Real spectral values coexistence and their effect on the stability of time-delay systems: Vandermonde matrices and exponential decay

This work exploits structural properties of a class of functional Vandermonde matrices to emphasize some qualitative properties of a class of linear autonomous n − th order diﬀerential equation with forcing term consisting in the delayed dependent-variable. More precisely, it deals with the stabilizing eﬀect of delay parameter coupled with the coexistence of the maximal number of real spectral values. The derived conditions are necessary and suﬃcient and represent a novelty in the litterature. Under appropriate conditions, such a conﬁguration characterizes the spectral abscissa corresponding to the studied equation. A new stability criterion is proposed. This criterion extends recent results in factorizing quasipolynomial functions. The applicative potential of the proposed method is illustrated through the stabilization of coupled oscillators


Introduction
Matrices arising from a wide range of problems in mathematics and engineering typically display characteristic structures.In particular, exploiting such a structure in problems from dynamical systems is known to be an engaging aperture for understanding of complex qualitative behaviors and for characterizing system's properties, see, for instance, 1 and references therein.This study is a crossroad between the investigation of the invertibility of a class of such structured matrices which is related to Multivariate Interpolation Problems (namely, the wellknown Lagrange Interpolation Problem) and the localisation of spectral values of linear time-delay systems.The study of conditions on the time-delay systems parameters that guarantees the exponential stability of solutions is a question of ongoing interest and to the best of the authors' knowledge it remains an open problem.In particular, in frequency-domain, the problem reduces to the analysis of the distribution of the roots of the corresponding characteristic equation, which is an entire function called characteristic quasipolynomial ), see for instance 2,3,4,5,6,7,8 .
The starting point of the present work is a property, discussed in recent studies, called Multiplicity-Induced-Dominancy, see for instance 9,10 .As a matter of fact, it is shown that multiple spectral values for timedelay systems can be characterized using a Birkhoff/Vandermonde-based approach; see for instance 11,1,12,13 .
More precisely, in previous works, it is emphasized that the admissible multiplicity of the real spectral values is bounded by the generic Polya and Szegö bound (denoted P S B ), which is nothing but the degree of the corresponding quasipolynomial (i.e the number of the involved polynomials plus their degree minus one), see for instance 14 Problem 206.2, page 144 and page 347.It is worth mentioning that such a bound were recovered using structured matrices in 1 rather than the principle argument as done in 14 .It is important to point out that 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.To the best of the authors' knowledge, the first time an analytical proof of the dominancy of a spectral value for the scalar equation with a single delay was presented and discussed in the 50s, see 15 .The dominancy property is further explored and analytically shown in scalar delay equations in 13 , then in second-order systems controlled by a delayed proportional controller is proposed in 16,17 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 18,19 where the dominancy property is parametrically characterized and proven using the argument principle.See also 19,18 which exhibit an analytical proof for the dominancy of the spectral value with maximal multiplicity for second-order systems controlled via a delayed proportional-derivative controller.Recently, in 20 it is shown that under appropriate conditions the coexistence of exactly P S B distinct negative zeros of quasipolynomial of reduced degree guarantees the exponential stability of the zero solution of the corresponding time-delay system.The dominancy of such real spectral values is shown using an extended factorization technique which generalizes the one provided in 20 .To the best of the authors's knowledge the necessary and sufficient conditions derived in the present paper as well as corresponding control strategy represent a novelty.
