Wirtinger-based integral inequality: Application to time-delay systems

Transcription

1 Wirtinger-bsed integrl inequlity: Appliction to time-dely systems Alexndre Seuret, Frédéric Gouisbut To cite this version: Alexndre Seuret, Frédéric Gouisbut. Wirtinger-bsed integrl inequlity: Appliction to timedely systems. Automtic, Elsevier, 2013, 49 (9), pp <hl > HAL Id: hl Submitted on 29 Aug 2013 HAL is multi-disciplinry open ccess rchive for the deposit nd dissemintion of scientific reserch documents, whether they re published or not. The documents my come from teching nd reserch institutions in Frnce or brod, or from public or privte reserch centers. L rchive ouverte pluridisciplinire HAL, est destinée u dépôt et à l diffusion de documents scientifiques de niveu recherche, publiés ou non, émnnt des étblissements d enseignement et de recherche frnçis ou étrngers, des lbortoires publics ou privés.

2 Wirtinger-Bsed Integrl Inequlity: Appliction to Time-Dely Systems A. Seuret,b F. Gouisbut,c CNRS, LAAS, 7 venue du Colonel Roche, Toulouse, Frnce. (e-mil: seuret,fgouisb@ls.fr). b Univ de Toulouse, LAAS, F Toulouse, Frnce c Univ de Toulouse, UPS, LAAS, F-31400, Toulouse, Frnce. Abstrct In the lst decde, the Jensen inequlity hs been intensively used in the context of time-dely or smpled-dt systems since it is n pproprite tool to derive trctble stbility conditions expressed in terms liner mtrix inequlities (LMI). However, it is lso well-known tht this inequlity introduces n undesirble conservtism in the stbility conditions nd looking t the literture, reducing this gp is relevnt issue nd lwys n open problem. In this pper, we propose n lterntive inequlity bsed on the Fourier Theory, more precisely on the Wirtinger inequlities. It is shown tht this resulting inequlity encompsses the Jensen one nd lso leds to trctble LMI conditions. In order to illustrte the potentil gin of employing this new inequlity with respect to the Jensen one, two pplictions on time-dely nd smpled-dt stbility nlysis re provided. Key words: Jensen Inequlity, stbility nlysis, time-dely systems, smpled-dt systems 1 Introduction The lst decde hs shown n incresing reserch ctivity on time-dely nd/or smpled-dt systems nlysis nd control due to both emerging dpted theoreticl tools nd lso prcticl issues in the engineering field nd informtion technology (see?? ) nd references therein). In the cse of liner systems, mny techniques llow to derive efficient criteri proving the stbility of such systems. Among them, two frmeworks, different in their spirits hve been recognized s powerful methodologies. The first one relies on Robust Anlysis. In this frmework, the time dely/smpled dt system is trnsformed into closed loop between stble nominl system nd perturbtion element depending either on the dely or the smpling process (which is lso modeled by time vrying dely). The perturbtion element is then embedded into some norm-bounded uncertinties nd the use of scled smll gin theorem??, Integrl Qudrtic Constrints (IQC)?, or qudrtic seprtion pproch? llows then to derive efficient stbility criteri. The chllenge is then to reduce the conservtism either by constructing elborted interconnections which generlly include stte ugmenttion? or by using finer L 2 induced norm upperbound evlution?, often bsed on Cuchy-Schwrtz