S De Smet

D Corpet

M Meurillon

G Caderni

S Rohrmann

S Sasazuki

P Weijenberg

M Cantwell

P Vineis

F A Beland

E Cho

D M Klurfeld

L Le Marchand

R Sinha

M Stern

Coffee and some other hot beverages IARC Monograph Working Group Members B W Stewart (Australia)-meeting chair

Observers

D D Alexander, for the EpidStat Institute, USA; B L Booren, for the North American Meat Institute, USA; J Carretier, for the Léon Bérard Centre, France;

Carcinogenicity of consumption of red and processed meat

In October, 2015, 22 scientists from ten countries met at the International Agency for Research on Cancer (IARC) in Lyon, France, to evaluate the carcinogenicity of the consumption of red meat and processed meat. These assessments will be published in volume 114 of the IARC Monographs. 1 Red meat refers to unprocessed mammalian muscle meat-for example, beef, veal, pork, lamb, mutton, horse, or goat meat-including minced or frozen meat; it is usually consumed cooked. Processed meat refers to meat that has been transformed through salting, curing, fermentation, smoking, or other processes to enhance fl avour or improve preservation. Most processed meats contain pork or beef, but might also contain other red meats, poultry, off al (eg, liver), or meat byproducts such as blood.

Red meat contains high biologicalvalue proteins and important micronutrients such as B vitamins, iron (both free iron and haem iron), and zinc. The fat content of red meat varies depending on animal species, age, sex, breed, and feed, and the cut of the meat. Meat processing, such as curing and smoking, can result in formation of carcinogenic chemicals, including N-nitroso-compounds (NOC) and polycyclic aromatic hydrocarbons (PAH). Cooking improves the digestibility and palatability of meat, but can also produce known or suspected carcinogens, including heterocyclic aromatic amines (HAA) and PAH. High-temperature cooking by panfrying, grilling, or barbecuing generally produces the highest amounts of these chemicals. [START_REF] Alaejos | Factors that aff ect the content of heterocyclic aromatic amines in foods[END_REF][START_REF] Alomirah | Concentrations and dietary exposure to polycyclic aromatic hydrocarbons (PAHs) from grilled and smoked foods[END_REF] Depending on the country, the proportion of the population that consumes red meat varies worldwide from less than 5% to up to 100%, and from less than 2% to 65% for processed meat. The mean intake of red meat by those who consume it is about 50-100 g per person per day, with high consumption equalling more than 200 g per person per day. 4 Less information is available on the consumption of processed meat.

The Working Group assessed more than 800 epidemiological studies that investigated the association of cancer with consumption of red meat or processed meat in many countries, from several continents, with diverse ethnicities and diets. For the evaluation, the greatest weight was given to prospective cohort studies done in the general population. High quality population-based case-control studies provided additional evidence. For both designs, the studies judged to be most informative were those that considered red meat and processed meat separately, had quantitative dietary data obtained from validated questionnaires, a large sample size, and controlled for the major potential confounders for the cancer sites concerned.

The largest body of epidemiological data concerned colorectal cancer. Data on the association of red meat consumption with colorectal cancer were available from 14 cohort studies. Positive associations were seen with high versus low consumption of red meat in half of those studies, including a cohort from ten European countries spanning a wide range of meat consumption and other large cohorts in Sweden and Australia. [START_REF] Norat | Meat, fi sh, and colorectal cancer risk: the European Prospective Investigation into cancer and nutrition[END_REF][START_REF] Larsson | Red meat consumption and risk of cancers of the proximal colon, distal colon and rectum: the Swedish Mammography Cohort[END_REF][START_REF] English | Red meat, chicken, and fi sh consumption and risk of colorectal cancer[END_REF] Of the 15 informative case-control studies considered, seven reported positive associations of colorectal cancer with high versus low consumption of red meat. Positive associations of colorectal cancer with consumption of processed meat were reported in 12 of the 18 cohort studies that provided relevant data, including studies in Europe, Japan, and the USA. [START_REF] Norat | Meat, fi sh, and colorectal cancer risk: the European Prospective Investigation into cancer and nutrition[END_REF][START_REF] Oba | The relationship between the consumption of meat, fat, and coff ee and the risk of colon cancer: a prospective study in Japan[END_REF][START_REF] Bernstein | Processed and unprocessed red meat and risk of colorectal cancer: analysis by tumor location and modifi cation by time[END_REF][START_REF] Cross | A large prospective study of meat consumption and colorectal cancer risk: an investigation of potential mechanisms underlying this association[END_REF][START_REF] Chao | Meat consumption and risk of colorectal cancer[END_REF] Supporting evidence came from six of nine informative case-control studies. A meta-analysis of colorectal cancer in ten cohort studies reported a statistically signifi cant dose-response relationship, with a 17% increased risk (95% CI 1•05-1•31) per 100 g per day of red meat and an 18% increase (95% CI 1•10-1•28) per 50 g per day of processed meat. [START_REF] Chan | Red and processed meat and colorectal cancer incidence: meta-analysis of prospective studies[END_REF] Data were also available for more than 15 other types of cancer. Positive associations were seen in cohort studies and population-based casecontrol studies between consumption of red meat and cancers of the pancreas and the prostate (mainly advanced prostate cancer), and between consumption of processed meat and cancer of the stomach.

On the basis of the large amount of data and the consistent associations of colorectal cancer with consumption of processed meat across studies in different populations, which make chance, bias, and confounding unlikely as explanations, the majority of the Working Group concluded that there is sufficient evidence in human beings for the carcinogenicity of the consumption of processed meat. Chance, bias, and confounding could not be ruled out with the same degree of confi dence for the data on red meat consumption, since no clear association was seen in several of the high quality studies and residual confounding from other diet and lifestyle risk is difficult to exclude. The Working Group concluded that there is limited evidence in human beings for the carcinogenicity of the consumption of red meat.

There is inadequate evidence in experimental animals for the carcinogenicity of consumption of red meat and of processed meat. In rats treated with colon cancer initiators and promoted with low calcium diets containing either red meat or processed meat, an increase in the occurrence of colonic preneoplastic lesions was reported in three and four studies, respectively. [START_REF] Pierre | Beef meat and blood sausage promote the formation of azoxymethaneinduced mucin-depleted foci and aberrant crypt foci in rat colons[END_REF][START_REF] Pierre | Beef meat promotion of dimethylhydrazine-induced colorectal carcinogenesis biomarkers is suppressed by dietary calcium[END_REF][START_REF] Santarelli | Meat processing and colon carcinogenesis: cooked, nitrite-treated, and oxidized high-heme cured meat promotes mucin-depleted foci in rats[END_REF] The mechanistic evidence for carcinogenicity was assessed as strong for red meat and moderate for processed meat. Mechanistic evidence is mainly available for the digestive tract. A meta-analysis published in 2013 reported a modest but statistically significant association between consumption of red or processed meat and adenomas (preneoplastic lesions) of the colorectum that was consistent across studies. [START_REF] Aune | Red and processed meat intake and risk of colorectal adenomas: a systematic review and meta-analysis of epidemiological studies[END_REF] For genotoxicity and oxidative stress, evidence was moderate for the consumption of red or processed meat. In human beings, observational data showed slight but statistically significant associations with APC gene mutation or promoter methylation that were identifi ed in 75 (43%) and 41 (23%) of 185 archival colorectal cancer samples, respectively. [START_REF] Gay | Dietary, lifestyle and clinicopathological factors associated with APC mutations and promoter methylation in colorectal cancers from the EPIC-Norfolk study[END_REF] Consuming well done cooked red meat increases the bacterial mutagenicity of human urine. In three intervention studies in human beings, changes in oxidative stress markers (in urine, faeces, or blood) were associated with consumption of red meat or processed meat. [START_REF] Pierre | Calcium and alpha-tocopherol suppress cured-meat promotion of chemically induced colon carcinogenesis in rats and reduce associated biomarkers in human volunteers[END_REF] Red and processed meat intake increased lipid oxidation products in rodent faeces. [START_REF] Pierre | Beef meat and blood sausage promote the formation of azoxymethaneinduced mucin-depleted foci and aberrant crypt foci in rat colons[END_REF] Substantial supporting mechanistic evidence was available for multiple meat components (NOC, haem iron, and HAA). Consumption of red meat and processed meat by man induces NOC formation in the digestive tract. High red meat consumption (300 or 420 g/day) increased levels of DNA adducts putatively derived from NOC in exfoliated colonocytes or rectal biopsies in two intervention studies. [START_REF] Leu | Butyrylated starch intake can prevent red meat-induced O6-methyl-2-deoxyguanosine adducts in human rectal tissue: a randomised clinical trial[END_REF][START_REF] Lewin | Red meat enhances the colonic formation of the DNA adduct O6-carboxymethyl guanine: implications for colorectal cancer risk[END_REF] Few human data, especially from intervention studies, were available for processed meat. Haem iron mediates formation of NOC, and of lipid oxidation products in the digestive tract of human beings and rodents. Haem iron effects can be experimentally suppressed by calcium, supporting its contribution to carcinogenic mechanisms. Meat heated at a high temperature contains HAA. HAA are genotoxic, and the extent of conversion of HAA to genotoxic metabolites is greater in man than in rodents. Meat smoked or Overall, the Working Group classifi ed consumption of processed meat as "carcinogenic to humans" (Group 1) on the basis of sufficient evidence for colorectal cancer. Additionally, a positive association with the consumption of processed meat was found for stomach cancer.