The present work investigates the effect of structural properties of a class of functional Vandermonde matrices and its effect on qualitative properties of a corresponding linear autonomous time-delay system of retarded type.
More precisely, the aim of this work is two-fold: firstly, it emphasizes the link between the invertibility of a class of structured functional Vandermonde matrices and the coexistence of distinct real spectral values of linear time-delay systems, which allows to recover the maximal number of distinct real spectral values that may coexist for a given time-delay system.Furthermore, if the number of coexistent real spectral values reaches the P S B , then a necessary and sufficient condition for the asymptotic stability is provided (which is equivalent to the exponential stability 21 p79), see also 22 for an estimate of the exponential decay rate for stable linear delay systems.Notice also that the constructive approach we propose, which consists in providing an appropriate factorization of a given quasipolynomial function and then to focus on the location of zeros of one of its factors, gives further insights on such a qualitative property.Namely, it furnishes the exact exponential decay rate rather than just counting the number of the quasipolynomial roots on the left-half plane as may be done by using the principle argument, see for instance 5 .
The class of dynamical systems we consider is an n−th order linear autonomous ordinary differential equations with a forcing term consisting in the delayed dependent variable.This class of systems has an applicative interest particularly in control design problems.As a matter of fact, the forcing term may be seen as a delayedinput able to stabilize the system's solutions.The idea of exploiting the delay effect in controllers design was first introduced in 23 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 24 .Furthermore, it was stressed in 25 that time-delay has a stabilizing effect in the control design.
Indeed, the closed-loop stability is guaranteed precisely by the existence of the delay.Also in 26 it is shown that a chain of n integrators can be stabilized using n distinct delay blocks, where a delay block is described by two parameters: a "gain" and a"delay".The interest of considering control laws of the form m k=1 γ k y(t − τ k ) lies in the simplicity of the controller as well as in its easy practical implementation.
From a control theory point of view, the problem we consider and the approach we propose give rise to an exponential decay assignment method using two parameters a "gain" and a"delay".Notice that the idea of using roots assignment for controller-design for time-delay system is not new.For instance, in 27 a feedback law yields a finite spectrum of the closed-loop system, located at an arbitrarily preassigned set of points in the complex plane.In the case of systems with delays in control only, a necessary and sufficient condition for finite spectrum assignment is obtained.Notice that the resulting feedback law involves integrals over the past control.In case of delays in state variables it is shown that a technique based on the finite Laplace transform leads to a constructive design procedure.The resulting feedback consists of proportional and (finite interval) integral terms over present and past values of state variables.In 28 , a similar finite pole placement for time-delay systems with commensurate delays is proposed.Feedback laws defined in terms of Volterra equations are obtained thanks to the properties of the Bezout ring of operators including derivatives, localized and distributed delays.Other analytical/numerical placement methods for retarded time-delay systems are proposed in 29,30 , see also 31 for further insights on pole-placement methods for retarded time-delays systems with proportional-integral-derivative controller-design.
The remaining paper is organized as follows.In Section 2, the problem formulation is presented and some technical lemmas are derived.Section 3 is devoted to the main results of the paper.Section 4 gives an illustrative example showing the applicative perspectives of the derived results.Some concluding remarks end the paper.
Finally, the reader finds proofs of the technical lemmas in the Appendix.

