits a double spectral value at s = s 0 if, and only if,

s 0 = - a 0 τ + 1 τ and a 1 = e -a0τ -1 τ . (6)
Furthermore, it was stressed that s 0 is the corresponding rightmost root and if s 0 < 0 then the zero solution of system ( 5) is asymptotically stable. One knows that s = s 0 is a spectral value of (5) if, and only if, s 0 is a root of the characteristic equation

∆(z, τ ) = s + a 0 + a 1 e -sτ = 0. (7)
The main ingredient of the dominancy proof of s 0 is an integral equation which cannot be satisfied for any spectral value s with (s) > s 0 . Namely, it was shown that if a 1 satisfies (6), then the characteristic function reads:

∆(s, τ ) = (s -s 0 ) 1 - 1 0 e -τ (s-s0) t dt . (8)
As a matter of fact, if s 1 = ζ + j η = s 0 is a root of (8) then s 1 is a root of its second factor. Hence, we obtain

1 = 1 0 e -τ (ζ-s0) t dt. (9)
But, e -τ (ζ-s0) t < 1 for ζ -s 0 > 0 and 0 < t < 1, thereby exhibiting the dominancy of s 0 . Remark 1. The rightmost root s 0 corresponding to equation [START_REF] Sipahi | Stability and stabilization of systems with time delay[END_REF] where system ( 6) is satisfied varies in the interval s 0 ∈ ]-∞, -a 0 [. Figure 2 illustrates the behavior of the rightmost root with respect to the time-delay variation.

IV. SECOND-ORDER SYSTEMS Second-order linear systems capture the dynamic behavior of many natural phenomena and have found wide applications in a variety of fields, such as vibration and structural analysis. In the sequel, we recall some hints, recent results and examples motivating the use of delay in controller-design for stabilizing the steady state solution corresponding to such a class of systems. In its generic form, equation ( 4) is written as:

∆(s, τ ) = s 2 + c 1 s + c 0 + (β 0 + sβ 1 ) e -τ s . ( 10
)
The case β 1 = 0 yields a rightmost root with maximal multiplicity as characterized in [START_REF] Boussaada | Further remarks on the effect of multiple spectral values on the dynamics of time-delay systems. application to the control of a mechanical system[END_REF]. Consider the standard linear change of variables

s = c 1 λ 2 , (11)
leading to the normalized characteristic function

   ∆(λ, τ ) = λ 2 + 2 λ + a 0 + α e -λτ , where α = 4 c 2 1 β 0 , τ = c 1 2 τ and a 0 = 4 c 0 c 1 2 . ( 12
)
If α = 0, the spectral abscissa is minimized at a 0 = 1 which corresponds to to the rightmost root located at λ 0 = -1, see for instance [START_REF] Kirillov | Robust stability at the swallowtail singularity[END_REF]. By exploiting the delay effect, the following proposition proved in [START_REF] Boussaada | Further remarks on the effect of multiple spectral values on the dynamics of time-delay systems. application to the control of a mechanical system[END_REF] asserts that the solution's decay rate can be further improved by decreasing the corresponding rightmost root. Assume that a 0 > 1, then the following proposition holds.

Proposition 2.

i) The multiplicity of any given root of the quasipolynomial function [START_REF] Boussaada | Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: A Vandermonde-based approach[END_REF] is bounded by 3. ii) The quasipolynomial (12) admits a real spectral value at λ 0 = -1 -1 τ with algebraic multiplicity 3 if, and only if,

τ = 1 a 0 -1 and α = - 2 e -(1+τ ) τ 2 . ( 13
)
iii) If equations (13) are satisfied then λ = λ 0 is the rightmost root of function [START_REF] Boussaada | Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: A Vandermonde-based approach[END_REF].

Remark 2. If equations ( 13) are satisfied then the trivial solution of the second order equation ẍ(t)+2 ẋ(t)+a 0 x(t)+ α x(t -τ ) = 0 is asymptotically stable with λ 0 as the corresponding exponential decay.