The Working Group classified consumption of red meat as "probably carcinogenic to humans" (Group 2A). In making this evaluation, the Working Group took into consideration all the relevant data, including the substantial epidemiological data showing a positive association between consumption of red meat and colorectal cancer and the strong mechanistic evidence. Consumption of red meat was also positively associated with pancreatic and with prostate cancer.

We declare no competing interests.

Véronique Bouvard, Dana Loomis, Kathryn Z Guyton, Yann Grosse, Fatiha El Ghissassi, Lamia Benbrahim-Tallaa, Neela Guha, Heidi Mattock, Kurt Straif, on behalf of the International Agency for Research on Cancer Monograph Working Group

a research project funded by the World Cancer Research Fund, a registered charity. All other working group members declare no competing interests. Invited Specialists None Representatives A Christodoulidou, for the European Food Safety Authority (EFSA), Italy; I Margaritis, for the French Agency for Food, Environment and Occupational Health and Safety (ANSES), France; Y Totsuka, for the National Cancer Center Research Institute,


Christophe Fonte
email: christophe.fonte@univ-lorraine.fr.

Houda Meddeb
email: houda.meddeb@univ-lorraine.fr.

Michel Zasadzinski
email: michel.zasadzinski@univ-lorraine.fr.

Stabilization of polytopes of plants by their vertices

Keywords: polytope of systems, simultaneous stabilization, robust stabilization, common interlacing property

come

Stabilization of polytopes of plants by their vertices

Introduction

This paper concerns the stabilization of polytopes of SISO plants with a fixed order controller. Until now the problem of deciding of the stability or stabilizability of a polytope of linear systems with a linear time-invariant (LTI) controller stays an open question. We do not know which are the conditions so that any polytope of systems is stabilizable by a LTI controller. In this article, we show that this question is equivalent to that of the stability of the set of segments of systems linking the vertices of the polytope with a LTI controller. The issue is therefore about the stability of this set of segments with a LTI controller. Remark that this problem is also an open question. This is even an undecidable mathematical issue that may be classified as an NP-hard problem, in the sense that it is not possible to find necessary and sufficient conditions to stabilize this family of systems with a LTI controller when the number of plants to stabilize, is greater than two, see [START_REF] Blondel | NP-hardness of some linear control design problems[END_REF]. Accordingly, there exist only sufficient or necessary conditions for the robust stabilization of a polytope of systems. This paper proposes some sufficient conditions based on the Edge theorem [START_REF] Bartlett | Root locations of an entire polytope of polynomials: it suffices to check the edges[END_REF]. More precisely, the question of stabilization of a polytope of systems is treated as a simple simultaneous control design of a family of LTI systems submitted to an additional constraint on the closed-loop characteristic polynomials associated with the vertices of this polytope. Furthermore, the controller is shown to not only simultaneously stabilize the vertices of the polytope of systems, i.e. a simultaneous control purpose, but also the set of systems belonging to the polytope, i.e. a robust control purpose. That is hence a simultaneous and robust controller for the polytopes of systems. Let us review briefly the main results in simultaneous and robust control.

Concerning the simultaneous control, [START_REF] Vidyasagar | Algebraic design techniques for reliable stabilization[END_REF] and [START_REF] Vidyasagar | Control System Synthesis: A Factorization Approach[END_REF] have proved that there exits a necessary and sufficient condition to stabilize simultaneously two LTI systems. Moreover, [START_REF] Vidyasagar | Algebraic design techniques for reliable stabilization[END_REF] and [START_REF] Vidyasagar | Control System Synthesis: A Factorization Approach[END_REF] have shown 1 that this question is equivalent to the stabilization of one plant by a stable compensator. This issue called "strong stabilization" has been originally treated in [START_REF] Youla | Single-loop feedback stabilization of linear multivariable dynamical plants[END_REF]. The problem of the simultaneous stabilization of three or more LTI plants was initially tackled by [START_REF] Ghosh | Transcendental and interpolation methods in simultaneous system design[END_REF] and [START_REF] Ghosh | Transcendental and interpolation methods in simultaneous stabilization and simultaneous partial pole placement problems[END_REF]. Otherwise [START_REF] Blondel | The simultaneous stability of three plants is rationally undecidable[END_REF] has shown that there exists no tractable and complete conditions to check the simultaneous stabilizability of three or more systems. About the simultaneous stabilization of a segment of LTI systems, i.e. a continuum of LTI systems described by a parameter and its two endpoints, [START_REF] Abdallah | Controller synthesis for a class of interval plants[END_REF] has provided existence conditions which depend on the choice of the initial compensator stabilizing the extremities of the segment. For this class of system, transcendental conditions have been given by [START_REF] Ghosh | Some new results on the simultaneous stabilizability of single input, single output systems[END_REF] and [START_REF] Ghosh | Simultaneous partial pole-placement: a new approach to multimode system design[END_REF]. In [START_REF] Chockalingam | Minimality, stabilizability and strong stabilizability of uncertain plants[END_REF] and [START_REF] Chockalingam | Strong stabilizability of systems with multiaffine uncertainties and numerator denominator coupling[END_REF] the authors address the question of the simultaneous strong stabilizability of a segment of systems. They are conditions that have been stated to stabilize each element of this family with a stable compensator. That does not imply existence conditions of a single controller stabilizing the whole set of systems belonging to this segment. Conditions for the simultaneous stabilizability of a segment of systems have been given in [START_REF] Fonte | Conditions for the simultaneous stabilizability of a segment of polynomials[END_REF].

Concerning the robust control, Kharitonov's theorem and its generalization, [START_REF] Chapellat | A generalization of the Kharitonov's theorem: robust stability of interval plants[END_REF] and [START_REF] Chapellat | An alternative proof of Kharitonov's theorem[END_REF] have been at the origin of many results. Others analysis methods have been studied in the literature but those dedicated to the synthesis of robust control with polynomial methods are only a few. In [START_REF] Sideris | An efficient algorithm for checking the robust stability of a polytope of polynomials[END_REF], the zero-exclusion principle has given conditions that are used to test the stability and allow to design robust controllers. More recently, [START_REF] Henrion | Positive polynomials and robust stabilization with fixed-order controllers[END_REF], [START_REF] Henrion | An LMI condition for robust stability of polynomial matrix polytopes[END_REF], [START_REF] Karimi | Robust control of polytopic systems by convex optimization[END_REF] and [START_REF] Khatibi | Fixed-order controller design for polytopic systems using LMIs[END_REF] propose fixed-order controllers for systems with polytopic uncertainty. The methods described in these papers relied on a criterion of stability which necessitates exactly proper and strict positive realness (EP-SPR) functions. These rational functions are particular units in H ∞ which also verify the properties of the SPR functions, see [START_REF] Abdallah | Controller synthesis for a class of interval plants[END_REF], [START_REF] Bianchini | Synthesis of robust strictly positive real systems with l 2 parametric uncertainty[END_REF] and [START_REF] Patel | A note on interpolation with a strictly positive real function[END_REF]. Nevertheless, derived from [START_REF] Patel | Classification of units in H ∞ and an alternative proof Kharitonov's theorem[END_REF], it may be easily deduced that to stabilize a polytope of systems, it is only required to synthesize units in H ∞ whose Nyquist plots do not cut the negative real axis. Then the conditions of the EP-SPR functions are not necessary. Consequently, the constraint to make EP-SPR functions introduces sufficiency and conservatism in regard to the necessary and sufficient condition to stabilize a polytope of systems. In this paper, we propose a new approach to synthesize units in H ∞ for stabilizing the polytopes of systems which are not SPR functions and which remain units in H ∞ if one interchanges even (or odd) part of the numerator and denominator polynomials. The Nyquist plots of these units in H ∞ do not cut the negative real axis, see [START_REF] Patel | Classification of units in H ∞ and an alternative proof Kharitonov's theorem[END_REF].

In this manuscript, we focus on the stabilization of polytope of single input single output LTI systems. Based on the Hermite-Biehler theorem and the Edge theorem, we prove that to stabilize a polytope of systems it is sufficient to stabilize all its vertices with a simultaneous controller giving characteristic polynomials that have a common even (or odd) part. Moreover, we show that the synthesized units in H ∞ do not cut the real negative axis of the Nyquist plot. In addition a controller design for stabilizing the polytopes of systems is provided mixing polynomial techniques and linear optimization.

The paper is organized as it follows. After preliminaries in section 2, the problem of the robust stabilization of a polytope of systems is stated in section 3 by applying the Edge theorem. Then an approach to stabilize the vertices of a polytope of systems with a simultaneous compensator is given in section 4 that leads to a simple linear programming (LP) problem. The issue of the stabilization of a polytope of systems by a simultaneous controller is formulated as positivity conditions if particular pole placement constraints hold. Finally in section 5, examples are given to illustrate this framework. In the appendix, our approach is extended to the case where characteristic polynomials in closed-loop have a common odd part.

Preliminaries

Notations: The degree of a real polynomial X is denoted δ(X). P m denotes the set of real polynomials of degree m. The set of Hurwitz-stable polynomials is denoted

H. R is the real field. C is the complex field. R n ++ = { x ∈ R n : x i > 0, ∀i } and R -{0} = { x ∈ R : x < 0 or x > 0}.
The notation x > 0 with x ∈ R n means that each component of the vector x is positive. If and only if is defined as iff. The exactly proper and strict positive realness functions are denoted as EP-SPR functions. H ∞ is the set of proper stable rational functions. ∥∥ denotes the operator norm. U is the set of units in

H ∞ i.e. U ∈ U iff U ∈ H ∞ and U -1 ∈ H ∞ .
In the sequel of this section, concepts on Hurwitz-stability of a polytope of real polynomials are presented. These results are used in section 3 to study the stability of a polytope of systems. The notion of Hurwitz-stability highlighted by [START_REF] Gantmacher | The Theory of Matrices[END_REF] is approached hereafter under the angle of interlacing of real zeros.

