Stability of linear systems with time-varying delays using Bessel-Legendre inequalities

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1 Stability of linear systems wit time-varying elays using Bessel-Legenre inequalities Alexanre Seuret, Frééric Gouaisbaut To cite tis version: Alexanre Seuret, Frééric Gouaisbaut Stability of linear systems wit time-varying elays using Bessel-Legenre inequalities IEEE Transactions on Automatic Control, Institute of Electrical an Electronics Engineers, 8, 63 (, pp5-3 <9/TAC773485> <al-53784> HAL I: al ttps://allaasfr/al Submitte on Jun 7 HAL is a multi-isciplinary open access arcive for te eposit an issemination of scientific researc ocuments, weter tey are publise or not Te ocuments may come from teacing an researc institutions in France or abroa, or from public or private researc centers L arcive ouverte pluriisciplinaire HAL, est estinée au épôt et à la iffusion e ocuments scientifiques e niveau recerce, publiés ou non, émanant es établissements enseignement et e recerce français ou étrangers, es laboratoires publics ou privés

2 Stability of linear systems wit time-varying elays using Bessel-Legenre inequalities Alexanre Seuret an Frééric Gouaisbaut Abstract Tis paper aresses te stability problem of linear systems wit a time-varying elay Hierarcical stability conitions base on linear matrix inequalities are obtaine from an extensive use of te Bessel inequality applie to Legenre polynomials of arbitrary orers Wile tis inequality as been only use for constant iscrete an istribute elays, tis paper generalizes te same metoology to time-varying elays We take avantages of te epenence of te stability criteria on bot te elay an its erivative to propose a new efinition of allowable elay sets It is sown tat a ligt an smart moification in te efinition of tis set leas to relevant conclusions on te numerical results Keywors Integral inequality, stability analysis, time-varying elay systems, allowable elay sets, linear matrix inequalities I INTRODUCTION Tis paper is evote to te stability analysis of linear timeelay systems, wic represent a wie class of systems arising in many applications suc as in biology, in traffic control, in engineering, in cyber-pysical systems or in networke control systems (see for instance te books an survey papers, 3, 7, Compare to te elay-free case, te ifficulties wen stuying time-elay systems mainly come from te infinite imensional nature of tis class of systems, wic require a eicate analysis Te aim of tis paper is to provie stability conitions of time-elay systems were te elay is assume to be a boune continuous time-varying function wit boune erivatives In te literature, tere exist several major results on tis topic, see for instance te Lyapunov-Krasovskii approac 4, 8, 5, 5, 3,, te Input-to-State or Input -to-output approaces 6, wic are relate to te robust analysis, Focusing on te Lyapunov-Krasovskii approac, several tecniques ave been consiere to reuce te conservatism of te stability conitions, often formulate in terms of Linear Matrix Inequalities (LMI Te conservatism of tese LMIs, formulate in terms of elay upperbouns, generally comes from several manipulations allowing to express te infinite imensional stability problem into a finite LMI test A list of tecnical tools tat ave been consiere in te literature to solve tis problem inclues matrix inequalities 3, te reciprocally convex combination lemmas 6, 7 Te objective of tis paper is te esign of generic stability conitions for linear systems subject to a continuous timevarying elay by using te Bessel-Legenre inequality Tese conitions generalize te original works on Bessel-Legenre *Tis work was supporte by ANR project SCIDiS contract number 5- CE3-4 A Seuret an F Gouaisbaut are wit LAAS - CNRS, Université e Toulouse, CNRS, UPS, 7 av u Colonel Roce, F-34 Toulouse, France