A. Multiple spectral values for time-delay systems are not necessarily dominant

The problem of stabilization of a chain of integrators is considered in [START_REF] Niculescu | Stabilizing a chain of integrators using multiple delays[END_REF] where a single integrator can be stabilized by a single delay state-feedback. Indeed, a positive gain guarantees the closed-loop stability of the system free of delay, and, by continuity, there exists a (sufficiently small) delay in the output preserving the stability of the closedloop system. However, the situation is completely different for a chain of integrators of order n when n > 1. For instance, consider the time-delay system characterized by the following quasipolynomial function:

∆(s, τ ) = s 2 + α e -τ s . ( 14
)
It can be checked that the maximal admissible multiplicity is 2 and it can be attained if, and only if,

α = -4 e -2 τ 2 , s = - 2 τ . (15)
However, the main result from [START_REF] Niculescu | Stabilizing a chain of integrators using multiple delays[END_REF] asserts that either n distinct delays or a proportional+delay compensator with n-1 distinct delays are sufficient to stabilize a chain including n integrators. In [START_REF] Kharitonov | Static output feedback stabilization: necessary conditions for multiple delay controllers[END_REF], a like assertion is shown to be also necessary to stabilize the chain of n integrators. Hence, in our case, either 2 distinct delays or a proportional+delay are necessary and sufficient to stabilize the double integrator. In conclusion, there exists at least a spectral value for [START_REF]Computing the codimension of the singularity at the origin for delay systems: The missing link with Birkhoff incidence matrices[END_REF] with a positive real part. As a result, s 0 = -2 τ , while being a multiple root cannot be dominant. Indeed, consider ( 14)-( 15) with τ = 1, that is

∆(s, 1) = s 2 -4e -(s+2) . (16)
As illustrated in Figure 3, the dominancy property is lost since s 1 ≈ 0.557 is a root of function [START_REF] Pólya | Problems and Theorems in Analysis[END_REF]. This is justified by the sparsity of ( 16).

B. Stabilizing a delayed proportional-derivative controller for generic second order systems

Let us consider again the quasipolynomial function (10):

∆(s, τ ) = s 2 + c 1 s + c 0 + (β 0 + sβ 1 ) e -τ s .
The following result generalizes Proposition 2 which is restricted to β 1 = 0.

Proposition 3. Considering equation [START_REF] Atay | Balancing the inverted pendulum using position feedback[END_REF], the following assertions hold: i) The multiplicity of any given root of the quasipolynomial function [START_REF] Atay | Balancing the inverted pendulum using position feedback[END_REF] is bounded by 4, it can be attained only on the real axis. Each intersection between the blue/red curves corresponds to a spectral value of function [START_REF] Pólya | Problems and Theorems in Analysis[END_REF].

ii) The quasipolynomial (10) admits a real spectral value at s = s ± with algebraic multiplicity 4 if, and only if, either

               s + = -2 + √ -2 + c 0 τ 2 τ , β 0 = 2 e -2+ √ -2+c0τ 2 -5 + √ -2 + c 0 τ 2 τ 2 , β 1 = -2 e -2+ √ -2+c0τ 2 τ , c 1 = -2 √ -2 + c 0 τ 2 τ , (17)
or                s -= -2 - √ -2 + c 0 τ 2 τ , β 0 = 2 e -2- √ -2+c0τ 2 -5 - √ -2 + c 0 τ 2 τ 2 , β 1 = -2 e -2- √ -2+c0τ 2 τ , c 1 = 2 √ -2 + c 0 τ 2 τ , ( 18
)
where τ is arbitrarily chosen satisfying c 0 τ 2 ≥ 2. iii) If either [START_REF] Vanbiervliet | A nonsmooth optimisation approach for the stabilisation of time-delay systems[END_REF] or (18) is satisfied, then s = s ± is the rightmost root of (4).

A complete proof of the main result will be presented in an extended version of the paper; its sketch is summarized below.

Proof: The degree of the quasipolynomial function is equal to 4 as defined above. First, the vanishing of the quasipolynomial ∆ yields the elimination of the exponential term as a rational function in s. The substitution of the obtained equality in the first three derivatives gives a system of algebraic equations. Solving them, one obtains the two solutions [START_REF] Vanbiervliet | A nonsmooth optimisation approach for the stabilisation of time-delay systems[END_REF] and [START_REF] Ramirez | Design of proportional-integral-retarded (pir) controllers for second-order lti systems[END_REF]. Next using the argument principle one shows the dominancy of s ± ; see Figure 6. Further explanation can be found in the next section. For an effective implementation in complex integral computations we refer the reader to [START_REF] Xu | Delay-dependent stability analysis by using delay-independent integral evaluation[END_REF]. Remark 3. It is worth noting that including information on the acceleration in the control loop allows to a timedelay system of neutral type characterized by the following quasipolynomial function of degree 5:

∆(s, τ ) = s 2 + c 1 s + c 0 + (β 0 + sβ 1 + sβ 2 ) e -τ s . (19)
Since we are dealing with the asymptotic stability analysis, one assumes that |β 2 | = 1, see for instance [START_REF] Hale | Introduction to functional differential equations[END_REF]. More precisely, if one assumes that -1 < β 2 < 0 then function (19) admits a negative root at s 0 = ln(-β2) τ with multiplicity 5 if, and only if, the function parameters satisfy:

                         β 0 = β 2 12 -6 ln (-β 2 ) + (ln (-β 2 )) 2 τ 2 , β 1 = 2 β 2 (3 -ln (-β 2 )) τ , c 0 = 6 ln (-β 2 ) + (ln (-β 2 )) 2 + 12 τ 2 , c 1 = 2 -3 -ln (-β 2 ) τ .
Furthermore, the spectrum distribution of function ( 19) consists of a chain of roots with real parts close to s 0 . However, the dominancy of multiple spectral values for neutral type remains an open question.

V. A PARAMETERIZED DOMINANCY ANALYSIS IN DELAYED-FEEDBACK UNDAMPED OSCILLATORS

Roughly speaking, sparsity of a quasipolynomial may preclude a given spectral value to attain the maximal admissible multiplicity, which is indeed the degree of the quasipolynomial. This section is devoted to the analysis of the parameters' effect on the admissible multiplicity as well as the dominancy of spectral values.

Consider the undamped oscillator controlled by a delayed proportional-derivative controller

ξ(t) + γξ(t) = u(t), u(t) = -βξ(t -τ ) -α ξ(t -τ ), ( 20
)
where γ is a real parameter, α and β are the gains of the delayed proportional-derivative controller. The corresponding quasipolynomial function is given by:

∆(s, τ ) = s 2 + γ + (β + α s) e -s τ . (21)
If γ = 0, then the control problem (20) reduces to the stabilization of the double integrator using a delayed proportional-derivative controller. Otherwise, using a linear transformation, it is sufficient to study the two cases γ = 1 and γ = -1 to get a complete picture of the effect of the parameter γ on the dominancy of admissible multiple roots.

A. The double integrator stabilized by a delayed proportional-derivative controller

A result from [START_REF] Niculescu | Stabilizing a chain of integrators using multiple delays[END_REF] and [START_REF] Kharitonov | Static output feedback stabilization: necessary conditions for multiple delay controllers[END_REF], mentioned in Section IV-A, asserts that a delayed proportional controller (with a single delay) is not able to stabilize a double integrator. In [START_REF] Boussaada | On the dominancy of multiple spectral values for time-delay systems with applications[END_REF] investigate the effect of the additional derivative term equipped with the same delay is investigated its stabilizing effect through the multiplicity induced-dominancy property is emphasized. Consider the quasipolynomial function

∆(s, τ ) = s 2 + (β + α s) e -s τ , ( 22
)
where α = 0.

                 α = 2 -1 - √ 2 e -2- √ 2 τ , β = 2 e -2- √ 2 -7 -5 √ 2 τ 2 , s -= -2 - √ 2 τ , (23)
or                  α = 2 √ 2 -1 e -2+ √ 2 τ , β = 2 e -2+ √ 2 -7 + 5 √ 2 τ 2 , s + = -2 + √ 2 τ , (24)
iii) A spectral value of function [START_REF] Boussaada | Further remarks on the effect of multiple spectral values on the dynamics of time-delay systems. application to the control of a mechanical system[END_REF] with maximal multiplicity (equal to 3) is dominant if, and only if, s = s + .

Proof: The degree of the quasipolynomial function is equal to 4 as defined in Section 2. First, the vanishing of the quasipolynomial ∆ yields the elimination of the exponential term as a rational function in s. The substitution of the obtained equality in the first three derivatives gives a system of algebraic equations. Solving them, one shows that the solutions set is empty. Thus, the maximal multiplicity is less than or equal to 3. Solving the two first derivatives yields solutions [START_REF] Boussaada | On the coalescence of spectral values and its effect on the stability of timedelay systems: Application to active vibration control[END_REF] and [START_REF] Boussaada | On the dominancy of multiple spectral values for time-delay systems with applications[END_REF].

