Exponential stability of time-delay systems via new weighted integral inequalities

Transcription

1 Exponentil tbility of time-dely ytem vi new weighted integrl inequlitie LV Hien nd H Trinh rxiv:54679v [mthoc] 7 My 5 Abtrct In thi pper, new weighted integrl inequlitie (WII re firt derived by refining the Jenen ingle nd double inequlitie It i hown tht the newly derived inequlitie in thi pper encomp both the Jenen inequlity nd it mot recent improvement bed on Wirtinger integrl inequlity The potentil cpbility of the propoed WII i demontrted through ppliction in exponentil tbility nlyi for ome cle of time-dely ytem in the frmework of liner mtrix inequlitie (LMI The effectivene nd let conervtivene of the derived tbility condition uing WII re hown by vriou numericl exmple Index Term Exponentil etimte, time-dely ytem, integrl-bed inequlitie, liner mtrix inequlitie I INTRODUCTION The problem of tbility nlyi nd it ppliction to control of time-dely ytem i eentil nd of gret importnce for both theoreticl nd prcticl reon [] Thi problem h ttrcted coniderble ttention during the lt decde [] [5] Mny importnt reult on ymptotic tbility of time-dely ytem hve been etblihed uing the Lypunov-Krovkii functionl (LKF method in the frmework of liner mtrix inequlitie (LMI [6] It i fct tht ymptotic tbility i ynonym of exponentil tbility [7], nd in mny ppliction, it i importnt to find etimte of the trnient decying rte of time-dely ytem [8] Therefore, gret del of effort h been devoted to tudy exponentil tbility of time-dely ytem [7] [9] To derive n etimte, lo referred to α-tbility, of the exponentil convergence rte of timedely ytem, vriou pproche hve been propoed in the literture For exmple, tte trnformtion ξ = e αt x combined with the Lypunov-Krovkii functionl method [8] [], model trnformtion [3], contructing modified LKF with exponentil weighted function [7], [5] [9], etimting the Lypunov component [4] or modified comprion principle [4], [] However, looking t the literture, it cn be relized tht the propoed method in the forementioned work uully introduce conervtim in exponentil tbility condition not only on the exponentil convergence rte but lo on the mximl llowble dely nd the number of mtrix vrible Therefore, iming t reducing conervtivene of exponentil LV Hien i with the the School of Engineering, Dekin Univerity, VIC 37, nd the Deprtment of Mthemtic, Hnoi Ntionl Univerity of Eduction, Hnoi, Vietnm (e-mil: hienlv@hnueeduvn H Trinh i with the School of Engineering, Dekin Univerity, VIC 37, Autrli (e-mil: hieutrinh@dekineduu tbility condition, nt importnt nd relevnt iue i to improve ome integrl-bed inequlitie In thi pper, we firt propoe ome new weighted integrl inequlitie (WII which re uitble to ue in exponentil tbility nlyi for time-dely ytem We how tht the newly derived inequlitie in thi pper encomp both the Jenen inequlity [] nd ome of it recent improvement bed on Wirtinger integrl inequlity [6], [] We then employ the propoed WII to derive new exponentil tbility condition for ome cle of time-dely ytem in the frmework of liner mtrix inequlitie Numericl exmple re provided in thi pper to how the efficiency nd potentil cpbility of the newly derived WII The ret of thi pper i orgnized