The Hermite-Biehler's stability.

Let f ∈ P m expanded as

f (s) = f e (s 2 ) + sf o (s 2 ) = m ∑ i=0 σ i s i
where σ i are real coefficients and f e and f o denote the even and odd parts of f respectively. The polynomials f e and f o are given by

• if δ(f ) = m = 2ℓ + 1 f e (u) = σ 0 + σ 2 u + . . . + σ (m-1) u ℓ , ( 1a
)
f o (u) = σ 1 + σ 3 u + . . . + σ m u ℓ . (1b) • if δ(f ) = m = 2ℓ f e (u) = σ 0 + σ 2 u + . . . + σ m u ℓ , ( 2a
)
f o (u) = σ 1 + σ 3 u + . . . + σ (m-1) u (ℓ-1) . ( 2b
)
Let us recall the Zeros Interlacing Property.

Definition 1 ([25],

The Zeros Interlacing Property ) Let f (s) = f e (s 2 ) + sf o (s 2 ) be a real polynomial with δ(f ) = m. Assume that the roots of f e and f o are defined by the following sets

• if δ(f ) = m = 2ℓ + 1 and δ(f e ) = ℓ, δ(f o ) = ℓ roots(f e ) = {a 1 , . . . , a ℓ }, roots(f o ) = {b 1 , . . . , b ℓ }, • if δ(f ) = m = 2ℓ and δ(f e ) = ℓ, δ(f o ) = ℓ -1 roots(f e ) = {a 1 , . . . , a ℓ }, roots(f o ) = {b 1 , . . . , b ℓ-1 }.
Then f e and f o interlace iff

• the roots of f e and f o are real, negative, distinct and simple,

• the leading coefficients of f e and f o have the same sign,

• the ℓ roots of f e alternate with the ℓ or ℓ -1 roots of f o as it follows.

If

δ(f ) = m = 2ℓ + 1 then b 1 < a 1 < b 2 . . . < b ℓ < a ℓ < 0. If δ(f ) = m = 2ℓ then a 1 < b 1 < . . . < b ℓ-1 < a ℓ < 0.
Let us note that the cases m = 2ℓ + 1 or m = 2ℓ are considered indistinctly in this paper but to simplify the writing, the theoretical results are presented for only m = 2ℓ + 1 and an example is given section 5.1 for the two cases m = 2ℓ and m = 2ℓ + 1. It should be noticed that the case m = 2ℓ may be directly deduced from the case m = 2ℓ + 1.

The relationship between the zeros interlacing property of a real polynomial and the Hurwitzstability is emphasized by the Hermite-Biehler's theorem that we remind.

Lemma 1 ([25], Hermite-Biehler's theorem)

The real polynomial f (s) = f e (s 2 ) + sf o (s 2 ) is Hurwitz-stable (i.e. all the real parts of the roots of f are strictly negative) iff f e and f o verify the Zeros Interlacing Property.

Let us write the partial fraction expansion of the real rational function f o /f e with f e and f o satisfying (1). Consequently, from lemma 1, simple conditions for testing the Hurwitz-stability of f are deduced.

Lemma 2 ([26], [27])

The polynomial f (s) = f e (s 2 ) + sf o (s 2 ) with f ∈ P m is Hurwitz-stable iff the two following conditions hold.

1. f e has real, simple and negative roots (a 1 , . . . , a ℓ ).

2. The real coefficients c k (k = 0, . . . , ℓ) given by the expansion of f o as the following sum of products of factors of f e

f o (s 2 ) = c 0 f e (s 2 ) + ℓ ∑ k=1 c k f e (s 2 ) s 2 -a k . ( 3
)
are strictly positive.

Note that c 0 = 0 in the case where m = 2ℓ.

Mention that all the next developments are made in function of the writing (3) with f e given a priori. The developments in function of f o are presented in the appendix section.

Let us consider n real polynomials f i , (i = 1, . . . , n) with a a same even part f e . Then we have f i (s) = f e (s 2 ) + sf o i (s 2 ), (i = 1, . . . , n).

In association with f i , a generalization of the Zeros Interlacing Property is now given.

Definition 2 (Common even interlacing property ) Consider a finite collection of polynomials f o i , (i = 1, . . . , n) and a polynomial f e with their leading coefficients of same sign. Assume that for each pair (f o i , f e ), (i = 1, . . . , n), the roots of f o i and f e are all real, negative, distinct and simple defined by the following sets

roots(f o i ) = {b i 1 , . . . , b i ℓ }, roots(f e ) = {a 1 , . . . , a ℓ }.
The polynomials f o i have a common even interlacing f e iff the ℓ roots of each f o i alternate with the ℓ roots of f e as below

b i 1 < a 1 < b i 2 < . . . < b i ℓ < a ℓ < 0.
Remark that if the polynomials f o i , (i = 1, . . . , n) have a common interlacing f e this implies that for any jth root, we have

max i∈{1,...,n} b i j < a j < min i∈{1,...,n} b i j+1
By using the common even interlacing property given above, an extension of lemma 2 is deduced for testing the Hurwitz-stability of polynomials f i , (i = 1, . . . , n).

Corollary 1

The polynomials f i (s) = f e (s 2 ) + sf o i (s 2 ), (i = 1, . . . , n) with f i ∈ P m are Hurwitzstable iff the two following conditions hold.

1. f e has real, simple and negative roots (a 1 , . . . , a ℓ ).

2. The real coefficients c i,k (i = 1, . . . , n) and (k = 0, . . . , ℓ) given by the expansions of f o i (i = 1, . . . , n) as the following sums of products of factors of f e

f o i (s 2 ) = c i,0 f e (s 2 ) + ℓ ∑ k=1 c i,k f e (s 2 ) s 2 -a k . ( 4
)
are strictly positive.

Proof: That is a consequence of lemma 2.

Matrix formulation of f o i and f e .

Let us set f e (s 2 ) = γ e ℓ ∏ j=1 (s 2 -a j ) with γ e ∈ R -{0} . Consequently relation (4) can be simplified as it follows

f o i (s 2 ) = γ e c i,0 ℓ ∏ j=1 (s 2 -a j ) + γ e ℓ ∑ k=1 c i,k ℓ ∏ j=1 j̸ =k (s 2 -a j ) = ℓ ∑ k=0 c i,k ℓ-1 ∑ j=0 υ j,k s 2j , = V A e Ψ e i , i = 1, . . . , n. ( 5a
)
f e (s 2 ) = V H e , ( 5b
)
where

Ψ e i = [ c i,0 . . . c i,ℓ ] T (6a) V = [ 1 s 2 . . . s 2ℓ ] (6b
)

A e =      υ 0,0 υ 0,1 . . . υ 0,ℓ υ 1,0 υ 1,1 . . . υ 1,ℓ . . . . . . . . . . . . υ ℓ,0 υ ℓ,1 . . . υ ℓ,ℓ      (6c
)

H e = [ h e 0 h e 1 . . . h e ℓ ] T (6d)
Note that matrix A e only depends on the even part f e . By using the above notations, a reformulation of corollary 1 is now deduced.

Corollary 2

The polynomials f i ∈ P m , (i = 1, . . . , n) are Hurwitz-stable iff the two following conditions hold 1. f e is a polynomial with real, simple and negative roots (a 1 , . . . , a ℓ ).

2. There exist Ψ e i ∈ R ℓ+1 ++ for i = 1, . . . , n, given by (6a).

Proof: That is a consequence of the writing (4) as (5a).

Hurwitz-stability of a polytope of real polynomials.

Let S f be a convex polytope of real polynomials described by the set of n vertices f i ∈ P m as it follows

S f { f : f = n ∑ i=1 λ i f i , λ i ≥ 0, n ∑ i=1 λ i = 1 } (7)
The set of the exposed edges1 connecting two vertices f i and f j in P m of the polytope of polynomials S f is defined by S f i,j where

S f i,j { f i,j : f i,j = αf i + (1 -α)f j , α ∈ [0, 1], i = 1, . . . , n -1, j = i + 1, . . . , n } (8)
Recall theorem 1 in [START_REF] Bartlett | Root locations of an entire polytope of polynomials: it suffices to check the edges[END_REF] that shows that the stability of a polytope of real polynomials can be inferred from its exposed edges.

Theorem 1 ([2])

The polytope of real polynomials S f is Hurwitz-stable iff the set of exposed edges S f i,j are Hurwitz-stable.

The Hurwitz-stability issue of S f may be simplified by considering corollary 3 in [START_REF] Fonte | Wronskian-based tests for stability of polynomial combinations[END_REF].

Lemma 3 ([28])

Let f i ∈ P m and f j ∈ P m be two vertices of S f with their leading coefficients of same sign and with a same even part f e , then the exposed edge f i,j defined by ( 8) is Hurwitz-stable iff f i and f j are Hurwitz-stable.

Corollary 3

If the vertices f i ∈ P m , (i = 1, . . . , n) of a polytope of real polynomials S f are Hurwitz-stable with a same even part f e and if the leading coefficients of f i , (i = 1, . . . , n) have a same sign then the polytope of real polynomials S f is Hurwitz-stable.

Proof: A consequence of theorem 1 and lemma 3.

In connection with corollary 2, a reformulation of corollary 3 is now deduced.

Corollary 4 Consider a convex polytope of real polynomials S f defined by the set of vertices f i ∈ P m verifying (7) with a same even part f e . If the two following conditions hold then S f is Hurwitz-stable.