Problem settings and prerequisites
In this paper, we are interested in studying the stabilizing effect of the coexistence of the maximal number of real spectral values for the generic n-order ordinary differential equation perturbed by a forcing term depending in the delayed dependent variable under appropriate initial conditions belonging to the Banach space of continuous functions C([−τ, 0], R) which is an infinite-dimensional differential equation with a single constant delay τ > 0.
From a control theory point of view, the aim is to establish a delayed-output-feedback controller u(t) = −α y(t − τ ) able to stabilize solutions of the following control system: The particular cases of first and second order equations are considered in 20 , where a stabilizing effect of the coexistence of respectively 2 and 3 negative real roots is shown.By this paper, one generalizes such a result for arbitrary order n.
In the Laplace domain, the corresponding quasipolynomial characteristic function defined by ∆ n : One can prove that the quasipolynomial function (3) admits an infinite number of zeros, see for instance the references 2,32,33 .The study of zeros of an entire function 33 of the form (3) plays a crucial role in the analysis of asymptotic stability of the zero solution of Equation (1).Indeed, the zero solution is asymptotically stable if, and only if, all the zeros of (3) are in the open left-half complex plane 7 .

Counting quasipolynomial roots in horizontal strips
The following result was first introduced and claimed in the problems collection published in 1925 by G.
Pólya and G. Szegö.In the fourth edition of their book that are contained in the horizontal strip α ≤ Im(z) ≤ β.Assuming that Setting α = β = 0, the above theorem yields P S ≤ D + N − 1 where D stands for the sum of the degrees of the polynomials involved in the quasipolynomial function f and N designates the associated number of polynomials.
This gives a sharp bound for the number of f 's real roots.Notice that D + N − 1 corresponds to the degree of the quasipolynomial f .1 Let's investigate the coexistence of n + 1 real (negative) roots for the quasipolynomial ∆ n (., τ ).Due to the linearity of ∆ n with respect to its coefficients (a k ) 0≤k≤n−1 and α, one reduces the system ∆ n (s In the sequel, such a matrix is called structured functional Vandermonde type matrix due to its form and its structural properties.

Structured matrices appearing in the control of dynamical systems
Initially, Birkhoff and Vandermonde matrices are derived from the problem of polynomial interpolation of some unknown function g, this can be presented in a general way by describing the interpolation conditions in terms of incidence matrices, see for instance 35 .For given integers n ≥ 1 and r ≥ 0, the matrix , is called an incidence matrix if e i,j ∈ {0, 1} for every i and j.Such a matrix contains the data providing the known information about a sufficiently smooth function g : R → R. Let x = (x 1 , . . ., x n ) ∈ R n such that x 1 < . . .< x n , the problem of determining a polynomial P ∈ R[x] with degree less or equal to ι (ι + 1 = 1≤i≤n, 1≤j≤r e i,j ) that interpolates g at (x, E), i.e. which satisfies the conditions: is known as the Birkhoff interpolation problem.Recall that e i,j = 1 when g (j) (x i ) is known, otherwise e i,j = 0.
Furthermore, an incidence matrix E is said to be poised if such a polynomial P is unique.This amounts to saying that, if, n = n i=1 r j=1 e i,j then the coefficients of the interpolating polynomial P are solutions of a linear square system with associated square matrix Υ that we call Birkhoff matrix in the sequel.This matrix is parametrized in x and is shaped by E. It turns out that the incidence matrix E is poised if, and only if, the Birkhoff matrix Υ is non singular for all x such that x 1 < . . .< x n .The characterization of poised incidence matrices is solved for interpolation problems of low degrees.As a matter of fact, the problem is still unsolved for any degree greater than six, see for instance 36,37 .
Remark 2.1.Unlike Hermite interpolation problem, for which the knowledge of the value of a given order derivative of the interpolating polynomial at a given interpolating point impose the values of all the lower orders derivatives of the interpolating polynomial at that point, the Birkhoff interpolation problem release such a restriction.Thereby justifying the qualification of "lacunary" to describe the Birkhoff interpolation problem.
In the spirit of the definition of functional confluent Vandermonde matrices introduced in 38 , the following functional Birkhoff matrices were introduced in 1 Definition 2.1.The square functional Birkhoff matrix Υ is associated to a sufficiently regular function and an incidence matrix E (or equivalently an incidence vector V) and is defined by: where Analogously to the Birkhoff interpolation problem, in 1 the non degeneracy of such functional Birkhoff matrices represent a fondamental assumption for investigating the codimension of the zero spectral values for time-delay systems.
To the best of the author's knowledge, the first time the Vandermonde matrix appears in a control problem is reported in 39 p. 121, where the controllability of a finite dimensional dynamical system is guaranteed by the invertibility of such a matrix, see also 38,40 .Next, in the context of time-delay systems, the use of the standard Vandermonde matrix properties was proposed by 26,7 when controlling one chain of integrators by delay blocks.
Here we further exploit the algebraic properties of such structured matrices into a different context.