follow In Section, ome preliminry reult re preented New weighted integrl inequlitie nd their ppliction in exponentil tbility nlyi for ome cle of time-dely ytem re preented in Section 3 nd Section 4, repectively Section 5 provide numericl exmple to demontrte the effectivene of the obtined reult The pper end with concluion nd reference II PRELIMINARIES It cn be relized in mny contribution tht, to derive the exponentil etimte for time-dely ytem, widely ued pproch i the ue of weighted exponentil Lypunov- Krovkii functionl [7] For exmple, functionl of the form V(x t = τ t+ e α(u t ẋ T (urẋ(udud ( where x i the tte vector, clr α >,τ > nd mtrix R >, h been ued in mny work in the literture [5] [9] The derivtive of V(x t i given by V(x t = τẋ T Rẋ t τ e α( t ẋ T (Rẋ(d ( In order to generte LMI condition, n etimte on the econd term of ( i obviouly needed The problem ried here i how to find tighter lower bound of weighted integrl of qudrtic term in the following form I w (ϕ,α = e α( b ϕ T (Rϕ(d where α > i clr, ϕ C([,b],R n nd R i ymmetric poitive definite mtrix in R n n, R S + n When α = we write I(ϕ inted of I w (ϕ,

2 Inpired from the proof of the Jenen inequlity [], we hve the following reult which referred in thi pper to Jenen-bed weighted integrl inequlitie (WII in ingle nd double form Lemm For given mtrix R S n +, clr b >, α >, nd function ϕ C([,b],R n, the following inequlitie hold I w (ϕ,α α ( TR ( ϕ(d ϕ(d, (3 γ e α(u b ϕ T (urϕ(udud α γ ( TR ( ϕ(udud where γ k = e α(b k α j (b j j= j!, k ϕ(udud, Proof: By tking integrl of the inequlity [ ] e α( b ϕ T (Rϕ( ϕ T ( ϕ( e α(b R we obtin [ ] b I w (ϕ,α ϕt (d ϕ(d ρ(αr which implie (3 by Schur complement The proof of (4 i imilr nd thu it i omitted here Remrk Obviouly α γ > e α(b for ll α >,b > Therefore, (3 give new lower bound in comprion to the common etimte I w (ϕ,α e α(b I(ϕ Remrk When α pproche zero the previou inequlitie led to the Jenen inequlity in ingle nd double form, repectively More preciely, from the fct thtlim γ k α + = (b k α k k! we redily obtin the following reult ϕ T (Rϕ(d ( TR ( ϕ(d ϕ(d, b (5 ϕ T (urϕ(udud ( (b TR ( ϕ(udud III NEW WEIGHTED INTEGRAL INEQUALITIES (4 ϕ(udud In thi ection, ome new weighted integrl inequlitie re derived by refining (3, (4 In the following, let u denote Jw(ϕ,α g = I w (ϕ,α α ( TR ( ϕ(d ϕ(d γ the gp of (3 By refining (3 we find new lower bound for J w (ϕ,α other thn zero Firt, let u introduce the following nottion for given clr b >, α >, nd ϕ C([,b],R n l = b, A α = γ α (+γ l γ, (6 L = [ { ] b αγ γ, ζ = col ϕ(d, ϕ(udud By uing the Tylor erie expnion of exponentil function, it cn be verified tht A α > for ny α > We lo ue the notion of Kronecker product A B for mtrice A R n m,b R q r For more detil bout the Kronecker product, we refer the reder to [3] Lemm For given n n mtrix R >, clrα >, nd function ϕ C([,b],R n, the following inequlity hold J g w(ϕ,α α ρ ζ T (L T L Rζ (7 ( Aα where ρ = αγ [ γ = γ γ γ (αl e αl] Proof: For ny ϕ C([,b],R n, we define n pproximtion function ψ C([,b],R n follow ψ = ϕ αeα(b t ϕ(d + hχ (8 γ where h i rel vlued function on [,b] nd χ R n i contnt vector which will be defined lter For brevity we let w = e α(b t nd predefine h = wp, where p