1. f e is a polynomial with real, simple and negative roots (a 1 , . . . , a ℓ ).

There exist Ψ e

i ∈ R ℓ+1 ++ (i = 1 . . . n) given by (6a).

Proof: As a consequence of corollaries 2 and 3.

Lemma 4

Let S f be a convex polytope of real polynomials described by a set of n vertices f i ∈ P m verifying [START_REF] Ghosh | Transcendental and interpolation methods in simultaneous stabilization and simultaneous partial pole placement problems[END_REF]. The number of exposed edges of

S f is n = ∑ n-1 i=1 (n -i).
Proof:

Consider f i , (i = 1, . . . , n), vertices of a convex polytope S f . Count the number of different exposed edges connected to each vertex with f i,j = f j,i . Each vertex f i , (i = 1, . . . , n -1), is connected by n -i different exposed edges to n -i vertices. In consequence, the total number of exposed edges of S f is n = ∑ n-1 i=1 (n -i).
Note that the problem of Hurwitz-stability of a polytope of real polynomials S f is significantly reduced by considering corollary 3 and lemma 4, with which we only need to verify the Hurwitzstability of n vertices of S f instead to verify the Hurwitz-stability of n exposed edges of S f , n < n if n > 3.

Robust stabilization of a polytope of plants.

Now a robust stability analysis for a polytope of systems may be carried out.

Definition 3 (A polytope of plants) The convex polytope of plants Ω

G generated by n SISO LTI systems G i = N i D -1 i (i = 1 . . . n) is defined as Ω G { G : G = N D -1 , N = n ∑ i=1 λ i N i , D = n ∑ i=1 λ i D i , λ i 0, n ∑ i=1 λ i = 1 } (9)
where N i and D i are real polynomials with δ(N i ) δ(D i ) and

D i ∈ P d .
All pairs of vertices (G i , G j ), with i = 1, . . . , n -1 and j = i + 1, . . . , n, of a polytope of systems Ω G are connected by an exposed edge G i,j . Each exposed edge is a segment of systems. The set of all segments of systems of Ω G is defined as the set Ω G i,j given below.

Definition 4 (The set of segments of systems

Ω G i,j )
The set of segments of systems Ω G i,j connecting any two vertices of the polytope of systems Ω G is defined as

Ω G i,j { G i,j : G i,j = N i,j D -1 i,j , N i,j = αN i + (1 -α)N j , D i,j = αD i + (1 -α)D j , ( 10
) α ∈ [0, 1], i = 1, . . . , n -1, j = i + 1, . . . , n } .
The objective of this paper is to propose a proper controller C = XY -1 that stabilizes the polytope of systems Ω G where X and Y are real polynomials. Consequently, the existence conditions of a compensator C satisfying Φ(G, C) = (N X + DY ) ∈ H [START_REF] Ghosh | Simultaneous partial pole-placement: a new approach to multimode system design[END_REF] are studied.

Lemma 5

The compensator C stabilizes the polytope of systems Ω G iff C stabilizes the set of segments of systems Ω G i,j .

Proof: The compensator C stabilizes the polytope of systems

Ω G iff Φ(G, C) = N X + Y D = ( n ∑ i=1 λ i (N i X + D i Y ) ) ∈ H or iff ( n ∑ i=1 λ i Φ(G i , C) ) ∈ H with λ i ≥ 0, n ∑ i=1 λ i = 1.
Hence, the compensator C stabilizes the polytope of systems Ω G iff the polytope of polynomials ϕ is Hurwitz-stable where

ϕ { Φ(G, C) : Φ(G, C) = n ∑ i=1 λ i Φ(G i , C), λ i ≥ 0, n ∑ i=1 λ i = 1 } (12)
By applying theorem 1, we know that Φ(G, C) ∈ H iff Φ(G i,j , C) ∈ H. This implies that ϕ is Hurwitz-stable iff ϕ i,j is Hurwitz-stable with ϕ i,j defined as the set of exposed edges of ϕ where

ϕ i,j { Φ(G i,j , C) : Φ(G i,j , C) = αΦ(G i , C) + (1 -α)Φ(G j , C), ( 13
) α ∈ [0, 1], i = 1, . . . , n -1, j = i + 1, . . . , n } .
Consequently, the controller C stabilizes the polytope of plants Ω G iff C stabilizes the set of segments of systems Ω G i,j defined by [START_REF] Ghosh | Some new results on the simultaneous stabilizability of single input, single output systems[END_REF]. Now to conclude this section, we can observe that by applying corollary 3 to the polytope of polynomials ϕ, we can simplify the issue of the Hurwitzness of ϕ. That is, if the vertices Φ(G i , C) (i = 1, . . . n) of the polytope ϕ are Hurwitz-stable with a same degree and with their leading coefficients of same sign and a same even part Φ(G i , C) e where

Φ(G i , C)(s) = Φ(G i , C) e (s 2 ) + sΦ(G i , C) o (s 2 )
then the polytope of polynomials ϕ is Hurwitz-stable.

Computational conditions for robust stabilization of a polytope of plants with a simultaneous compensator.

In this section, the problem of the stabilization of the polytope of systems Ω G is posed as that of the stabilization of its vertices G i with a simultaneous compensator C that guarantees a same even part Φ(G i , C) e for the closed-loop characteristic polynomials associated with G i for i = 1, . . . , n.

Theorem 2 Consider a convex polytope of plants Ω G defined by [START_REF] Abdallah | Controller synthesis for a class of interval plants[END_REF]. If there exists a simultaneous compensator C satisfying the following relations

Φ(G i , C) ∈ H, (14a) Φ(G i , C) e = Φ(G j , C) e , i = 1, . . . , n -1, j = i + 1, . . . , n. (14b)
with the leading coefficients of Φ(G i , C) of same sign, then C stabilizes Ω G .

Proof: This theorem is a consequence of lemma 5 and corollary 3.

Henceforth, the issue of the stabilization of the polytope of systems Ω G is reduced to the design of a simultaneous compensator that must satisfy the particular poles placements required by theorem 2. This yields to the following lemma.

Lemma 6 Consider a convex polytope of plants Ω G defined by [START_REF] Abdallah | Controller synthesis for a class of interval plants[END_REF]. Let V , A e , H e and Ψ e i be given by (6b), (6c), (6d) and (6a) respectively. If there exist Ψ e i ∈ R ℓ+1 ++ (i = 1, . . . , n) and a compensator C verifying the following relations

Φ(G i , C) e = V H e , i = 1, . . . , n (15a) Φ(G i , C) o = V A e Ψ e i (15b) then C stabilizes Ω G .
Proof: This lemma is a consequence of corollary 4 and theorem 2.

Notice that the simultaneous controller C proposed in lemma 6 has a fixed order since the degree of the closed-loop characteristic polynomials Φ(G i , C) associated with the vertices of the polytope of polynomials ϕ given in [START_REF] Chockalingam | Minimality, stabilizability and strong stabilizability of uncertain plants[END_REF], are chosen a priori.

Consider the simultaneous compensator C = XY -1 and the vertices G i = N i D -1 i (i = 1, . . . , n) of a polytope of systems Ω G where X, Y , N i and D i be real polynomials expressed in terms of their even and odd parts as follows

X(s) = X e (s 2 ) + sX o (s 2 ), Y (s) = Y e (s 2 ) + sY o (s 2 ) N i (s) = N e i (s 2 ) + sN o i (s 2 ), D i (s) = D e i (s 2 ) + sD o i (s 2 )
Then we obtain

Φ(G i , C) e = Γ e i C p , i = 1, . . . , n (16a) Φ(G i , C) o = Γ o i C p (16b)
with

C p (s 2 ) = [X e (s 2 ) Y e (s 2 ) X o (s 2 ) Y o (s 2 )] T Γ e i (s 2 ) = [N e i (s 2 ) D e i (s 2 ) s 2 N o i (s 2 ) s 2 D o i (s 2 )] Γ o i (s 2 ) = [N o i (s 2 ) D o i (s 2 ) N e i (s 2 ) D e i (s 2 )]
Reformulate Φ(G i , C) e and Φ(G i , C) o in a matrix form. For that consider ] T [START_REF] Sideris | An efficient algorithm for checking the robust stability of a polytope of polynomials[END_REF] with p e = δ(X e ), p o = δ(X o ), q e = δ(Y e ) and q o = δ(Y o ). Define two matrices Λ e i and Λ o i such that

C υ = [ x e 0 . .
Φ(G i , C) e = V Λ e i C υ , i = 1, . . . , n (18a) Φ(G i , C) o = V Λ o i C υ (18b)
where Λ e i and Λ o i are a rewriting of Γ e i and Γ o i according to the coefficients of polynomials

N e i , D e i , N o i and D o i .
Lemma 6 and relations [START_REF] Henrion | Positive polynomials and robust stabilization with fixed-order controllers[END_REF] permit to deduce the following lemma.

Lemma 7 Consider a convex polytope of plants Ω G defined by [START_REF] Abdallah | Controller synthesis for a class of interval plants[END_REF]. Let A e , H e and Ψ e i given by (6c), (6d) and (6a) respectively. If there exist C υ and Ψ e i ∈ R ℓ+1 ++ , (i = 1, . . . , n) satisfying the following relations

Λ e i C υ = H e i = 1, . . . , n, (19a)
Λ o i C υ = A e Ψ e i (19b) then C stabilizes Ω G .
Proof: This result is a consequence of lemma 6 and relations (18a) and (18b). Now rewrite relations ( 19)

B e = Θ e χ e ( 20
)
where

B e =             0 . . . 0 H e H e . .
.