The determinant of a structured functional Vandermonde type matrix
The following auxiliary result gives explicitly the determinant of the structured functional Vandermonde type matrix (6).Its proof is presented in the Appendix.In the following we adopt the notation [x, y] t to designate the t−convex combination of the real (or complex) numbers x and y, that is: Theorem 2. For any distinct real numbers s n+1 < • • • < s 2 < s 1 , and τ > 0, the structured functional Vandermonde type matrix V n X n+1 , τ is invertible.Moreover, its determinant is which is always positive and where F τ,n : R n+1 → R * + is defined as follows: It is worth mentioning that the product in the expression of Q n given by (10) corresponds to the determinant of the standard Vandermonde matrix, see for instance 41 .

Symmetry property
The multivariate function F τ,n admits an invariance property that will be emphasized in the following Lemma which will be used in the proof of the main results.Its proof is presented in the appendix.
Lemma 2.1.For any positive delay τ the functional F τ,n is invariant for any permutation of the finite sequence , namely, for any permutation σ of X n+1 , we have For instance, for n = 2, Lemma 2.1 allows to say that for all (x, y, z Remark 2.3.The symmetry property emphasized in the above Lemma 2.1 is justified by the convexity property on the argument of the exponential kernel.Its proof which can be found in the appendix relies on simple change of coordinates.

Shifting properties
The following Lemmas exhibit some shifting properties which will be used in the proof of the main results.
Their proofs are presented in the appendix.
⊂ R be any subsequence from i=1 be a sequence of distinct real numbers.For any subsequence • Lemma 2.2 and Lemma 2.3 remain valid even if the elements of the sequence (s i ) 1≤i≤n+1 are distinct and complex.
• Under the conditions of Lemma 2.3 it is obvious that F τ,n > 0 for any τ > 0.

Factorization property
The following Lemma provides a way to factorize a given quasipolynomial function (3) having at least n distinct real roots.This will be used in the proof of the main results.
Lemma 2.4.Assume that the quasipolynomial (3) admits n distinct real roots s n < . . .< s 1 then it can be written under the following factorized form:

Main results
In this section, we provide mainly two theorems exploiting the structural properties of the considered class of functional Vandermonde matrices to give some qualitative properties of the solutions of (1).Namely, the first theorem gives conditions on the coexistence of real roots of the quasipolynomial ∆ n .The next theorem emphasizes the effect of the coexistence of such real roots on the remaining roots of ∆ n .Finally, the combination of those results allows to give some important insights on the exponential stability of the solutions of (1).

Coexistence of n + 1 real roots of ∆ n
The following Theorem 3 allows to recover P S B as a bound of the admissible number of coexisting real roots for the quasipolynomial (3), see for instance 14 .This provides an alternative constructive analytical proof based on factorization technique.Furthermore, explicit conditions on the parameters guaranteeing the coexistence of such a number of real roots is provided allowing to Vieta's-like formulas for quasipolynomials.
i) The maximal number of coexisting real roots of the quasipolynomial ii) For a fixed τ > 0, Equation (3) admits n + 1 distinct real spectral values s n+1 , s n , • • • , s 2 and s 1 with , and only if, the coefficients (a k ) 0≤k≤n−1 and α are respectively given by the following functions in τ and X n+1 = (s 1 , . . ., s n+1 ) and for 1 ≤ k ≤ n − 1 one has: and Remark 3.1.
• From a control theory point of view, let recall the design problem presented in (2), which consists in tuning the controller gain α and the delay parameter τ such that the closed-loop system's solution is asymptotically stable.In such a problem the sign of the controller gain is important with respect to the system structure.
Here one has to emphasize that in the design induced from the result we propose, the coefficient α is of alternate sign with respect to the parity of the derivative order n.
• One can observe that the asymptotic expansion of the coefficients a k allows to recover the well-know Vieta's formulas.This comes from the fact that when τ → ∞ the quasipolynomial ∆ n reduces to a polynomial of degree n.So here the important fact to emphasize is the disappearance of the (n + 1)-th real root of the quasipolynomial ∆ n .
Proof of Theorem 3. Let us start by the proof of ii) and we conclude by i).
ii) Assume that (3) admits n + 1 real spectral values . This means that the coefficients (a k ) 0≤k≤n−1 and α satisfy the linear system Thanks to the invertibility of structured functional Vandermonde type matrix V n X n+1 , τ as asserted in Theorem 2, one deals with a Cramer system with respect to the coefficients (a k ) 0≤k≤n−1 and α.So that, one easily computes these coefficients with the standard formulas allowing to ( 12), ( 13) and ( 14) respectively.In particular, the expression of α X n+1 , τ is reduced to showing the alternating sign of α.
i) Let proceed by contradiction in assuming the coexistence of n + 2 real roots of (3).We shall use the factorization of (3) derived in Lemma 2.4, that is: Since we assumed that s n+1 and s n+2 are two distinct real roots of ∆ n then one has Hence, which, using the shift property given in Lemma 2.3, proves the inconsistency in assuming the coexistence of n + 2 distinct real roots.