aseuret,fgouais@laasfr inequality for systems wit constant (iscrete or istribute elay presente in 9, to te case of time-varying elays Tis work is also a generic extension of te stability conitions for time-varying elay systems of former papers wic use te Wirtinger-base inequality 8, te auxiliarybase integral inequalities 7, or te free-weigting matrix integral inequality from 5 We also stress tat anoter contribution of tis paper is to evaluate te impact of te allowable elay set caracterizing te elay functions as suggeste in Several numerical experiments sow tat te propose Lyapunov-Krasovskii functional is very competitive if an aequate elay set is constructe It suggests ten tat te conservatism inuce by te meto stems mainly from te robust analysis performe on te LMI Notations: Trougout te paper R n enotes te n- imensional Eucliean space wit vector norm, R n m is te set of all n m real matrices Te notation P, for P R n n, means tat P is symmetric an positive efinite Te set S n (S n + represents te set of symmetric (positive efinite matrices of R n n Te set of continuous functions from an interval I R to R n wic are square integrable is enote as space L (I, R n Te symmetric matrix A B C stans for A B iag(a, B stans for te iagonal matrix B T C A B Moreover, for any square matrix A Rn n, we efine He(A = A + A T For any function x :, + R n, x t (θ stans for x(t + θ, for all t an all θ, A System ata II PROBLEM FORMULATION Consier a linear time-elay system of te form: { ẋ(t = Ax(t + A x(t (t, t, x(t = φ(t, t,, were x(t R n is te state vector, φ is te initial conition function, A an A are constant matrices an is a given positive scalar Te elay function is assume to be continuous an ifferentiable an satisfies te following constraint ( t, ((t, ḣ(t H R+ R, ( were H is assume to be a convex boune subset of R + R, enote as te allowable elay set for system ( In te literature, te set H is often selecte as a polytope,,, for some appropriate values of,,, In tis paper we efine alternative polyeral allowable elay sets wit te same number of vertices We will sow tat oter possibilities can be consiere an lea to notable canges in te numerical experiments reporte in Section VI

3 B Legenre polynomials an teir properties Let us first recall te efinition of te Legenre polynomials, use in te efinition of te Lyapunov-Krasovskii functional employe tereafter Definition : Te sifte Legenre polynomials are te sequence of polynomials efine over te interval,, wic are given for all i N, L i (u = ( i i j= pi j uj were p i j = ( ( ( j i i+j j j, an ( k l refers to te binomial coefficients k! given by (k l!l! In te sequel, te notation L N refers to a polynomial matrix of imension (N + n n, were n an N are positive integers, an is given by L N (u := L (ui n L (ui n L N (ui n T (3 Te ortogonality property of te Legenre polynomials can be summarize by te following statement For any matrix R in S n +, it ols N N, L N (ur L T N(uu = R N (4 were R N = iag {R, 3R,, (N + R} In aition, te evaluation of tese polynomials at te bounaries of te interval, are simply given, for any integer N, by L N ( = I n I ṇ I n := N, L N ( = I n I n ( N I n := N (5 Since ortogonal polynomials usually verify recursive properties, Legenre polynomials satisfy te two following ifferentiation rules, tat will be te key tecnical tools of tis paper u L N(u = Γ N L N (u = Γ N L N (u u (ul N(u = L N (u + Θ N L N (u, were Γ N = Γ N n(n+,n, Γ N = γ N I n an Θ N = θ N I n, were refers to te classical Kronecker prouct Matrices γ N R (N+ N an θ N R (N+ (N+ are efine by { if k i, γ N (k, i = (k ( ( i+k if k < i θ N (k, i = if k > i, k if k = i, (k if k < i Tese two erivation rules are erive from classical properties of te Legenre polynomials Basically, it means tat teir erivative are expresse using Legenre polynomials of lower egree C From constant to time-varying elays In 9, a