inequlity?. Another populr pproch is the use of Lypunov function to prove stbility. For smpled-dt systems, two pproches hve been successfully proposed ltely. The first one relies on impulsive systems nd some piecewise liner Lypunov functions??. This pproch hs been then extended by considering discontinuous Lypunov functions which llow to consider periodic smpling??. In the second pproch, the smpled stte is modeled by time vrying delyed stte. In tht cse, the originl system is recsted into time vrying dely system where Lypunov-Krsovskii functionls? cn be used directly. Hence, for smpled-dt nd time dely systems, the lst decde hs seen tremendous emergence of reserch devoted to the construction of Lypunov-Krsovskii functionls which ims t reducing the inherent conservtism of this pproch. Severl ttempts hve been done concerning the structure of the functionl by extending stte bsed Lypunov-Krsovskii functionls (??? ), discretized Lypunov functionls (? ) or discontinuous Lypunov functions?. Aprt the choice of the functionl, n importnt source of conservtism relies lso on the wy to bound some cross terms risen when mnipulting the derivtive of the Lypunov-Krsovskii functionl. According to the literture on this subject (see??? for some recent ppers), common feture of ll Preprint submitted to Automtic 29 My 2013

3 these techniques is the use of slck vribles nd more or less refined Jensen inequlity???. At this point, it is cler tht for both frmeworks - Robust Anlysis nd Lypunov functionls-, prt of the conservtism comes from the use of Jensen inequlity?, usully used to get trctble criteri. Bsed on this observtion, the objective of the present pper is then to show how to use nother clss of inequlities clled Wirtinger inequlities, which re well known in Fourier Anlysis. Notice tht this clss of inequlities hs been recently used to exhibit new Lypunov functionls to prove stbility of smpleddt systems?. In the present pper, contrry to the work of?, we do not construct some new Lypunov functionls. We im rther t developing new inequlity used to reduce the conservtism when computing the derivtive of Lypunov-Krsovskii functionls. The Wirtinger inequlity llows to consider more ccurte integrl inequlity which encompsses the Jensen one. The resulting inequlity depends not only on the stte x(t) nd the delyed or smpled stte but lso on the integrl of the stte over the dely or smpling intervl. This new signl is then directly integrted into suitble Lypunov function, highlighting so the fetures of Wirtinger inequlity. Hence, it results new stbility criteri for time-dely systems nd smpled-dt systems directly expressed in terms of LMIs. Nottions: Throughout the pper R n denotes the n- dimensionl Eucliden spce, R n m is the set of ll n m rel mtrices. The nottion P 0, for P R n n, mens tht P is symmetric nd positive definite. The nottion A B A B stnds for. For ny squre mtrices A C B T C A 0 nd B, define dig(a, B) =. Moreover, for ny B squre mtrix A R n n, we define He(A) = A + A T. The nottion I stnds for the identity mtrix. 