On qualitative properties of s 1 as a root of ∆ n
To study the stability of solutions of Equation ( 3), one needs to study the negativity as well as the dominancy of the root s 1 by using an adequate factorization of the quasipolynomial ∆ n (s, τ ) in (3).
Theorem 4. The following assertions hold: i) (Negativity) The spectral value s 1 is negative if, and only if, there exists τ * > 0 such that ii) (Dominancy) The spectral value s 1 is the spectral abscissa of Equation (1).
Proof of Theorem 4.
i) Let assume that s 1 < 0. Since the parameter a n−1 given by ( 13) is a continuous function with respect to the delay τ and thanks to the l'Hospital's rule one asserts that its asymptotic behavior is described by: lim τ →0 a n−1 (τ ) = −∞ and lim τ →∞ a n−1 (τ ) = − n k=1 s k > 0, which proves the existence of τ * > 0 such that a n−1 (τ * ) + n k=2 s k = 0. Conversely, to show the negativity of s 1 , one exploites determinant expressions provided in Theorem 2, allowing to write for any τ > 0 one has: In particular Using (18) and the positivity of τ * as well as the positivity of both F n,τ * and F n−1,τ * one concludes ii) The proof is based on the quasipolynomial factorization established in the proof of Theorem 3, more precisely, see formula (11).
To prove dominancy property for s 1 , let us assume that there exists some s 0 = ζ + jη a root of ∆ n (s, τ ) = 0 such that ζ > s 1 .This means that P (s 0 , τ ) = 0. Hence Denote by x 2,n the quantity Then, using the following estimates we get from ( 19) and Lemma 2.1 which is inconsistent.This proves the dominancy of s 1 .The proof of Theorem 4 is achieved.

Exponential stability
Note that for a linear retarded functional differential equations the exponential stability is equivalent to the uniform asymptotic stability, 21 p79.Further, for the linear autonomous retarded functional differential equations, asymptotic stability implies uniform asymptotic stability and, hence, exponential stability.Recall that Theorem 3 gives necessary and sufficient conditions for the coexistence of n + 1 real roots of (3).Theorem 4 gives a necessary and sufficient conditions for the negativity of all such real roots and asserts that the roots of (3) have necessarily (s) < s 1 .So the following result which is a direct consequence of Theorems 3-4 allows to the exponential stability.then the trivial solution of (1) is exponentially stable with s 1 as a decay rate.