H e             , Θ e =              A e 0 . . . 0 -Λ o 1 . . . . . . . . . . . . . . . 0 0 . . . A e -Λ o n 0 0 . . . 0 Λ e 1 0 0 . . . 0 Λ e 2 . . . . . . . . . . . . . . . 0 0 . . . 0 Λ e n              , χ e = [ ψ eT C T υ ] T , ψ e = [ Ψ e 1 T Ψ e 2 T Ψ e 3 T . . . Ψ e n T ] T ∈ R n(ℓ+1) ++ .
There is solutions ψ e ∈ R n(ℓ+1) and C υ to the equation ( 20) iff [START_REF] Ben-Israel | Generalized Inverses: Theory and Applications[END_REF].

rank

([ B e Θ e ]) = rank(Θ e ). (21)
By considering equation ( 19), the robust stabilization of the polytope of systems Ω G given by lemma 7 is now set as the issue of feasibility of a linear programming problem (LP).

Theorem 3

Let Ω G be a convex polytope of plants defined by [START_REF] Abdallah | Controller synthesis for a class of interval plants[END_REF]. Assume that (21) holds then the controller C stabilizes Ω G if there exists a solution χ e to the linear programming (LP) problem given by [START_REF] Karimi | Robust control of polytopic systems by convex optimization[END_REF].

Proof: Since (20) is a rewritten of the relations [START_REF] Henrion | An LMI condition for robust stability of polynomial matrix polytopes[END_REF], then theorem 3 is equivalent to lemma 7.

This LP problem can be expressed as a LP computational procedure. Readily available, linear programming software allows to solve this type of system [START_REF] Karimi | Robust control of polytopic systems by convex optimization[END_REF] in an efficient way. This possibility is demonstrated through some numerical examples that are reported in the section 5.

After focusing initially on the controller design for stabilizing a polytope of systems, we are again going to review some literature results. Derived from [START_REF] Patel | Classification of units in H ∞ and an alternative proof Kharitonov's theorem[END_REF], an equivalent condition to the one given in lemma 5 may be formulated.

Corollary 5

To stabilize a polytope of plants Ω G with a LTI compensator, it is necessary and sufficient that the units in H ∞ formed by the ratio of every two closed-loop characteristic polynomials associated with the vertices of the polytope of systems have the modulus of their argument less than 180 • .

Proof: This necessary and sufficient condition is a direct consequence deduced from lemma 5 and theorem 2.3 in [START_REF] Patel | Classification of units in H ∞ and an alternative proof Kharitonov's theorem[END_REF].

The condition given in corollary 5 may be rewritten as

arg ( Φ(G i , C)(jw) Φ(G j , C)(jw) ) < 180 o , i = 1, . . . , n -1, j = i + 1, . . . , n (22)
with

Φ(G i , C) Φ(G j , C) ∈ U (23)
The approach outlined here, permits to synthesize units in H ∞ which remain units in H ∞ if one interchanges even (or odd) part of the numerator and denominator polynomials. The rational functions Φ(G i , C) Φ(G j , C) may have the modulus of their argument more than 90 • , see [START_REF] Patel | Classification of units in H ∞ and an alternative proof Kharitonov's theorem[END_REF]. Consequently the framework described in this article for the stabilization of polytopes of systems allows to envisage other solutions than those allowed by [START_REF] Henrion | Positive polynomials and robust stabilization with fixed-order controllers[END_REF].

Illustrative examples

Several examples are now provided to illustrate the applicability of our approach. Since the interval plants are an important class of systems considered in many applications, we begin in the sections 5.1 and 5.2 to apply our approach on these particular uncertain systems. In the last part, section 5.3, an application is given for a polytope of systems that does not belong to the class of interval systems. That is a system whose the polynomial coefficients are not bound by some minimum and maximum values. On the other hand these coefficients can be dependant.

Although our analysis and our conditions of robust stability given in the sections 3 and 4 do not fall within the framework of the so-called "16-plant theorem", see [START_REF] Barmish | New Tools for Robustness of Linear Systems[END_REF], [START_REF] Bhattacharyya | Robust Control, The Parametric Approach[END_REF], we show by these examples that we can use our approach to stabilize these polytopes of systems.

A detailed example

To illustrate this design as defined in section 4, we propose to stabilize an interval plant G with a proper LTI compensator where

G(s) = s 2 + n 0 2s 2 + d 1 s + 1 ( 24
)
with n 0 ∈ [ϵ min , ϵ max ] and

d 1 ∈ [η min , η max ].
First of all, if we consider a simple gain K as compensator, the characteristic equation of the closed-loop system is

Φ( G, K)(s) = (2 + K)s 2 + d 1 s + (1 + Kn 0 )
We observe that a constant gain K cannot stabilize a such system G if d 1 has not always the same sign.

Before developing our approach to synthesize a robust controller stabilizing G, the following result is stated.

Proposition 1: Consider the interval plant G given by [START_REF] Patel | Classification of units in H ∞ and an alternative proof Kharitonov's theorem[END_REF]. For all n 0 ∈ [ϵ min , ϵ max ] and d 1 ∈ [η min , η max ], there exists a polytope of plants Ω G defined by its vertices

G i = N i D -1 i (i = 1 . . . 4) where G 1 (s) = s 2 + ϵ min 2s 2 + η min s + 1 , G 2 (s) = s 2 + ϵ min 2s 2 + η max s + 1 (25a) G 3 (s) = s 2 + ϵ max 2s 2 + η min s + 1 , G 4 (s) = s 2 + ϵ max 2s 2 + η max s + 1 (25b) such that G ∈ Ω G . Proof: Let G = N D -1 with N = 4 ∑ i=1 λ i N i , D = 4 ∑ i=1 λ i D i , 4 ∑ i=1 λ i = 1 and λ i ≥ 0,
Or equivalently, we have

N (s) = s 2 + (λ 1 + λ 2 )ϵ min + (λ 3 + λ 4 )ϵ max D(s) = 2s 2 + (λ 1 + λ 3 )η min s + (λ 2 + λ 4 )η max s + 1 with 4 ∑ 1 λ i = 1 and λ i ≥ 0, i = 1 . . . 4
Using the above relations, we can define the scalars α j ∈ [0, 1] (j = 1, 2) that satisfy the following relations

       α 1 = λ 1 + λ 2 1 -α 1 = λ 3 + λ 4 α 2 = λ 1 + λ 3 1 -α 2 = λ 2 + λ 4 (26)
Now let us show that for any α j ∈ [0, 1] (j = 1, 2), there exist λ i (i = 1 . . . , 4) with 0 ≤ λ i and 4 ∑ i=1 λ i = 1. In terms of (26), let us remark that for any α j ∈ [0, 1] (j = 1, 2), we can always choose 0 ≤ λ 1 ≤ 1 such that the following inequalities hold.

   0 ≤ λ 1 ≤ α 1 ≤ 1 0 ≤ λ 1 ≤ α 2 ≤ 1 -1 ≤ α 1 + α 2 -1 ≤ λ 1 ≤ 1 (27)
From relations ( 26) and ( 27), we deduce that for any α j ∈ [0, 1] (j = 1, 2), there exist λ 2 ≥ 0, λ 3 ≥ 0 and λ 4 ≥ 0 with ∑ 4 1 λ i = 1 that the following relationships hold.

   0 ≤ λ 2 = α 1 -λ 1 ≤ 1 0 ≤ λ 3 = α 2 -λ 1 ≤ 1 0 ≤ λ 4 = 1 -α 1 -α 2 + λ 1 ≤ 1 (28)
Then we can rewrite G = N D -1 with

N (s) = s 2 + ( α 1 ϵ min + (1 -α 1 ) ϵ max ) D(s) = 2s 2 + ( α 2 η min + (1 -α 2 ) η max ) s + 1 α 1 ∈ [0, 1], α 2 ∈ [0, 1]
Notice that for any α j ∈ [0, 1], (j = 1, 2), there exist always scalars λ i , (i = 1, 4) such that 0 ≤ λ i ≤ 1, (i = 1, . . . , 4) that satisfy ( 27) and ( 28) by taking λ 1 = min(α 1 , α 2 ). Now consider G defined by [START_REF] Patel | Classification of units in H ∞ and an alternative proof Kharitonov's theorem[END_REF] with ϵ min = 1, ϵ max = 2, η min = -1, η max = 1. By applying Proposition 1, we know that if a compensator C exists that stabilizes the polytope of plants Ω G defined by its four vertices [START_REF] Gantmacher | The Theory of Matrices[END_REF] then this compensator stabilizes G. We can note that for stabilizing the polytope of systems Ω G , it is sufficient to stabilize simultaneously only the four vertices (25) instead of six segments of systems as in the case of lemma 5. Consequently, to synthesize a robust controller that stabilizes Ω G , theorem 3 and theorem 5 will be considered successively.

Firstly, let us solve this robust stabilization problem guaranteeing that the closed-loop characteristic polynomials associated with the four vertices of Ω G have the same even part. To that goal, we can take a controller of the following form Now solve the same problem that the one that we have just treated but by considering theorem 5. With this formulation detailed in the appendix section, we can compute a controller C o guaranteeing that the closed-loop characteristic polynomials associated with the four vertices of Ω G have the same odd part. To that goal, we can consider a controller of the form

C o (s) = x 2 s 2 + x 0 y 3 s 3 + y 1 s
The odd part of the characteristic polynomials can be fixed by the two negative distinct real roots b 1 = -2.3704, b 2 = -0.