first stability analysis base on Legenre-base Lyapunov-Krasovskii functionals as been provie to cope wit linear systems subject to constant time-elay In tis paper, te following functional as been employe W N (x t, ẋ t = χ T N (tp N χ N (t+ t xt (ssx(ss + t θ ẋt (srẋ(ssθ, (6 (7 were N N, P N S (N+n, S, R S n + an te augmente vector χ T N (t = x T t ( φ T N (t an φ N (t = ( s + L N x t (ss, N Guie by tis functional (7 eicate to te analysis of systems wit a constant elay, we propose to consier te following extension wic aims at ealing wit te case of time-varying elays It consists in using te following Lyapunov-Krasovskii functional V N (x t, ẋ t = x T N (tp N x N (t + t xt (ssx(ss + t x T t (sqx(ss + t θ ẋt (srẋ(ssθ, (8 were, in tis case, te matrix P N is now in S (N+3n an were S, Q, R are in S n + Tis new functional is efine using te augmente vector x N efine by were x T N(t = x T t ( (tφ T,N (t ( (tφ T,N (t φ,n (t = (t φ,n (t = (t (t L s+(t N (t x t (ss, ( (t s+ L N (t x t (ss Tis functional generalizes te one tat as been efine in 8 for time-varying elay systems Inee selecting N = in (8 allows retrieving te same functional as in 8 Te propose extension to time-varying elays is not an easy task since te time-varying elay appears in te efinition of φ,n an φ,n To acieve tis goal, several tecnical problems arise for te consieration of te time-varying elay Te first ifficulty refers to te exibition of LMI conitions to prove te positive efiniteness of V N A first solution woul be to impose tat P N, S, Q an R are positive efinite However, tis migt be too conservative an we look forwar a tigter conition wic relies on an extensive use of te Bessel Legenre inequality 9 togeter wit an improve version of te reciprocally convex combination lemma 6 In orer to satisfy te requirements of te Lyapunov- Krasovskii teorem, we must also sow tat te functional V N is upper boune by some function of te state of te time-elay systems Te consieration of φ,n an φ,n in te first quaratic term of V N makes tis step unclear Te computation of te time-erivative of V N requires to express te time-erivative of φ,n an φ,n Tis step is also not straigtforwar an requires a eicate evelopment III PRELIMINARY RESULTS A Delay-epenent reciprocally convex inequality In te two first steps, we will use an improve version of te reciprocally convex combination lemma originally presente in 6 Te following lemma relaxes te inequality provie in 6 troug te introuction of aitional matrices variables (9

4 3 Lemma : Let n be a positive integer an R, R be in S n + If tere exist X, X in S n an Y, Y in R n n suc tat R X Y α Y R Y T ( α Y T ( X ols for α =,, ten, te following inequality α R R X Y α R +( α Y R Y T +α Y T X ols for all α (, Proof : Te proof is given in te appenix Remark : As notice in 8, te aitional variables X an X can be remove by taking X = R Y R Y T an X = R Y T R Y, wic obviously verifies conitions ( Tis manipulation allows a notable reuction of te numerical complexity Te particular selection in te previous lemma wit X = X = an Y = Y = Y leas to te original reciprocally convex combination lemma 6 Lemma refines te original convex combination lemma since it allows obtaining a lower boun wic epens explicitly on α, wic will be relate to te time-varying elay (t B Bessel-Legenre inequality Te Bessel-Legenre inequality is state in te next lemma Lemma 3: For a given function x in L (a, b R n, any scalars a < b an any matrix R S n + te inequality b a x T (srx(ss (b aφ T N R N φ N ( ols, for any positive integer N were φ N = b b a a L s a N b a x(ss R N = iag(r, 3R,, (N R Proof : As in 9,, te proof of tis lemma is base on te function( z n : a b R n given by z N (s = Rx(s L T s a N b a RN φ N, were we recall tat φ N = b b a a L s a N b a x(ss Tis function z N can be viewe as te error between te function x an te best polynomial approximation of