2 Preliminries 2.1 Necessity of integrl inequlities Diverse methodss re provided in the literture to ssess stbility of time-dely systems using Lypunov- Krsovskii functionls. Among them, one of the most relevnt term introduced for the first time in? is The min issue relted to (2) is tht the integrl is not pproprite to the LMIztion process. It consists in trnsforming the previous expression into n pproprite form to derive n LMI formultion of the stbility conditions. In the following, the problem under considertion is to providing new lower bound of integrl qudrtic terms of the form I R (ω) = ω T (u)rω(u)du, where < < b < + re sclrs, ω is continuous function from, b R n nd, consequently integrble. The first method to tret this problem is bsed on the Jensen inequlity formulted in the next lemm Lemm 1 For given n n-mtrix R 0 nd for ll continuous functions ω in, b R n, the following inequlity holds: ( ) ( I R (ω) 1 b ) b b ωt (u)du R ω(u)du. (3) The proof is omitted nd cn be found in?. Nturlly, the Jensen inequlity is likely to entil some inherent conservtism nd severl works hve been devoted to the reduction of such gp??. In the present pper, we propose to use different clss of inequlities clled Wirtinger inequlities to obtin more ccurte lower bound of this integrl. 2.2 Wirtinger inequlity In the literture?, Wirtinger inequlities refer to inequlities which estimte the integrl of the derivtive function with the help of the integrl of the function. Often proved using Fourier theory, it exists severl versions which depend on the chrcteristics or constrints we impose on the function. Let us focus on the following inequlity dpted to our purpose. Lemm 2 Consider given n n-mtrix R 0. Then, for ll function z in C 1 (, b R n ) which stisfies z() = z(b) = 0, the following inequlity holds V (x t ) = t h s ẋ T (θ)rẋ(θ)dθds, (1) ż T (u)rż(u)du π 2 (b ) 2 z T (u)rz(u)du, (4) where x represents the stte of time-dely system, R 0 nd h > 0. Differentiting this term with respect to the time vrible t, we get V (x t ) = hẋ T (t)rẋ(t) ẋ T (s)rẋ(s)ds. (2) t h Proof : The proof is omitted but cn be found in?. It is worth noting tht (4) is not relted to the Jensen inequlity in its essence. Indeed, the function z hs to meet severl constrints wheres the function ω is ssumed to be continuous function in the Jensen inequlity. 2

4 2.3 Reciproclly convex combintion inequlity Recll useful lemm inspired from the reciproclly convex combintion lemm provided in?. Lemm 3? For given positive integers n, m, sclr α in the intervl (0, 1), given n n-mtrix R 0, two mtrices W 1 nd W 2 in R n m. Define, for ll vector ξ in R m, the function Θ(α, R) given by: Θ(α, R) = 1 α ξt W T 1 RW 1 ξ α ξt W T 2 RW 2 ξ. Then, if there exists mtrix X in R n n such tht R X 0, then the following inequlity holds R T W1 ξ R X W1 ξ min α (0, 1) Θ(α, R). W 2 ξ R W 2 ξ Proof : The proof is omitted but cn be found in?. This lemm will be useful to derive stbility conditions for liner systems with time-vrying delys. 3 Appliction of the Wirtinger inequlity The objective of this section is to provide n inequlity bsed on Lemm 2, which, on the first hnd, is implementble into convex optimiztion scheme nd, on the other hnd, which reduces the conservtism of the Jensen inequlity. To do so, the function z hs to be constructed such tht I R (ω) ppers nturlly in the future developments. Thus necessry condition is tht z(u) = u ω(s)ds y(u),where y is continuous function in, b R n to be defined. Corollry 4 Consider given mtrix R 0. Then, for ll continuous function ω in, b R n the following inequlity holds: ( 1 ) b T ( ) I R (ω) b ω(u)du b R ω(u)du + 3 b ΩT RΩ, where Ω = ω(s)ds 2 b s b ω(r)drds. (5) Proof : For ny continuous function ω nd which dmits continuous derivtive, define the function z given, for ll u, b, by z(u) = u ω(s)ds u b (b u)(u ) ω(s)ds (b ) 2 Θ, where Θ is constnt vector of R n to be defined. The difference between this function z nd the one proposed in? remins in the ddition of the third term. By construction, the function z stisfies the conditions of the Wirtinger inequlity given in Lemm 2, tht is z() = z(b) = 0. The computtion of the left-hnd-side of the inequlity stted in Lemm 2 leds to: żt (u)rż(u)du = ωt (u)rω(u)du ( ) 1 b T ( ) b ω(u)du b R ω(u)du + ( ) b 2 (b+ 2u) (b ) duθ T RΘ 2 2Θ T R ( b b+ 2u +2 ( (b+ 2u) (b ) 2 ) ω(u)du (b ) 2 ) duθ T R ( ω(u)du ). (6) By noting tht (b + 2u)du = 0 nd by use of n integrtion by prts, it yields żt (u)rż(u)du = ωt (u)rω(u)du ( ) 1 b T ( ) b ω(u)du b R ω(u)du + 1 3(b ) ΘT RΘ + 2 (b ) ΘT RΩ. (7) Rewriting the two lst terms s sum of squres leds to żt (u)rż(u)du = ωt (u)rω(u)du ( ) 1 b T ( ) b ω(u)du b R ω(u)du 3 (b ) ΩT RΩ + 1 3(b ) (Θ + 3Ω)T R(Θ + 3Ω) (8) Consider now the right-hnd side of the inequlity (4). Applying the Jensen inequlity nd noting tht (b ) z(u)du = 6 (Θ + 3Ω), Lemm 2 ensures tht ( I R (ω) 1 b b + 3 (b ) ΩT RΩ + ) T ( ) ω(u)du b R ( ω(u)du π (b ) ) (Θ + 3Ω) T R(Θ + 3Ω). Since 12 π 2, the right-hnd side of the previous inequlity is definite negtive independently of the choice of Θ. Hence, its mximum is reched nd is 0 when Θ = 3Ω. This concludes the proof. Remrk 1 The previous corollry refines the inequlity proposed in?, in which the lst term of the right-hndside of (9) is multiplied by π2 4 which is less thn 3. This proves tht the proposed inequlity delivers more ccurte lower bound of I R (ω) thn the one proposed in?. 3

5 Remrk 2 Since R 0, the second term of (9) is definite positive. It thus implies tht the Jensen inequlity is included in the inequlity proposed (5). It is lso worth noting tht this improvement is llowed by using n extr signl u ω(s)dsdu nd not only ω(u)du. As it ws mentioned previous section, the differentition of the Lypunov-Krsovskii functionl proposed in (2) requires to find lower bound of I R ( ω).in such sitution, the previous lemm is rewritten s follows. Corollry 5 For given mtrix R 0, the following inequlity holds for ll continuously differentible function ω in, b R n : I R ( ω) 1 b (ω(b) ω())t R(ω(b) ω()) + 3 (b ) Ω T R Ω, where Ω = ω(b) + ω() 2 b b ω(u)du. (9) The previous inequlity will be employed for the stbility nlysis of time-dely nd smpled-dt systems. 4 Appliction to the stbility nlysis of timedely systems 4.1 Systems with constnt nd known dely Consider liner time-dely system: { ẋ(t) = Ax(t) + Ad x(t h) + A D x(s)ds, t 0, t h x(t) = φ(t), t h, 0, (10) where x(t) R n is the stte vector, φ is the initil condition nd A, A d, A D R n n re constnt mtrices. The dely is ssumed to be known nd constnt. The following stbility theorem is provided. Theorem 6 Assume tht, for given h > 0, there exist 2n 2n-mtrix P 0, nd n n-mtrices S 0, R 0 such tht the following LMI is stisfied where Ψ(h) = Ψ 0 (h) 1 h F T 2 RF 2 0, (11) Ψ 0 (h) = He(F1 T (h)p F 0 (h)) + S + hf0 T (h) RF 0 (h), A Ad ha D I 0 0 F 0 (h) =, F 1 (h) =, I I hi I I 0 F 2 =, S = dig(s, S, 0n ), I I 