Stabilizing coupled oscillators using delayed output feedback
To show the applicative potential of the obtained results, let consider as an illustrative example a system consisting in two coupled oscillators.Coupled oscillations occur when two or more oscillating systems are connected in such a way the motion energy is transferred between them.The dynamics of coupled oscillators plays an important role in a variety of systems in nature and technology, see for instance 42 and references therein.Their ability to display complex self-organized dynamical phenomena makes them an important tool to explain fundamental mechanism of emergent dynamics in coupled systems.It is known that when the coupling is small then each oscillator operates at its natural frequency and the system is then said to be incoherent.
However, when the coupling exceeds a certain threshold then the system spontaneously synchronizes.Here we consider the mechanical system of two coupled oscillators as depicted in Figure 1 and we aim to design a stabilizing delayed controller, which corresponds to oscillation quenching.Using the fundamental principle of dynamics and the standard assumption about the linearity of the damping lead to the following differential equations governing the motion of the system: where the parameters values are chosen accordingly to an experimental settings: If forcing term f acts on the system as an input and takes a proportional-minus-delay structure as suggested in 23 ; that is f (t) = −α 0 x 2 (t)−α 1 x 2 (t−τ ) and by setting ξ(t) = (x 1 (t), ẋ1 (t), x 2 (t), ẋ2 (t)) the above system writes where .
The corresponding characteristic quasipolynomial function has the form (3) and writes explicitely as follows: The aim is to establish values for controller's gains α 0 and α 1 as well as the value of the delay parameter τ enabling us to assign P S B = 5 real roots of the quasipolynomial (22) guaranteeing the exponential stability of the trivial solution of the closed-loop system as asserted in Theorem 4. To simplify the design task, we consider the case of equidistributed negative spectral values where the distance between two consecutive roots . By setting a targeted decay rate or equivalently the rightmost root, for instance at 2 for k = 2, . . ., 5 one then applies Theorem 4 and a simple parameter identification to recover the gains values α 0 ≈ 5.29, α 1 ≈ −4.54 and the delay value τ ≈ 0.81, the spectrum distribution illustrated in Figure 2.

Concluding remarks
By this paper, we investigated conditions on the coefficients of the n − th order linear ordinary differential equations with delayed-state forcing term guaranteeing the coexistence of the maximal number of real spectral values, which itself corresponds to the well-known Polya and Szegö bound for quasipolynomial's real roots.Such a bound was recovered using an analytical constructive approach.Furthermore, an easy to check criterion was provided, which allows to characterize the stabilizing effect of the coexistence of such spectral values.It is worth noting that such a configuration guarantees the exponential stability and explicitly describes the corresponding exponential decay rate.The applicative potential of the presented results is illustrated through the problem of stabilizing controller design for the system of coupled oscillators.
where 1 ≤ i ≤ n.Any other permutation of sets of indices is none other than the composition of such permutations.For example, if σ i,j , with j − i > 1, is such that It is then necessary to introduce a suitable change of variable, that switches the coefficient of s i with the coefficient of s i+1 , without affecting the other coefficients.Let for all 1 ≤ i ≤ n − 1, and the following properties are satisfied.
On the other hand, from Hence, thanks to the two properties and Lemma 2.1, we get This achieves the proof of the lemma.
Proof of Theorem 2. The calculation is done in n steps.The idea is to have at each step k in the penultimate column only "1".Then, a linear combination of the lines makes it possible to reduce the size of the determinant of a unit, as well as to recover the factors (s i − s j ), with i − j = k, using Lemmas 2.3 and 2.2.To do so, denoting by Using the mean value theorem as follows, we get using the same linear combination as above Repeating the same process as above.In the last step, only the term (s 1 − s n+1 ) remains to be recovered.The determinant is reduced to the following: Thanks to Lemma 2.3, we get which is always positive since F τ,n is positive and s i > s j .
Proof of Lemma 2.4.We start first by carrying out the following factorization of ∆ n by writing ∆ n (s, τ ) = n i=1 (s − s i ) .P n (s, τ ) with Let introduce the following coefficients which come from the standard Vieta's formulas Then by performing an Euclidean division in (24) one gets: .
Let B n be defined as follows: Then one performs the partial fractions corresponding to B n (s) The expression of P n given by ( 27) becomes: From Lemma 2.3

Figure 2 :
Figure 2: Spectrum distribution of the closed-loop system (21) using a proportional-minus-delay controller.The parameters values are given in Section 4.