x accoring te inner prouct Te inequality results from te expansion of b a zt N (sr z N (ss, wic is positive Remark : Te previous inequality inclues Jensen s inequality 7, te Wirtinger-base integral inequality an te auxiliary-function base integral inequality 7 as particular cases wen N =,,, respectively It is also wort noting tat te Parseval ientity ensures tat te propose inequality becomes asymptotically non conservative wen N goes to infinity IV STABILITY ANALYSIS In orer to simplify te exposure of tis section, te stability analysis is ivie into tree parts Te first part eals wit te positive efiniteness of te functional Te secon part aims at obtaining LMI conitions to guarantee te negative efiniteness of te erivative of te functional along te trajectories of te system Te last part resumes te main result of te paper A Positive efiniteness of V N Lemma 4: For a given positive integer N, assume tat tere exist P N S (N+3n, S, Q, R S n + an a matrix U R n(n+ n(n+ suc tat te following LMIs Φ N ( = P N + Φ N SN, Φ N = U T U ( QN are satisfie, were S N = iag(s, 3S,, (N + S, an Q N = iag(q, 3Q,, (N + Q Ten, tere exist ɛ > an ɛ > suc tat te functional V N (x t, ẋ t satisfies ɛ x t ( V N (x t, ẋ t ɛ x t W, (3 ( were x t W = max s, x t (s + (see eg 3 ẋ t (s s Proof : Te objective of te next evelopments is to fin a lower an an upper bouns of V N In orer to erive te first lower boun (ie te existence of ɛ, one may apply Lemma 3 to te secon an tir terms of te functional, wic can be one since te matrices S an Q are symmetric positive efinite Hence, applying Lemma 3 to te orer N yiels t xt (ssx(ss φ T,N (t S N φ,n (t, t x T (sqx(ss ( φ T,N (t Q N φ,n (t, wic ensures te following inequality ( V N (x t, ẋ t x N (t P N + N N + t θ ẋt (srẋ(ssθ x N (t If tere exists a matrix U in R n(n+ n(n+ suc tat SN U, ten Lemma, wit X U T QN = X = an Y = Y = U ensures, togeter wit R tat V (x t, ẋ t x T N (tφ N( x N (t Terefore, if Φ N ( an R, ten tere exists ɛ > suc tat V N (x t, ẋ t ɛ x t (, wic conclues on te first inequality To prove te secon inequality, tere exists a sufficiently large scalar η > suc tat P N ηiag(i, I N, I N were I N = iag(i, 3I,, (N + I It tus ols ξn T (tp N ξ N (t η x t ( + η (tφ T,N (ti Nφ,N (t +η( (t φ T,N (ti N φ,n (t Using Lemma 3, we obtain ξn T (tp Nξ N (t η x t ( + η(t t (t x(s s +η( (t (t t x(s s η x t ( t + η t x(s s

5 4 Re-injecting tis inequality into te efinition of te functional V N, we can easily obtain tat tere exists ɛ >, suc tat V N (x t, ẋ t ɛ x t W B Negative efiniteness of te erivative of V N Te main result of tis section is given below Lemma 5: For a given positive integer N, assume tat tere exist P N S (N+3n, S, Q, R S n + an Y, Y R (N+n (N+n, suc tat Ψ N (, ḣ W N T Ψ N (, ḣ = T Y W N Y T at te vertices of H, were Ψ N (, ḣ = He (G T N (P N Y W T N RN+ T W N ( J N + ḣh N Y T RN+ (4 +Σ N (ḣ + FN T RF N WN T Ξ N(W N, R N+ = iag(r, 3R,, (N + R, (N + 3R, F N = A A, G N ( = H N = J N = Σ N I I nn ( I nn,, N ΓN Θ N N Θ N A A N N Γ N N N Γ N, (ḣ = iag(s, ( ḣ(s Q, Q,,, N+ N+ Γ N+ N+ N+ Γ N+ W N =, Ξ N ( = RN+ + RN+ Y RN+ + Y Y T (5 wit N+, N+, Γ N, ΓN an Θ N are given in (5, (6 an (6, respectively Ten, tere exists ɛ 3 > suc tat te functional V N (x t, ẋ t satisfies V N (x t, ẋ t ɛ 3 x t ( Proof : Before entering into te etails of tis proof, two preliminary results are propose for te sake of simplicity Tese results use an augmente vector, efine for N by Y T R N+, ξ T N (t = x T t ( x T t ( x T t ( φ T,N (t φt,n (t Proposition 6: For a given N >, te time-erivative of te vector x N is expresse using matrices H N an J N given in (5 by x N (t = (J N + ḣh Nξ N (t Proposition 7: For any function x suc tat ẋ is in L, any scalar >, any integer N an any matrix R S n +, ten te inequality ẋ T t (srẋ t (ss ξn(tw T N T Ξ N ( Ξ N (W N ξ N (t Conitions (4 are formally strict inequalities for all elays