2I R = dig(r, 0 2n ), R = dig(r, 3R), Then the system (10) is symptoticlly stble for the constnt nd known dely h > 0. Proof : Consider the functionl given by V (h, x t, ẋ t ) = x T (t)p x(t) + t h xt (s)qx(s)ds + t h θ ẋt (θ)rẋ(θ)dθds. (12) x(t) where x(t) = t h x(s)ds. This functionl is positive definite since P 0, S 0 nd R 0. Differentiting (17) long the trjectories of (10) leds to: V (h, x t, ẋ t ) = ζ0 T (t)ψ 0 (h)ζ 0 (t) ẋ T (s)rẋ(s)ds, t h where ζ 0 (t) = x T (t) x T (t h) 1 t h This eqution hs been obtined by noting tht t h xt (s)ds x(t) = F 1 (h)ζ 0 (t), x(t) = F 0 (h)ζ 0 (t). T. Then the ppliction of Corollry 5 to the integrl defined over the intervl t h, t leds to t h ẋt (s)rẋ(s)ds 1 h ζt 0 (t)f T 2 RF 2 ζ 0 (t). It yields V (h, x t, ẋ t ) ζ0 T (t)ψ(h)ζ 0 (t). Then if (11) is stisfied for given h > 0, then the system (10) is symptoticlly stble for this dely h. Remrk 3 It is worth noting tht the previous theorem only dels with the cse of constnt nd known delys. It does not men tht the considered system is symptoticlly stble for ny dely belonging to the intervl 0, h. Remrk 4 This cse is very specil nd the cse of of different vlue for the discrete nd distributed delys would be more relevnt to study. However, the gol of the present rticle is to show tht the two different problems of distributed nd discrete delys cn be tckled by using unique clss of Lypunov-Krsovskii functionls. 4.2 Systems with time-vrying dely Consider the following clss of systems { ẋ(t) = Ax(t) + Ad x(t h(t)), t 0, x(t) = φ(t), t h, 0, (13) where the dely function h is unknown or time-vrying nd stisfies the following constrints h(t) h m, h M, ḣ(t) d m, d M, t 0, (14) 4

6 where 0 h m h M nd d m d M 1. In such sitution, the following stbility theorem is provided. Theorem 7 Assume tht there exist 3n 3n-mtrix P 0, n n-mtrices S 0, Q 0, R 0 nd 2n 2nmtrix X such tht the following LMIs re stisfied for h = {h m, h M } nd for ḣ = {d m, d M } where Φ 1 (h, ḣ) = Φ 0(h, ḣ) 1 h M Γ T Φ 2 Γ 0, R X Φ 2 = R 0, Φ 0 (h, ḣ) = He(GT 1 (h)p G 0 (ḣ)) + Ŝ + ˆQ(ḣ) nd +h M G T 0 (ḣ) ˆRG 0 (ḣ), T Γ = G T 2 G T 3 G T 4 G T 5, G 2 = I I 0 0 0, G 3 = G 4 = 0 I I 0 0, G 5 = ˆQ(ḣ) = dig(q, (1 ḣ)q, 0 3n), (15) I I 0 2I 0 0 I I 0 2I Ŝ = dig(s, 0, S, 0 2n ), ˆR = dig(r, 0 3n ), R = dig(r, 3R), A A d G 0 (ḣ) = I (1 ḣ)i 0 0 0, 0 (1 ḣ)i I 0 0 I G 1 (h) = hi (h M h)i,, (16) Then the system (13) is symptoticlly stble for ll dely function h stisfying (14). Proof : Consider the functionl given by V (h, x t, ẋ t ) = x T (t)p x(t) + t h(t) xt (s)qx(s)ds + x T (s)sx(s)ds + θ ẋt (s)rẋ(s)dsdθ, (17) where x(t) = x T (t), t h(t) xt (s)ds, T t h(t) x T (s)ds. This functionl is positive definite since P 0, Q 0, S 0 nd R 0. Differentiting the functionl (17) long the trjectories of (13) leds to: V (h, x t, ẋ t ) = ζ1 T (t)φ 0 (h)ζ 1 (t) ẋ T (s)rẋ(s)ds, (18) where ζ 1 (t) = 1 t h(t) t h(t) x(s)ds t h(t) 1 h M h(t) x(t) x(t h(t)) x(t h M ) x(s)ds This eqution hs been obtined by noting tht x(t) = G 1 (h)ζ 1 (t) nd x(t) = G 0 (ḣ)ζ 1(t). The next step consists in splitting the integrl into two integrls, tken over the two intervls t h M, t h(t) nd t h(t), t nd in pplying Corollry 5 to ech of them. It yields ẋ T (s)rẋ(s)ds ζ1 T (t) where G 23 = G 2 G 3 nd G 45 = ( T. 1 h(t) GT 23 RG h M h(t) GT 45 RG 45 ) ζ 1 (t), G 4 G 5. Providing tht there exists mtrix X such tht Φ 2 0, Lemm 3 ensures tht ẋ T (s)rẋ(s)ds 1 h M ζ T 1 (t)γ T Φ 2 Γζ 1 (t), which leds to V (x t, ẋ t ) ζ1 T (t)φ(h, ḣ)ζ 1(t). Finlly, V is negtive definite if there exists mtrix X such tht Φ 2 0 nd if Φ(h, ḣ) 0, for ll (h, ḣ) 0, h M d m, d M. Since the mtrix Φ(h, ḣ) is ffine, nd consequently convex, with respect to h(t) nd ḣ(t), it is necessry nd sufficient to ensure tht Φ(h, ḣ) 0 t the vertices of the intervls 0, h M d m, d M, which concludes the proof. 