Φ(G 1 , C o )(s) = 54s 5 + 194s 4 + 155s 3 + 177s 2 + 64s + 20 Φ(G 2 , C o )(s) = 54s 5 + 194s 4 + 155s 3 + 398s 2 + 64s + 40 Φ(G 3 , C o )(s) = 54s 5 + 248s 4 + 155s 3 + 305s 2 + 64s + 20 Φ(G 4 , C o )(s) = 54s 5 + 248s 4 + 155s 3 + 526s 2 + 64s + 40 Consider U o i,j = Φ(G i , C o ) Φ(G j , C o ) and U e i,j = Φ(G i , C e ) Φ(G j , C e ) with i = 1, . . . , 3, j = i + 1, . . . , 4.
Let us plot the modulus of the argument of the rational functions U i,j (jw) e and U i,j (jw) o , see figures 1 and 2. We observe that all these units have the modulus of their argument less than 180 • and some of these units have modulus of their argument more than 90 • . Hence, they are not all EP-SPR functions.

The conditions given in the theorems 2 and 4 are all satisfied then C e and C o stabilize the polytope of systems Ω G . Accordingly, C e and C o stabilize G.

A numerical benchmark

In this part, a collection of interval plants are now presented showing the efficiency of our numerical design when the order of these systems grows. Consequently, the number of constraints treated in these configurations by the linear programming procedure becomes important. It is about pointing out that in spite of this numerical complexity, the algorithm can find feasible solutions. The allowed maximum uncertainty is defined by a parameter τ max which is linked to the size of the uncertainty intervals with 0 ≤ τ max . To simplify the presentation, only simultaneous compensators guaranteeing a same even part for the characteristic polynomials are given.

Example 1

A similar example to the one given previously in section 5.1 but by considering a polytope of systems of 8 vertices instead of 4.

System G(s) = n 2 s 2 + n 0 2s 2 + d 1 s + 1 n 0 ∈ [1.25 -0.25τ 0 , 1.25 + 0.25τ 0 ], n 2 ∈ [1.25 -0.25τ 2 , 1.25 + 0.25τ 2 ] d 1 ∈ [-τ 1 , τ 1 ] with τ i, (i=0,...,2) ∈ [0, τ max ]
From G is deduced a polytope of plants Ω G defined by the set of its vertices G i , (i = 1 . . . 8) associated with the 8 combinations of the bounds of the uncertainty intervals.

Controller 1

A stabilizing controller C e allowing a maximum uncertainty τ max = 1

C e (s) = 81s 1.85s 2 + 8
The roots of the even part of the 8 characteristic polynomials are a i = {-4.3243, -0.5}.

Controller 2

A stabilizing controller C e allowing a maximum uncertainty τ max = 1.5 C e (s) = 13.4277s 3 + 1.1379s 0.28s 4 + 0.65s 2 + 0.05

The roots of the even part of the 8 characteristic polynomials are a i = {-2.2418, -0.5, -0.0797}.

Example 2

A 3rd order unstable system is considered with four uncertain coefficients.

System

G(s) =

s 3 + n 2 s 2 + n 0 d 3 s 3 -2s 2 + d 1 s + 1 n 0 ∈ [1 -0.5τ 0 , 1 + 0.5τ 0 ], n 2 ∈ [3 -0.5τ 2 , 3 + 0.5τ 2 ] d 1 ∈ [-τ 1 , τ 1 ], d 3 ∈ [1 -0.5τ 3 , 1 + 0.5τ 3 ] with τ i, (i=0,...,3) ∈ [0, τ max ]
From G is deduced a polytope of plants Ω G defined by the set of its vertices G i , (i = 1 . . . 16) associated with the 16 combinations of the bounds of the uncertainty intervals.

Controller

A stabilizing controller C e allowing a maximum uncertainty τ max = 1 C e (s) = 4.5049s 3 + 0.44035s 1.7524s 4 + 0.1914s 2 + 0.0017

The roots of the even part of the 16 characteristic polynomials are a i = {-1.7, -0.1, -0.01}

Example 3

A 4th order unstable system is treated with five uncertain coefficients.

System G(s) = n 4 s 4 + s 3 + n 2 s 2 + n 0 0.1s 4 + d 3 s 3 -2s 2 + d 1 s + 1 n 0 ∈ [0.75 -0.5τ 0 , 0.75 + 0.5τ 0 ], n 2 ∈ [2.25 -0.5τ 2 , 2.25 + 0.5τ 2 ] n 4 ∈ [0.11 -0.1τ 4 , 0.11 + 0.1τ 4 ], d 1 ∈ [-τ 1 , τ 1 ], d 3 ∈ [1.5 -τ 3 , 1.5 + τ 3 ] with τ i, (i=0,...,4) ∈ [0, τ max ]
From G is deduced a polytope of plants Ω G defined by the set of its vertices G i , (i = 1 . . . 32) associated with the 32 combinations of the bounds of the uncertainty intervals.

Controller

A stabilizing controller C e allowing a maximum uncertainty τ max = 0.5 C e (s) = 29.23s 3 + 395.65s 10s 4 + 138.7s 2 + 10.5

The roots of the even part of the 32 characteristic polynomials are a i = {-15, -7, -1, -0.1}

Example 4

This is a 5th order unstable system with four uncertain coefficients.

System

G(s) = 6s 3 + n 2 s 2 + 22s + n 0 -0.2s 5

-3s 4 + d 3 s 3 + 22s 2 + d 1 s -0.5 n 0 ∈ [-3 -τ 0 , -3 + τ 0 ], n 2 ∈ [3 -τ 2 , 3 + τ 2 ] d 1 ∈ [-11 -τ 1 , -11 + τ 1 ], d 3 ∈ [-16 -τ 3 , -16 + τ 3 ] with τ i, (i=0,...,3) ∈ [0, τ max ]
From G is deduced a polytope of plants Ω G defined by the set of its vertices G i , (i = 1 . . . 16) associated with the 16 combinations of the bounds of the uncertainty intervals.

Controller

A stabilizing controller C e allowing a maximum uncertainty τ max = 1.7 C e (s) = 50.612s 3 + 65.161s -4.6063s 4 -30.132s 2 -40.469

The roots of the even part of the 16 characteristic polynomials are a i = {-17.275, -3.1836, -0.68588, -0.038819}

Example 5

This is a 5th order unstable system with five uncertain coefficients.

System

G(s) = -2.5s 5 + n 4 s 4 -

11s 3 + n 2 s 2 + 10s + n 0 -0.2s 5 -31s 4 + d 3 s 3 -81s 2 + d 1 s -2 n 0 ∈ [30.5 -2.5τ 0 , 30.5 + 2.5τ 0 ], n 2 ∈ [24 -2τ 2 , 24 + 2τ 2 ] n 4 ∈ [2.15 -0.85τ 4 , 2.15 + 0.85τ 4 ], d 1 ∈ [-72.5 -4.5τ 1 , -72.5 + 4.5τ 1 ], d 3 ∈ [-4.1 -1.4τ 3 , -4.1 + 1.4τ 3 ] with τ i, (i=0,...,4) ∈ [0, τ max ]
From G is deduced a polytope of plants Ω G defined by the set of its vertices G i , (i = 1 . . . 32) associated with the 32 combinations of the bounds of the uncertainty intervals. The roots of the even part of the 32 characteristic polynomials are a i = {-28.8, -5.32, -1.57, -0.42, -0.034}

Comparison with the robust stabilization approach given in [32] and [33].

Well known H ∞ methods [START_REF] Glover | Robust stabilization of normalized coprime factor plant descriptions with H ∞ -bounded uncertainty[END_REF] and [START_REF] Mcfarlane | A loop-shaping procedure using H ∞ synthesis[END_REF] are used to establish robust stabilizability conditions for a family of uncertain plants and also to synthesize controllers that would stabilize the whole family. These regulators synthesis techniques can be implemented on the examples 1 to 5, in order to evaluate robustness of our approach in a simple way. For that, an interval system is associated to a family of systems with a central model G 0 also called nominal model defined through the average values of parameters of the interval model. The plant transfer function is written with a coprime factorization G 0 = T 0 S -1 0 where T 0 , S 0 are normalized such that

T 0 (s)T 0 (-s) + S 0 (s)S 0 (-s) = 1.
Then a family of plants G is characterized by H ∞ bounded perturbations to the numerator T 0 and denominator S 0 of the normalized coprime factorization of the nomimal system G 0 as

F G = { G : G = T S -1 , T = T 0 + ∆ T , S = S 0 + ∆ S : [ ∆ S ∆ T ] ∞ ≤ β, β > 0 } (29)
where ∆ T , ∆ S are stable unknown transfer functions representing the uncertainty. Then F G is robustly stabilizable by a single linear time invariant controller

C iff (see Lemma 3.1 in [32]) inf C [ C(1 -G 0 C) -1 S -1 0 (1 -G 0 C) -1 S -1 0 ] ∞ ≤ 1 β ( 30
)
where the infinimum is chosen over all compensators C that stabilizes G 0 . The problem of robust stabilization of normalized coprime factor plant descriptions with H ∞ bounded uncertainty is formulated as

ρ(G 0 , G) < β max (31)
where β max denotes a maximum β and ρ(G 0 , G) is the gap-metric between G 0 and G as defined in [START_REF] Georgiou | Optimal robustness in the gap metric[END_REF] and [START_REF] Georgiou | On the computation of the gap metric[END_REF] with

ρ(G 0 , G) = max ( inf Q∈H∞ [ S 0 T 0 ] - [ S T ] Q ∞ , inf Q∈H∞ [ S T ] - [ S 0 T 0 ] Q ∞ )
The issue of finding β max can be solved via standard H ∞ optimization techniques (see [START_REF] Glover | Robust stabilization of normalized coprime factor plant descriptions with H ∞ -bounded uncertainty[END_REF], [START_REF] Mcfarlane | A loop-shaping procedure using H ∞ synthesis[END_REF] and [START_REF] Georgiou | Optimal robustness in the gap metric[END_REF]). So, as for the gap metric, any plant G at a distance less than β from G 0 (β < β max ) will be stabilized by any compensator C stabilizing the nominal system with a stability margin of β.