ifferent from an Due to te cases = an =, some rows an columns migt be zero, wic implies te non strict inequality (4 ols for any elay, an for any matrices Y, Y R (N+n (N+n, were Ξ N ( an W N are efine in (5 Y R N+ Y T an were Ξ N ( = Y T R N+ Y Te proof of tese two propositions are provie in te appenix Te computation of te erivative of V N leas to V N (x t, ẋ t = He ( x T N (tp N x N (t + x T t (Sx t ( ( ḣxt t ( (S Qx t ( x T t ( Qx t ( + ẋ T t (Rẋ t ( ẋ T t (srẋ t (ss (6 Our objective is to erive an upper boun of V N expresse as a quaratic form using te augmente vector ξ N (t Using te matrices G N (, F N an Σ N (ḣ efine in (5, te erivative of V N along te trajectories of te system can be rewritten as V N (x t, ẋ t = He ( ξn T (tgt N (P N x N (t +ξn T (t FN T RF N + Σ N (ḣ ξ N (t ẋ T t (srẋ t (ss (7 Using Propositions 6 an 7, te erivative of te V N along te trajectories of te system satisfies ( V N (x t, ẋ t ξn He T (t G T N (P N(J N +ḣh N + Σ N (ḣ + F T N RF N W T N Ξ N ( Ξ N ( W N ξn (t = ξ T N (t ( Ψ N (, ḣ + W T N Ξ N(W N ξ N (t It is ten easy to see tat te matrix Ψ N (, ḣ + WN T Ξ N(W N is multi-affine wit respect to an ḣ an consequently convex in tese two parameters Hence, if te LMIs (4 are satisfie at te vertices of H, tere are also verifie for any (, ḣ H an tere exists a sufficiently small ɛ 3 suc tat Φ N (, ḣ ɛ 3 I, wic allows concluing te proof C Main result Lemma 4 an 5 allow us to state te main result of tis paper, given below Teorem 8: For a given positive integer N, assume tat tere exist a matrix P N S (N+3n, matrices S, Q, R S n + an two matrices Y, Y R n(n+ n(n+ an U R n(n+ n(n+ suc tat LMI Φ N (, Φ N an Ψ N(, ḣ are satisfie for all (, ḣ H, ten system ( is asymptotically stable for any time-varying elay, satisfying ( Proof : Accoring to Lemmas 4 an 5, provie tat te LMI conitions ol, te functional V N satisfies te following inequalities ɛ x t ( V N (x t, ẋ t ɛ x t W an V N (x t, ẋ t ɛ 3 x t ( Tese inequalities ensures te asymptotic stability of te elay system ( for any time-varying time elay, wic satisfies (, ḣ H, by application of te extene Lyapunov-Krasovskii teorem (see eg 3

6 5 D Hierarcy of LMI conitions Following te results presente in 9, ealing wit systems subject to constant iscrete an istribute elays an also taking avantages of te Bessel-Legenre framework, te LMI conitions presente in Teorem 8 form a ierarcy In oter wors, it means tat increasing N can only reuce te conservatism Tis is summarize in te following teorem Teorem 9: For a given time-elay system ( an a given allowable elay set H, if tere exists a positive integer N suc tat te LMI conitions of Teorem 8 are satisfie at te orer N, ten te same LMIs are also verifie for all integer N N Proof : Assume tat te LMI conition of Teorem 8 are verifie at a given orer N N Ten tere exist matrices P N, S, Q, R, U an Y, Y of appropriate imensions suc tat te following inequalities are verifie for all (, ḣ H Φ N (, Φ N, Ψ N (, ḣ To prove tat te same LMIs also ol at te orer N +, it suffices to keep te same matrices S, Q, R an to introuce P N+ = P N, U + = U an Y + i = Y i, for i =, Using tese efinitions an using some matrix congruence, we obtain te following equivalences ΦN ( Φ N+( (N +3S, (N +3Q Φ N Φ N + (N +3S, (N +3Q ΨN (,ḣ Ψ N+(, ḣ R N, R N were RN = iag(r, (N + 5(Γ N+ T N R(Γ + N + N+, wit (Γ N+ N+ being te n last lines of te matrix Γ N+ Terefore, tere exists a solution to te LMI problem at te orer N + By inuction, we can easily sow tat for any integer N N, tere also exists a solution to te same LMI problem at te orer N, wic conclues te proof Tis teorem can be interprete as an inclusion of te inner allowable elay sets obtaine troug Teorem 8 V ALLOWABLE DELAY SETS In Teorem 8, te resulting LMIs are multi-affine in te elay parameters an ḣ Ten, in orer to get tractable numerical conitions, it suffices to consier tat H is a