The dominancy of s + proof follows the same steps as that of Proposition 3. First, using Proposition 1, one establishes a generic supremum bound for the real and imaginary parts of roots of function [START_REF] Boussaada | Further remarks on the effect of multiple spectral values on the dynamics of time-delay systems. application to the control of a mechanical system[END_REF] such that system (24) is satisfied. Then define an integration contour γ = ∪ 6 k=1 C k which is taken as a counterclockwise closed curve, hence an integral over γ is defined as the sum of the integrals over the directed smooth curves that make γ up, as illustrated in Figures 6. Elementary calculations give a parametrization of γ on each C k . Since ∆ is analytic then the argument principle asserts:

1 2iπ γ ∂ s ∆(s, τ ) ∆(s, τ ) ds = Z, (25)
where Z designates the number of the quasipolynomial roots enclosed by γ. Furthermore, the left-hand side of (31) gives:

γ ∂ s ∆(s, τ ) ∆(s, τ ) ds = lim →0 6 k=1,k =4 1 0 ṡk (t) ∂ s ∆(s k (t), τ ) ∆(s k (t), τ ) dt + lim →0 π 2 -π 2 ṡ4 (t) ∂ s ∆(s 4 (t), τ ) ∆(s 4 (t), τ ) dt.
(26) where s k (t) designates the parametrization of s along C k for k ∈ {1, . . . , 6}. Some tedious but elementary computations lead to Z = 0 in case [START_REF] Boussaada | On the dominancy of multiple spectral values for time-delay systems with applications[END_REF], but Z = 1 in case [START_REF] Boussaada | On the coalescence of spectral values and its effect on the stability of timedelay systems: Application to active vibration control[END_REF]. Finally, Figure 4 illustrates the result.

B. Harmonic Oscillator stabilized by a delayed proportional-derivative controller

Consider the problem of stabilization of a classical harmonic oscillator using proportional-derivative controller:

∆(s, τ ) = s 2 + 1 + (β + α s) e -s τ . (27)
Proposition 5. Consider the quasipolynomial function [START_REF] Kirillov | Robust stability at the swallowtail singularity[END_REF] for which the following assertions hold: i) The multiplicity of any given root of the quasipolynomial function [START_REF] Atay | Balancing the inverted pendulum using position feedback[END_REF] is bounded by 4, it can be attained only on the real axis. ii) The quasipolynomial (10) admits a real spectral value at s = -√ 2 with algebraic multiplicity 4 if, and only if,

α = - √ 2e -2 , β = -5 e -2 , τ = √ 2. (28)
iii) If (28) is satisfied then s = -√ 2 is the rightmost root of (4).

Proof: Following Proposition 3, s 0 = -√ 2 is a root of the quasipolynomial function [START_REF] Kirillov | Robust stability at the swallowtail singularity[END_REF] of multiplicity 4 with parameters values:

α 0 = - √ 2e -2 , β 0 = -5 e -2 , τ 0 = √ 2 (29)
Furthermore, s 0 is the rightmost root of ( 27)-( 29). The dominancy proof follows the same steps as that of Proposition 3. First, using Proposition 1, one establish a generic supremum Fig. 5. Zoom on the dominant non resonant spectral value corresponding to ( 27)-( 29) located at λ = -√ 2. The solid blue line is the zero modulus manifold and the solid red line is the zero imaginary-part manifold of ( 27)- [START_REF] Kharitonov | Static output feedback stabilization: necessary conditions for multiple delay controllers[END_REF].

bound for the real part as well as the imaginary part of roots of ( 27)-( 29). Then define an integration contour γ = ∪ 6 k=1 C k which is taken as a counterclockwise closed curve, then an integral over γ is defined as the sum of the integrals over the directed smooth curves that make γ up, as illustrated in Figure 6. Elementary calculations leads to a parametrization of γ on each C k :

C 1 : s 1 (t) = 2 + √ 2 t - √ 2 -7 i where 0 ≤ t ≤ 1 C 2 : s 2 (t) = 2 + 7 i(2t -1) where 0 ≤ t ≤ 1 C 3 : s 3 (t) = -2 - √ 2
1 2iπ γ ∂ s ∆(s, τ ) ∆(s, τ ) ds = Z, (31)
where Z designates the number of the quasipolynomial roots enclosed by γ. Furthermore, the left-hand side of [START_REF] Manitius | Feedback controllers for a wind tunnel model involving a delay: Analytical design and numerical simulation[END_REF] gives:

γ ∂ s ∆(s, τ ) ∆(s, τ ) ds = lim →0 6 k=1,k =4 1 0 ṡk (t) ∂ s ∆(s k (t), τ ) ∆(s k (t), τ ) dt + lim →0 π 2 -π 2 ṡ4 (t) ∂ s ∆(s 4 (t), τ ) ∆(s 4 (t), τ ) dt.
(32) Some tedious but elementary computations lead to Z = 0. Figures 5 and6 illustrate the distribution of the spectrum of function ( 27)- [START_REF] Kharitonov | Static output feedback stabilization: necessary conditions for multiple delay controllers[END_REF].