In the case of our approach, an interval plant is represented by a polytope of systems Ω G defined by its vertices G i (i = 1, . . . , n). It is thus possible to measure for each vertex of the polytope the "distance" to the central plant G 0 by the gap metric. Denote by ρ m = min i=1,...,n ρ(G 0 , G i ) and ρ M = max i=1,...,n ρ(G 0 , G i ), the maximum and the minimum of the gap-metrics between G 0 and

G i (i = 1, . . . , n).
Consequently, for each interval plant, the radius of the maximal gap ball β max around the central plant G 0 associated with the family F G , may be compared to the distance measured by the gap metric between G 0 and the vertices G i (i = 1, . . . , n) associated with the polytope Ω G . This comparison between the H ∞ method given in [START_REF] Glover | Robust stabilization of normalized coprime factor plant descriptions with H ∞ -bounded uncertainty[END_REF] and [START_REF] Mcfarlane | A loop-shaping procedure using H ∞ synthesis[END_REF] and our approach is presented for the examples 1 to 5 in table 1 hereafter where 20

• n is the number of vertices G i (i = 1, . . . , n) associated with the polytope of plants Ω G , In table 1, we can observe for every example that ρ M > β max . We deduce that the robust stabilization approach developed in [START_REF] Glover | Robust stabilization of normalized coprime factor plant descriptions with H ∞ -bounded uncertainty[END_REF] and [START_REF] Mcfarlane | A loop-shaping procedure using H ∞ synthesis[END_REF] does not allow to take into account an uncertainty level as large as the one considered in the benchmark section 5.2. These results are due to the lost of the uncertainty structure with representation [START_REF] Ben-Israel | Generalized Inverses: Theory and Applications[END_REF]. This design limitation of H ∞ methods in regard to the uncertainty structure provides conservatism.

• n ρ is the number of vertices G i (i = 1, . . . , n) satisfying ρ(G 0 , G i )) ≤ β max . Section G 0 (s) β max n ρ m ρ M n ρ 5.2.
Finally, we conclude our numerical benchmark of section 5.2 by pointing out that like in the literature on the stabilization of interval plants, see [START_REF] Henrion | Positive polynomials and robust stabilization with fixed-order controllers[END_REF], [START_REF] Henrion | An LMI condition for robust stability of polynomial matrix polytopes[END_REF], [START_REF] Karimi | Robust control of polytopic systems by convex optimization[END_REF] and [START_REF] Khatibi | Fixed-order controller design for polytopic systems using LMIs[END_REF], our approach is based on sufficient conditions which generate some inevitable conservatism. In other words, if the linear programming problem defined by equation [START_REF] Karimi | Robust control of polytopic systems by convex optimization[END_REF] has no solution, it does not mean that there is no robust controller that stabilizes the polytope of plants. In our examples, the limit of stability is reached for a given controller when τ i, (i=0,...,n) = τ max where n is the number of uncertainty coefficient for a given interval plant. It does not imply that there is no robust stabilizing controllers if τ i, (i=0,...,n) > τ max . This is illustrated by example 1 where a 2nd order controller is get with τ max = 1 while a 4th order controller permits τ max = 1.5. The issue of τ max is the one of the compromise between the controller order and the robustness.

An example of a polytope of systems not belonging to the family of interval plants

Consider now a polytope of plants Ω G defined by the 8 vertices G i (i = 1, . . . , 8) given in table 2. That is a system whose the polynomial coefficients are not bound by some minimum and maximum values. On the other hand these coefficients can be dependant. The set of all convex combinations of vertices G i (i = 1, . . . , 8) forms a convex hull. Note that the systems G i (i = 1, . . . , 7) are unstable plants.

First of all, we show that this polytope Ω G is not stabilizable by a first order controller. For this purpose, consider the vertex G 1 of Ω G of the form [START_REF] Glover | Robust stabilization of normalized coprime factor plant descriptions with H ∞ -bounded uncertainty[END_REF] with K = 1, ã = 1 and b = 1 and the compensator [START_REF] Mcfarlane | A loop-shaping procedure using H ∞ synthesis[END_REF].

G 1 (s) = 1 s 3 -s 2 + s + 1 G 2 (s) = -0.
G (K,ã, b) (s) = K ãs 3 -ãs 2 + ãs + b , {K, ã, b} ∈ R 3 (32)
C(s) = x 1 s + x 0 y 1 s + y 0 (33)
The resulting closed-loop polynomial is Φ(G (K,ã, b) , C)(s) = ãy 1 s 4 + ã(y 0 -y 1 )s 3 + ã(y 1 -y 0 )s 2 + ( by 1 + ãy 0 + Kx 1 )s + ( by 0 + Kx 0 )

Let us remark that this characteristic equation will never be Hurwitz-stable because there are two coefficients of opposite sign, the coefficient of degree 3 and the coefficient of degree 2. Consequently, the polytope of plants Ω G can not be stabilized with a first order compensator. Therefore, a controller of higher order must be tested. A second order controller of the following general form

C(s) = x 2 s 2 + x 1 s + x 0 y 2 s 2 + y 1 s + y 0 (34)
is already considered to stabilize this polytope of plants Ω G .

To solve this robust stabilization problem, we desire that the closed-loop characteristic polynomials Φ(G i , C) (i = 1, . . . 8) have a same odd part. At this end, we select the two following negative distinct real roots, b 1 = -62.203 and b 2 = -2.7973. Then we find C(s) = 218s 2 + 82s -6 s 2 + 14s + 78 [START_REF] Georgiou | On the computation of the gap metric[END_REF] With this simultaneous compensator [START_REF] Georgiou | On the computation of the gap metric[END_REF], the eight closed-loop characteristic polynomials associated with the eight vertices of Ω G are Hurwitz-stable and have a same odd part.

Discussion

The considered numerical examples are not easy to stabilize because simple compensators like constant gain cannot stabilize these uncertain systems. Moreover unstable poles and unstable zeros of these uncertain plants may change which would constitute additional difficulties. It is also useful to observe that in literature this same problem requires the solution of bilinear matrix inequalities (BMIs) which lead to nonconvex optimization problems (see for example [START_REF] Henrion | Positive polynomials and robust stabilization with fixed-order controllers[END_REF], [START_REF] Henrion | An LMI condition for robust stability of polynomial matrix polytopes[END_REF]) while our approach generates a simple linear programming (LP) problem for which there are efficient solvers with low computational costs. This basic computational procedure founded on linear programming permits to find robust controllers of reasonable order in many cases when the plant order and the number of vertices of the polytope grow. These simulations also show that the order of the compensators is not higher than those used in the usual applications of automatic control: it is at most of the same order that the plant order plus one.

Conclusion

In this paper, a new approach for stabilizing a polytope of SISO plants has been proposed which yields to a simple controller design for these systems. The Hermite-Biehler theorem and Edge theorem have been the backbones of this framework. An alternative would be to use the Hermite-Fujiwara criterion, see [START_REF] Jury | Inners and stability of dynamic systems[END_REF] and [START_REF] Henrion | Algebraic approach to robust controller design: a geometric interpretation[END_REF] instead of the Hermite-Biehler criterion that we apply to stabilize the closed-loop system. So we would have a feedback that would not be based on the common interlacing property but on the properties of the Hermite-Fujiwara matrices. Thus [START_REF] Jury | Inners and stability of dynamic systems[END_REF] has proved that a polynomial f is Hurwitz-stable iff the Hermite-Fujiwara matrix associated with f is positive definite. Thereafter this property has been extended to define the Hurwitz-stability of a linear combination of several polynomials, see [START_REF] Henrion | Algebraic approach to robust controller design: a geometric interpretation[END_REF]. Assuming that we consider the Hermite-Fujiwara criterion instead of the Hermite-Biehler criterion, our approach would remain unchanged until theorem 2. Then the conditions (15) of lemma 6 would be modified in order to rewrite the closed-loop characteristic polynomials of the vertices of the polytope with the Hermite-Fujiwara matrices. In this situation, we would get bilinear matrix inequalities (BMIs) conditions which would give a non-convex optimization problem. From the literature, we know that the problem of checking the solvability of a BMI system is NP-hard, see [START_REF] Toker | On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback[END_REF]. It is the reason why, the BMIs are converted to linear matrix inequalities (LMIs) with a constraint rank. However this alteration in rank-one LMI optimization problem produces an heuristic algorithm without guaranteeing to find solutions when they exist. For example, in the case of the 2nd order system associated with a polytope of plants of 8 vertices studied in section 5.2, no solution has been found with the Yalmip and SeDuMi solvers for a fourth-order controller by using the Hermite-Fujiwara's criterion. Moreover, this approach generates a higher numerical complexity than the one developed with the Hermite-Biehler criterion. This complexity is linked to the size and to the number of Hermite-Fujiwara matrices. For example, the second order system regulated with a fourth-order controller given in the section 5.2 involves to calculate two hundred Hermite-Fujiwara's matrices. To summarize, use of the Hermite-Biehler criterion has reduced the stabilization problem of a polytope of systems to a linear programming problem for which there exists efficacious solvers. Our approach has been illustrated by examples showing that the controllers have a reasonable order in spite of the numbers of vertices of the polytope of plants to stabilize and the uncertainty regarded.