polyeral set or if it is not te case to embe it by a polytopic outer approximation In te literature, only a few attention as been taken on te efinition of te set H In te following we will recall, in a first step, te main caracterization tat as been consiere in te literature Ten, base on an iscussion regaring tis first set, an alternative allowable elay set is introuce in orer to provie a more natural an accurate caracterization of te elay function A consequence of tis selection is a notable moification of te numerical results (a Grap representing H (b Grap representing H Fig Grapical illustration of te elay set H (a an H (b A Usual assumption on te elay set H Te usual formulation of te elay set is as follows Consier some scalars an an assume tat te elay function satisfies: (, ḣ H =,, (8 Tis set is te usual caracterization of te elay function Inee, it allows te elay function to take any values in te interval, wile its erivative belongs to te interval, Notice tat it is implicitly assume tat an ḣ are inepenent parameters However tis efinition requires tat te LMI are satisfie wenever = an ḣ > or = an ḣ < Tese situations contraict te fact tat an are respectively te upper an te lower bouns of te elay B Refine caracterization of te allowable elay set As pointe out in te previous section, te polytopic moeling of te allowable elay set allows to exclue some subsets of H A first irection is to note tat wen = (or =, its erivative cannot be positive (negative Following tis comment, we introuce te new allowable elay set H escribe as follows Consier some scalars an < < an we assume tat te elay function satisfies: (, ḣ H = Co{(,, (,, (,, (, } (9 A grapical interpretation of te sets H an H are epicte in Figure, were it is sown tat, for fixe values of,,, H is inclue in H In te next section wic presents te application of Teorem 8 on an example, te use of an accurate selection of te allowable elay set notably moifies te numerical results, as expecte an incluing aitional information on te elay functions may enlarge tese results VI NUMERICAL EXAMPLE In tis section, we will consier te linear time-elay system ( wit te matrices ẋ(t = x(t+ x(t (t ( 9 Tis system is a well-known elay epenent stable system, tat is te elay free system is stable an te maximum allowable elay max = 675 can be easily compute by elay sweeping tecniques (see for instance 7 Of course, in te time-varying elay case, suc sweeping tecniques cannot be consiere

7 6 5 8 Number of ecision variables Number of LMIs 4, n + 3n / 5n + 35n / n + 3n / n + n n + n n + 9n n + 8n 4 6 (N = n + 5n n + 3n n + 7n n + n 6 7 (TC n + 4n 4 T8 N = n + 35n 5 T8 N = n + 5n 5 T8 N = n + 65n 5 T8 N = n + 8n 5 T8 N = n + 95n 5 TABLE I THE MAXIMAL ALLOWABLE DELAYS FOR EXAMPLE ( WITH VARIOUS VALUES OF AND = AND (, ḣ H A Numerical complexity Te two last columns of Table I summarize te numerical complexity of various stability conitions from te literature an te ones presente in Teorem 8 by inicating te number of ecision variables an te number of LMI conitions to be implemente For sake of simplicity, it oes not inclue te size of te LMIs At tis stage, one can note tat te complexity of Teorem 8 is similar to te one of existing conitions from te literature, for small values of N(=, For larger values of N, te number of ecision variables increases in a polynomial way B Numerical results for H Table I sows te numerical results obtaine troug several criteria using bot Lyapunov-Krasovskii an robust analysis for te example Te first rows of Table I present criteria base on Jensen s inequality Te last rows present te numerical results wic can be seen as te most recent works on improve integral inequalities incluing te Wirtinger-base ouble integral inequality, te free-weigting matrix inequality 5 Tese contributions are compare wit results issue from Teorem 8 Te numerical results presente