4.3 Exmples Constnt distributed dely cse Consider the liner time-dely systems (10) with the mtrices tken from? : A =, A d =, A D = (19) An eigenvlue nlysis provides tht the system remins stble for ll constnt delys in the intervl 0.200, In? nd?, stbility is gurnteed for delys over the intervl , nd , , respectively. Using our new inequlity, Theorem 6 ensures stbility for ll constnt delys which belong to the intervl 0.200, which encompsses these previous results nd shows the potentil of Corollry 5. 5

7 d M (= d m) N v? n n? n 2 + n?? n 2 + 3n? n 2 + n? (N=1) n n? n 2 + 9n + 9? n 2 + 8n Th n 2 + 2n Th n 2 + 3n Tble 1 The mximl llowble delys h M for system described in Exmple (20) Unknown time-vrying dely cse We consider the liner time-dely systems (13) with A =, A d =, A D = (20) This system is well-known dely dependent stble system where the mximum llowble dely h mx = cn be esily computed by dely sweeping techniques. To demonstrte the effectiveness of our pproch, results re compred to the literture nd re reported in Tble 1. Tble 1 shows tht our result is competitive with the most ccurte stbility conditions from the literture. For the cse of constnt nd known dely, Theorem 6 delivers the sme result s the one provided by the discretiztion method 1 of? with N = 1 nd with lower number of vribles. For the time-vrying cse, only the conditions provided in? nd in? re less conservtive thn the ones of Theorem 7 for slow vrying delys. However for fst vrying delys, Theorem 7 becomes less conservtive thn the conditions from these rticles. 5 Appliction to smpled-dt systems Let {t k } k N be n incresing sequence of positive sclrs such tht k N t k, t k+1 ) = 0, + ), for which there exist two positive sclrs T min T mx such tht k N, T k = t k+1 t k T min, T mx. (21) The sequence {t k } k N represents the smpling instnts. Consider the smpled-dt system given by t t k, t k+1 ), ẋ(t) = A c x(t) + A s x(t k ), (22) where x R n represents the stte. The mtrices A c nd A s re constnt, known nd of pproprite dimensions. 1 The stbility conditions providedby this method only concerns the cse of constnt nd known delys Adopting the method bsed on looped-functionls proposed in???, the following result is proposed Theorem 8 Let 0 < T min T mx be two positive sclrs. Assume tht there exist n n-mtrices P 0, R 0, S = S T, Q = Q T nd X = X T nd 3n n- mtrices Y 1 Y 2 nd Y 3 tht stisfy Θ 1 (T ) = Π 1 + T (Π 2 + Π 3 ) 0, Π 1 T (Π 3 + Π 4 ) T Y 1 3T Y 2 Θ 2 (T ) = T R 0 0, 3T R (23) for T {T min, T mx } nd where Π 1 = Π 0 1 He{(Y 1 + Y 3 )W 1 + 3Y 2 W 2 } Π 0 1 = He{M T 1 P M 0 W T 1 QM 2 } W T 1 SW 1, Π 2 = M T 0 RM 0 + He{M T 0 SW 1 + M T 0 QM 2 }, Π 3 = M T 2 XM 2, Π 4 = Y 3 M 4, (24) with M 0 = A c A s 0, M 1 = I 0 0, M 2 = 0 I 0, M 3 = 0 0 I, M 4 = 0 A s A c, W 1 = I I 0 nd W 2 = I I 2. Then the system (22) is symptoticlly stble for ll sequence {t k } k 0 stisfying (21). Proof : Consider given integer k 0 nd the ssocited T k T min T mx. The stbility nlysis cn be performed using the qudrtic function V (x) = x T P x where P 0 nd functionl V 0 of the form V 0 (t t k, x) = (t k+1 t) x T (t)(s x(t) + 2Qx(t k )) + (t k+1 t) t k ẋ T (s)rẋ(s)ds + (t k+1 t)(t t k )x T (t k )Xx(t k ), where x(t) = x(t) x(t k ). This functionl is clled looped functionls becuse it stisfies the boundry conditions V 0 (0, x) = V 0 (T k, x) = 0. Define the functionl W = V + V 0 (see??? for more detils). In 6

8 the following, the nottion τ = t t k is dopted. Introduce the vectors ν k (τ) = 1 t τ t k x(s)ds nd ξ k (τ) = x T (t) x T (t k ) ν Tk (τ). Since the vector x(t k ) is constnt over t k t t k t k+1, the following eqution is derived, for ll mtrix Y 3 R 3n n 2ξk T (τ)y 3 t k ẋ(s)ds = 2ξk T (τ)y 3(x(t) x(t k )) = 2τξk T (τ)y 3M 4 ξ k (τ). (25) This expression shows tht there exists reltion between the vectors x(t), x(t k ) nd ν k (τ). Hence, following the proof of Theorem 2 in?, the computtion of the derivtive of W together with the linking reltion (25) Ẇ(τ, x) = ξk T (τ) Π (T k τ)π 2 + (T k 2τ)Π 3 +τπ 4 ξ k (τ) t k ẋ T (s)rẋ(s)ds. Applying Corollry 5 yields t k ẋ T (s)rẋ(s)ds 1 τ ξt k (τ) W1 T RW 1 + 3W2 T RW 2 ξk (τ) Noting tht, for ll mtrices Y i, i {1, 2} in R n 3n, it holds 1 τ (RW i τy i ) T R 1 (RW i τy i ) 0 for ll i {1, 2}, the inequlity 1 τ W T i RW i Y T i holds for i = 1, 2. This ensures W i Wi T Y i + τyi T R 1 Y i Theorems Ex. (20) Ex. (26) Nb of vribles Th. bounds (0, (0, ? (0, 2.53 (0, n 2 + n? (0, 2.53 (0, n n? (0, 2.62 (0, n 2 + 2n Th. 8 (0, 2.87 (0, n 2 + 3n Tble 2 Intervl of llowble synchronous smplings. The theoreticl bounds hve been computed by n eigenvlue nlysis for the cse of synchronous smplings. For this exmple, when the smpling period is chosen constnt (i.e. T k = T min = T mx, for ll k 0), n eigenvlue nlysis of the trnsition mtrix ensures tht the system is stble for ll constnt smpling period in (0, Applying Theorem 8, we prove tht system (26) is symptoticlly stble for ll synchronous smpling over the intervl 0, 1.724, encompssing mny results of the literture s it cn be seen in the Tble 2 but t incresing numericl burden. 6 Conclusions In this pper, we hve provided new inequlity which encompsses the Jensen inequlity. In combintion with simple choice of Lypunov-Krsovskii functionls, this inequlity leds to new stbility criteri for liner timedely nd smpled-dt systems. These new results hve been expressed in terms of LMIs nd hs shown on numericl exmples lrge improvements of existing results using only limited number of mtrix vribles. Ẇ(τ, x) = ξ T k (τ)π(τ, T k )ξ k (τ), where Π(τ, T k ) = Π 1 + (T k τ)π 2 + (T k 2τ)Π 3 + τ Π 4 nd Π 4 = Π 4 + Y 1 R 1 Y1 T + 3Y 2 R 1 Y2 T. Since Π(τ, T k ) is ffine, nd consequently convex, with respect to τ 0, T k, it is sufficient to ensures tht Π(0, T k ) 0 nd Π(T k, T k ) 0, or equivlently Θ 1 (T k ) 0 nd Θ 2 (T k ) 0. The sme rgument on the prmeter T k T min T mx ensures tht Ẇ 0. According to?, the system (22) is symptoticlly stble. 5.1 Exmples Consider gin the system (22) provided in the exmple (20) with A c = A nd A s = A d. Additionlly, we will lso consider the following exmple tken from??. ẋ(t) = x(t) T x(t k ) (26) 7