Note that these results could be extended in several ways. Notably we could study the polytopes of plants with a similar controller design than the one given in this paper but in discrete-time or for other stability domains than the unit circle or than the left half-plane of the complex plane. Finally, we could research conditions to stabilize a polytope of systems with a compensator of any order.

7 Appendix: Stabilization of Ω G with a same Φ(G i , C) o

In the previous sections has been proposed a design of a controller C that guarantees the same even part for the closed-loop characteristic polynomials associated with the vertices of a polytope of systems Ω G . In this appendix are presented similar results but for guaranteeing a same odd part Φ(G i , C) o for the closed-loop characteristic polynomials associated with the vertices of Ω G . Based on this property of common odd interlacing, a corollary is deduced for testing the Hurwitzstability of f i (s) = f e i (s 2 ) + sf o (s 2 ), (i = 1 . . . , n).

Corollary 6 (see corollary 1)

The polynomials f i (s) = f e i (s 2 ) + sf o (s 2 ), (i = 1, . . . , n) with f i ∈ P m are Hurwitz-stable iff the two following conditions hold.

f o (s 2 ) = V H o (37b)
where

Ψ o i = [ c i,0 . . . c i,ℓ ] T (38a) V = [ 1 s 2 . . . s 2ℓ ] (38b) A o =      υ 0,0 υ 0,
H o = [ h o 0 , h o 1 . . . h o ℓ ] T (38d)
A reformulation of corollary 6 is now deduced.

Corollary 7 (see corollary 2)

The polynomials f i ∈ P m (i = 1, . . . , n) are Hurwitz-stable iff the two following conditions hold.

1. f o has real, simple and negative roots (b 1 , . . . , b ℓ ).

2. There exist Ψ o i ∈ R ℓ ++ for i = 1, . . . , n, given by (38a).

The Hurwitz-stability issue of a polytope of real polynomials S f defined by ( 7) may be simplified by considering corollary 3 in [START_REF] Fonte | Wronskian-based tests for stability of polynomial combinations[END_REF].

Lemma 8 (see lemma 3) Let f i and f j be in P m , two vertices of S f with their leading coefficients of same sign and with a same odd part f o then the exposed edge f i,j defined by ( 8) is Hurwitz-stable iff f i and f j are Hurwitz-stable.

Hence, we write.

Corollary 8 (see corollary 3) If the vertices f i ∈ P m (i = 1, . . . , n), of a polytope of real polynomials S f are Hurwitz-stable with a same odd part f o and if the leading coefficients of f i have a same sign then the polytope of real polynomials S f is Hurwitz-stable.

That implies theorem below. Hence the following result by considering C υ given by [START_REF] Sideris | An efficient algorithm for checking the robust stability of a polytope of polynomials[END_REF] and Λ e i and Λ o i defined by [START_REF] Henrion | Positive polynomials and robust stabilization with fixed-order controllers[END_REF]. There is solutions ψ o and C υ to (41) iff [START_REF] Ben-Israel | Generalized Inverses: Theory and Applications[END_REF].

rank

([ B o Θ o ]) = rank(Θ o ). ( 42
)
By considering equation (40), the robust stabilization of Ω G given by lemma 9 is now set as the issue of feasibility of a linear programming problem. Thus, we can deduce the next theorem.

Theorem 5 (see theorem 3) Let Ω G be a convex polytope of plants defined by [START_REF] Abdallah | Controller synthesis for a class of interval plants[END_REF]. Assume that (42) holds then the controller C stabilizes Ω G if there exists a solution χ o to the linear programming (LP) problem given by (41).

C e (s) = x 1 s y 2 s 2 + y 0

 0 We choose two negative distinct real roots:a 1 = -0.5, a 2 = -4. Then we find ψ = [3.06 0.01 2.04 1.03 3.06 1.01 2.04 2.03] and we get C e (s) = 3.57s 0.5s 2 + 2 The closed-loop characteristic polynomials associated with the four vertices of Ω G are Hurwitzstable where Φ(G 1 , C e )(s) = s 4 + 3.07s 3 + 4.5s 2 + 1.57s + 2 Φ(G 2 , C e )(s) = s 4 + 3.07s 3 + 4.5s 2 + 5.14s + 2 Φ(G 3 , C e )(s) = s 4 + 4.07s 3 + 4.5s 2 + 5.57s + 2 Φ(G 4 , C e )(s) = s 4 + 4.07s 3 + 4.5s 2 + 9.14s + 2

5 .

 5 This yields to ψ = [3.5926 0.198 6.8363 3.5926 1.0941 1.8476 4.5926 0.6980 6.8363 4.5926 1.5941 1.8476] and C o (s) = 221s 2 + 20 27s 3 + 64s The closed-loop characteristic polynomials associated with the four vertices of the polytope Ω G are Hurwitz-stable where

Fig 1 :

 1 Fig 1: Plot of arg U o i,j (jw) in regard to C o Fig 2: Plot of arg U e i,j (jw) in regard to C e

Controller A stabilizing controller C e allowing a maximum uncertainty τ max = 1 C

 1 e (s) = 50.612s 5 + 65.161s 3 -53.913s -4.6063s 6 -30.132s 4 -40.469s 2 -28.229

1 .

 1 f o has real, simple and negative roots (b 1 , . . . , b ℓ ).

2 .From corollary 6 , j=1 (s 2 - j=1 (s 2 -

 26j=12j=12 The real coefficients c i,k (i = 1, . . . , n) and (k = 0, . . . , ℓ) given by the expansions of f e i (i = 1, . . . , n) as the following sums of products of factors off o f e i (s 2 ) = c i,0 f o (s 2 )we can deduce a matrix formulation of f e i and f o .Sincef o (s 2 ) = γ o ℓ ∏ b j ) with γ o ∈ R -{0} then relation[START_REF] Jury | Inners and stability of dynamic systems[END_REF] can be simplified as it followsf e i (s 2 ) = γ o c i,0 ℓ ∏ b j ) -γ o ℓ k s 2j , = V A o Ψ o i , i = 1, . . . , n,(37a)

Theorem 4 (see theorem 2 )

 42 Consider a convex polytope of plants Ω G defined by[START_REF] Abdallah | Controller synthesis for a class of interval plants[END_REF]. If there exists a compensator C satisfying the two following relationsΦ(G i , C) ∈ H, (39a) Φ(G i , C) o = Φ(G j , C) o , i = 1, . . . , n -1, j = i + 1, . . . , n. (39b)with the leading coefficients of Φ(G i , C) of same sign, then C stabilizes Ω G .

Lemma 9 (see lemma 7 ) 1 T Ψ o 2 T Ψ o 3 T

 97123 Consider a convex polytope of plants Ω G defined by[START_REF] Abdallah | Controller synthesis for a class of interval plants[END_REF] and A o , H o given by (38c), (38d) respectively. If there exists C υ andΨ o i ∈ R ℓ ++ , (i = 1, . . . , n) satisfying Λ o i C υ = H o , i = 1, . . . , n,(40a)Λ e i C υ = A o Ψ o i (40b) then C stabilizes Ω G .Now rewrite relations (40)B o = Θ o χ o

Table 1 :

 1 -2.5s 5 + 2.15s 4 -11s 3 + 24s 2 + 10s + 30.5 -0.2s 5 -31s 4 -4.1s 3 -81s 2 -72.5s -2 Gap metric results applied to examples 1 to 5

	1	1.25s 2 + 1.25 2s 2 + 1	with τ max = 1	0.60964 8	0.6312	0.96777 0
	5.2.1	1.25s 2 + 1.25 2s 2 + 1	with τ max = 1.5	0.60964 8	0.70467	0.999	0
	5.2.2	s 3 + 3s 2 + 1 s 3 -2s 2 + 1	0.3329 16 0.25884 0.42977 2
	5.2.3	0.11s 4 + s 3 + 2.25s 2 + 0.75 0.1s 4 + 1.5s 3 -2s 2 + 1	0.24363 32 0.17906	0.2891 14
	5.2.4	6s 3 + 3s 2 + 22s -3 -0.2s 5 -3s 4 -16s 3 + 22s 2 -11s -0.5	0.1728 16 0.20878 0.79022 0
	5.2.5			0.16273 32 0.034146 0.17463 31

Table 2 :

 2 Vertices G i of the polytope of plants Ω G

	24771s + 2.0307
	s 3 + 3s 2 -s + 6

Definition 5 (Common odd interlacing property )

  Consider a finite collection of polynomials f e i and a polynomial f o with their leading coefficients of same sign. Assume that for each pair (f e i , f o ), (i = 1, . . . , n), the roots of f e i and f o are all real, negative simple and distinct defined by the following sets The polynomials f e i have a common odd interlacing f o iff the ℓ roots of each f e i alternate with the ℓ roots of f o as below

	roots(f e i ) = {a i 1 , . . . , a i ℓ }
	roots(f a i 1 < b 1 < a i 2 < . . . < a i ℓ < b ℓ < 0
	Note that if f e
	we have
	max i∈{1,...,n}	a i j < b j < min i∈{1,...,n} a i j+1 .

o ) = {b 1 , . . . , b ℓ } i , (i = 1 . . . , n) have a common interlacing f o , this implies that for any jth root,

  1 . . . υ 0,ℓ υ 1,0 υ 1,1 . . . υ 1,ℓ

					
	. . .	. . .	. . .	. . .	   	(38c)

υ ℓ,0 υ ℓ,1 . . . υ ℓ,ℓ

See[START_REF] Bartlett | Root locations of an entire polytope of polynomials: it suffices to check the edges[END_REF] for the definition of the exposed edges.