in Table I sow tat te conitions presente in Teorem 8 are less conservative tan existing results It is also wort noting te ifferent levels of conservatism between Teorem 8 are mainly ue to te conservatism inuce by te reciprocally convex combination lemma of 6 or presente in Lemma Tis observe conservatism is ten partially reuce by te propose metoology at te price of an increasing numerical buren C Numerical results for H Tis subsection presents te numerical results for an alternative allowable elay set, excluing te conflicting point (, an (, wic prevents for an to be te maximum an te minimum of te elay function (t Following te iscussion presente in Subsection V, we consier ere tat te elay function belongs in H, given in (9 Te numerical results presente in Table II sow a notable increase of te elay upper boun compare to Table I, obtaine for H One can also see tat tere is no brutal ecrease of te upper boun wen te boun on te erivative is small (see for instance, wen te erivative of te elay function become large, te jump from = 53 wit H to = 37 wit H Table II finally emonstrates tat te gap between te constant an te time-varying elay case is not only ue to te conservatism inuce by te coice of te Lyapunov- Krasovskii functional Te coice of te elay set H as also a ramatic effect on te results an te conservatism of te meto Inee, by an accurate moification of te allowable elay set efinition, it is possible to notably enlarge te upper boun of te elay upperboun Clearly, tere is a room for improvements if we consier a tinner escription of te elay set as it as been alreay note in 9, 4 for sample-ata systems It also suggests tat a particular attention soul be performe wen we consier te robust analysis of te LMI conitions In orer to motivate te notion of allowable elay set, we will consier in te next section, a particular case of elay function, for wic several allowable elay sets can be erive D Numerical results for sinusoial elays It as been sown previously tat te selection of te allowable elay set may ave a notable impact on te numerical results In orer to illustrate tis coice, let ( consier( te particular elay function given by (t = + cos t, for all t, so tat te ( elay function belongs to te convex ( ellipsoi set efine by (t + ḣ(t A first approac woul consier tat te elay function lies in a

8 7 T8: N/ M TABLE II THE MAXIMAL ALLOWABLE DELAYS FOR SYSTEM DESCRIBED IN EXAMPLE ( SOLVING THEOREM 8 FOR VARIOUS VALUES OF N AND AND = WITH (, ḣ H (,(θ ((θ, (,-(θ Fig Grapical illustration of te elay set H 3 (a an H (b θ (,(θ (t (,-(θ polytope of H -type given by,, However, it is possible to efine oter polytopes, still wit four vertices, wic allow caracterizing tis class of elay functions Figure epicts a possible solution obtaine by te geometric properties of ellipsois wic oes not increase te complexity of te LMI problem Using te equation of te tangent to an ellipsoi, anoter polytope is efine, for a given angle θ, π/, by H θ := Co{(, (θ, (, (θ, (, (θ, (, (θ}, were (θ = cos(θ sin(θ + cos(θ, (θ = sin(θ Figure 3 sows ow te upper boun of te elay varies wit te parameter θ wit efines te polytope for various values of It can also be seen tat te maximal elay boun is never obtaine wen θ = π/, corresponing to te H Tis emonstrates te relevance of te selection of te allowable elay set It woul be also possible to refine te selection of te elay set by incluing oter vertices For instance, one coul select te set H H θ, wic woul introuce two aitional vertices VII CONCLUSIONS In tis paper, we ave provie new stability tests for systems subject to a continuous time-varying elay wit boune erivatives Te novelty of tis contribution is ue to an extensive use of te Bessel-Legenre inequality Te set of LMI conitions forms a ierarcy, wic means tat increasing te orer of N of te Legenre polynomials in te analysis can only improve te numerical