C. Controlling an unstable second-order system via a delayed PD Controller Consider the problem of stabilization of the second-order sparse polynomial (having two real roots with different

                 s + = -2 + √ τ 2 + 2 τ , β = 2 e -2+ √ τ 2 +2 -7 + 5 √ τ 2 + 2 -τ 2 τ 2 , α = 2 -1 + √ τ 2 + 2 e -2+ √ τ 2 +2 τ , (34)
or                  s -= -2 - √ τ 2 + 2 τ , α = 2 -1 - √ τ 2 + 2 e -2- √ τ 2 +2 τ , β = 2 e -2- √ τ 2 +2 -7 -5 √ τ 2 + 2 -τ 2 τ 2 , (35)
Throughout the above result, one can illustrate various scenarios. In the first one, the multiple root is dominant and stable, for instance when condition (34) is satisfied and τ = 1, see Figure 7 (Up). In the second, the multiple root is dominant and unstable which occurs when condition (34) is satisfied and τ = 2, see Figure 7 (Down). In the last, the multiple spectral value is not dominant as exposed in Figure 8. Finally, Figure 9 illustrates the behavior of the rightmost root corresponding to function (33) with respect to the delay value in both cases (34) and (35).

VI. A THIRD-ORDER MACH NUMBER REGULATION IN A WIND TUNNEL MODEL

Transonic flows analysis is still a challenging problem in compressible fluid dynamic. In a stationary transonic flow, subsonic and supersonic regions live at the same time and are respectively governed by elliptical and hyperbolic equations. Furthermore, these two types of partial differential equations require completely different approaches, which often preclude solutions that are valid in the entire region.

In particular, the Mach number regulation in a wind tunnel is based on the Navier-Stokes equations for unsteady flow and contains control laws for temperature and pressure regulation. The following simplified model of Mach number regulation described in [START_REF] Manitius | Feedback controllers for a wind tunnel model involving a delay: Analytical design and numerical simulation[END_REF] consists of a system of three state equations with a delay in one of the state variables. It is stressed that in steady-state operating conditions, the dynamic response of the Mach number perturbations ξ 1 to small perturbations in the guide vane angle actuator ξ 2 are governed by:

     ξ1 (t) = -aξ 1 (t) + k a ξ 2 (t -τ ) ξ2 (t) = ξ 3 (t) ξ3 (t) = -ω 2 ξ 2 (t) -2ζωξ 3 (t) + ω 2 u(t) (36)
where a, ω, ζ, k and τ are parameters depending on the operating point and presumed constant when the perturbations ξ i are small. Moreover, following the experimental parameter values of the wind tunnel developed at NASA Langley Research Center, the parameters a, ω, ζ, τ are positive.

In [START_REF] Manitius | Feedback controllers for a wind tunnel model involving a delay: Analytical design and numerical simulation[END_REF], a feedback consisting of a linear combination of state variables and weighted integrals of some of the state variables over a period equal to the time delay, where the spectrum of the closed-loop system is finite (consists of three eigenvalues). However, our method does not render the closed-loop system finite dimensional but only involves controlling its rightmost root. Consider the control law: u(t) = -α ω 2 ξ 2 (t) -β0 ω 2 ξ 2 (t -τ ) -β1 ω 2 ξ 3 (t -τ ). In our case, the corresponding quasipolynomial function is given by: ∆(s, τ ) = (s+a) (sβ 1 +β 0 )e -sτ +s 2 +2 s ζ ω+ω 2 +α . (37) Since a is a positive parameter, our aim is to establish conditions on parameters such that the rightmost root of the second factor of (37) has a negative real part. Interestingly, the analysis of the second factor can be deduced directly from the result in Section IV-B. As a matter of fact, by denoting

ω 2 + α = c 0 and 2 ζ ω = c 1 , (38)
one may directly exploit condition [START_REF] Ramirez | Design of proportional-integral-retarded (pir) controllers for second-order lti systems[END_REF] from Proposition 3 to guarantee the exponential stability of the trivial solution by assigning its rightmost root as a stable root. Note that condition [START_REF] Vanbiervliet | A nonsmooth optimisation approach for the stabilisation of time-delay systems[END_REF] is not convenient as it imposes that c 1 < 0 which it cannot be applied here. Since the delay is intrinsic to the model, then the first step consists in finding the gain α such that

which satisfies the condition c 0 τ 2 ≥ 2. The gains β 0 and β 1 are easily computed from system [START_REF] Ramirez | Design of proportional-integral-retarded (pir) controllers for second-order lti systems[END_REF] and owing to the first equality from (39). Finally, s -= -2 -ζ ω τ τ is the rightmost root of the second factor of function (37), which insures the stability of the steady state solution.