results Compare to existing resulting from te literature, notable improvements of te numerical results ave been obtaine Te secon contribution of te paper relies on te efinition of allowable elay sets It is sown tat a sligt moification of te efinition of te elay set allows to obtain more accurate results In particular, te upper boun of te elays for slow varying elays is very close to te one obtaine in te constant case APPENDIX PROOF OF LEMMA Proof : Following 6, te proof consists in noting tat α R R α α R = + α R α R α R ( = =5 =8 = π/6 π/3 π/ θ Fig 3 Grapical illustration of te elay set H 3 (a an H (b Te objective is to fin a lower boun of te secon term of te rigt-an-sie of ( Using a convexity argument, if ( ols for α =,, it also ols for any α in, Ten, βi β I pre- an post-multiplying ( by yiels, for all α in (, α α R X Y ( α Y T α α R wic togeter wit ( yiels te result, wit β = + α PROOF OF PROPOSITION 6 α α Y Y T, X Proof : For simplicity, te time argument will be omitte in te next evelopments Consier φ N = b b a a L s a N b a x(ss were a an b are functions of time In orer to simplify te computations, we apply te cange of variable s(u = (b au+a to get φ N = L N(ux(s(uu an t (b aφ N = ȧψ,n + (ḃ ȧ(ψ,n + φ N ( were ψ,n = (b a L N(uẋ(s(uu, an ψ,n = (b a ul N(uẋ(s(uu An integration by parts yiels ψ,n = L N (x(b L N (x(a L N (ux(s(uu, ψ,n = L N (x(b u (ul N(ux(s(uu Rules (6 of te Legenre polynomials yiel ψ,n = N x(b N x(a Γ N φ N, ψ,n = N x(b Θ N φ N φ N Reinjecting tese equations into ( yiels, for all t R +, t (b aφ N = ȧ( N x(b N x(a Γ N φ N +(ḃ ȧ( Nx(b Θ N φ N φ N + φ N = ḃ Nx(b ȧ N x(a (ȧ Γ N + (ḃ ȧθ N φ N

9 8 Tis previous expression is translate to te vector φ,n an φ,n by consiering (a, b = (t (t, t an (a, b = (t, t (t, respectively Tis yiels to x N (t = ẋ T (t t (φt,n wit ẋ(t = A(t + A x(t an t (( φ T,N, t (φ,n = N x(t ( ḣ N x(t (( ḣ Γ N + ḣθ Nφ,N t (( φ,n = ( ḣ Nx(t N x(t ( Γ N ḣθ N φ,n Using matrices J N an H N, te previous equations can be summarize as x N = (J N + ḣh Nξ N PROOF OF PROPOSITION 7 Proof : As we mentione in te introuction, one of te most popular term employe in a Lyapunov-Krasovskii functional leas to an integral quaratic term of te timeerivative of x Tus, te first step of te proof consist in particularizing te inequality provie in Lemma 3 to tis situation Consier te integral t ẋt (srẋ(ss By application of te Bessel-Legenre inequality in Lemma 3, to te orer N +, we obtain ẋ T (srẋ(ss t φ,n+ φ,n+ T RN+ ( R N+ φ,n+ were φ,n+ = L ( s+ N+ ẋt (ss, s+ φ,n+ = L N+ ẋ t (ss φ,n+ (3 An integration by parts an te ifferentiation rule (6 yiel φ,n+ N+ x ( φ = t ( N+ x t ( Γ N+ φ,n,,n+ N+ x t ( N+ x t ( Γ N+ φ,n = W N ξ N were te matrix W N is given in te statement of te lemma Combining tis expression into (3 yiels ẋ T (srẋ(ss ξn(tw T N T RN+ W N ξ N (t t R N+ We are now in te situation to apply te convex inequality provie in Lemma It implies tat, if tere exist X, X S nn an Y, Y R nn nn suc tat te two conitions Ψ N an Ψ N ol, ten we ave R N R N Ξ N ( Reinjecting tis inequality into te previous equation conclues te proof of te lemma REFERENCES Y Ariba, F Gouaisbaut, an KH Joansson Robust stability of timevarying elay systems: Te quaratic separation approac Asian Journal of Control, 4(5:5 4, C Briat Linear Parameter-Varying an Time-Delay Systems Analysis, Observation, Filtering an Control, Avance in Delays an Dynamics, vol 3 Springer, Hielberg, 5 3 E Friman Introuction to Time-Delay Systems: Analysis an Control Birkauser, Basel, 4 4 E Friman an U Sake An improve stabilization meto for linear time-elay systems IEEE Trans on Automatic Control, 47:93 937, November 5 E Friman an U Sake Delay-epenent stability an control: constant an time-varying elays Int J of Contr, 76(:48 6, 3 6 E Friman an U Sake Input-output approac to stability an L - 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