VII. CONCLUDING REMARKS

Recent results by the authors on maximal multiplicity induced-dominancy for spectral value of time-delay systems of retarded type are overviewed. This note emphasizes a delayed controller-design based on the trivial solution's decay rate assignment. To illustrate the corresponding steps, a parameterized analysis of the dominancy property validity is established for generic second order oscillators. Finally, to demonstrate its concrete applicability, the regulation of the Mach number in a wind tunnel is considered. In future works, a generalization of the approach to arbitrary order dynamical systems will be studied and further applications treated.

Fig. 1 .

 1 Fig.1. The distribution of the spectrum corresponding to equation[START_REF] Sipahi | Stability and stabilization of systems with time delay[END_REF] and system (6) for a 0 = τ = 1. The roots' distribution is illustrated using QPmR toolbox from[START_REF] Vyhlidal | Mapping based algorithm for large-scale computation of quasi-polynomial zeros[END_REF]

Fig. 2 .

 2 Fig. 2. The rightmost root corresponding to equation (7) and system (6) as a function of the delay τ (red solid line) for a fixed value of a 0 = 1.

Fig. 3 .

 3 Fig. 3. Sparsity-induced loss of dominancy for the multiple spectral value.Each intersection between the blue/red curves corresponds to a spectral value of function[START_REF] Pólya | Problems and Theorems in Analysis[END_REF].

Fig. 4 .Proposition 4 .

 44 Fig. 4. (Up) Spectrum distribution corresponding to function (22) for τ = 1/3 with system (24) is satisfied. (Down) Spectrum distribution corresponding to function (22) for τ = 1/3 with system (23) is satisfied, the dominancy property is lost.

Fig. 6 .

 6 Fig. 6. The simplified contour used for applying the argument principle to investigate the dominancy of the multiple root in blue solid line. The dashed red line gives the generic spectrum envelope established in Proposition 1.

  signs): ∆(s, τ ) = s 2 -1 + (β + α s) e -s τ . (33) Proposition 6. Consider the quasipolynomial function (33) for which the following assertions hold: i) The multiplicity of any given root of the quasipolynomial function (33) is bounded by 3, it can be attained only on the real axis. ii) The quasipolynomial (33) admits a real spectral value at s = s ± with algebraic multiplicity 3 if and only if either

Fig. 7 .

 7 Fig. 7. (Up) The spectral distribution of function (33) under the conditions (34) with τ = 1. The multiple spectral value is dominant and stable. (Down) The spectral distribution of function (33) under the conditions (34) with τ = 2. The multiple spectral value is dominant but unstable.

Fig. 8 .

 8 Fig. 8. The spectral distribution of the quasipolynomial function (33) under conditions (35) with τ = 1. The multiple spectral value is not dominant.

Fig. 9 .

 9 Fig. 9. The dominant root corresponding to equation (33) and system (35) as a function of the delay τ (red solid line). The dominant root corresponding to equation (33) and system (34) as a function of the delay τ (blue dashed line).

c 1 = 2 √ - 2 + 2 .

 222 c 0 τ 2 τ which gives:α = 2 + ζ 2 ω 2 τ 2 -ω 2 τ 2 τSubstituting this last equality in (38) one obtainsc 0 = 2 + ζ 2 ω 2 τ 2 τ 2 and c 1 = 2 ζ ω,

  t + 2 + 7 i where 0 ≤ t ≤ 1

	C 4 : s 4 (t) = -C 5 : s 5 (t) = -C 6 : s 6 (t) = -	√ √ √	2 + e it where -2 + i (( -7) t + 7) where 0 ≤ t ≤ 1 π 2 π ≤ t ≤ 2 2 + i (( -7) t -) where 0 ≤ t ≤ 1
			(30)
	Since ∆ is an analytic function then argument principle
	asserts:

The quasipolynomial degree is exactly the number of the involved polynomials plus their degree minus one

The authors are partially financially supported by a grant from Hubert Curien (PHC) BRANCUSI 2017, project number 38390ZL and a grant from PHC BALATON 2018, project number 40502NM. The authors are also supported by Inria Saclay, DISCO Team.