belong to P k, the et of polynomil of order le thn k A prior computtion give J g w (ψ,α = Jg w (ϕ,α+j w(hχ T Rχ α h(dχ T R ϕ(d γ ( +χ T R p( ϕ(d+p ( ϕ(udud + p ( ϕ(vdvdud, (9 where J w (h = u [ w (h (d α γ ( h(d ] Now, for ny p P which we cn write p = c +c t, c Then h(d = p(eαl p(b + c γ α α, w (h (d = eαl p ( p (b α + c ( e αl p( p(b α + c γ α 3 nd thu J w (h = Aα α c From (9 we obtin J g w (ψ,α = Jg w (ϕ,α+ A αc α χt Rχ γ c αγ χ T R(L I n ζ By Lemm J g w(ψ,α which led to } ( J g w (ϕ,α A αc α χt Rχ+ γ c αγ χ T R(L I n ζ ( Herefter, we will denote by R(Jw(ϕ,α g the right-hnd ide of ( Now we define vectorχin the formχ = λ c (L I n ζ, where λ i clr, then R(Jw g (ϕ,α = ( γ λ A α λ ζ T (L T α γ L Rζ

3 The function f(λ = γ γ λ A α λ ttin it mximum γ A αγ t λ = γ A αγ Then it follow from ( tht J g w(ϕ,α αγ A α(αγ ζ T (L T L Rζ which complete the proof Remrk 3 It i intereting tht etimte (7 doe not depend on the election of firt-order polynomil p P In other word, inequlity (7 cn be derived from (8 for ny function h of the form (c +c te α(b t, c Of coure, when c = then (8 led to (3 Remrk 4 By repeting the proof of Lemm with the pproximtion ψ = ϕ α w γ ϕ(udud + wpχ ( where w = e α(b t nd p P then (4 led to double WII formulted in the following lemm Lemm 3 For given n n mtrix R >, clr α >, nd function ϕ C([,b],R n, the following inequlity hold where e α(u b ϕ T (urϕ(udud (3 α γ ˆζT (L T L Rˆζ + 4α ρ ˆζT (L T L Rˆζ (4 ˆζ { = col ϕ(udud, } b u ϕ(vdvdud, L = [ ], L = [ αγ γ ], ρ = B α = γ α [αl+(αl γ]l γ ( αγ γ Bα nd Remrk 5 The following fct cn be found by uing Tylor erie of the exponentil function α lim = 3 α + ρ b, lim L = ˆL = [ b ], α + α 4 lim = α + ρ (b, lim L = ˆL = [ 3 b ] α + Therefore, when α pproche zero we obtin the following reult which re the me thoe derived by the Wirtinger inequlity in ingle nd double form [6], [] ϕ T (Rϕ(d b ζt (L T L +3ˆL T ˆL Rζ (5 ϕ T (urϕ(udud (b ˆζ T (L T L +8ˆL T ˆL Rˆζ (6 Remrk 6 The reult obtined in Lemm nd Lemm 3 cn be extended uing (8 nd ( where h = e α(b t p nd p belong to the et of higher order polynomil, nd then new lower bound for (7, (4 cn be derived Thi will be ddreed in future work IV EXPONENTIAL ESTIMATES FOR TIME-DELAY SYSTEMS Thi ection im to demontrte the effectivene of the newly weighted integrl inequlitie propoed in thi pper through ppliction to exponentil tbility nlyi for two cle of time-dely ytem A Sytem with dicrete nd ditributed contnt dely Conider the following time-dely ytem ẋ = A x+a x(+a x(d, t, x = φ, t [ h,], (7 where A,A,A R n n re given contnt mtrice, h i known time-dely, φ C([ h,],r n i the initil condition We recll here tht [7], [3], for given σ >, ytem (7 i exponentilly tble with convergence rte σ if there exit β > uch tht ny olution x(t,φ of (7 tifie x(t,φ β φ e σt, t Let e i R n 4n defined by e i = [ n (i n I n n (4 in ], i =,,4 We denote A = A e + A e + A e 3 nd the following mtrice F = e A e 3, F = e e, e 4 he e 3 [ ] [ ] e e F = he e, F he e 3 = 3 3 h, /e e 4 Π = F T PF +F T PF +αf T PF, Π = e T Qe e αh e T Qe +A T (hr+h /ZA, Π = α γ (e e T R(e e + α ρ F T (L T L RF, Π 3 = α γ (he e 3 T Z(he e 3 + 4α ρ F T 3 (L T L ZF 3 wherel,l ndγ,γ,ρ,ρ re defined in (3, (4, (7, (4 with = h,b = Theorem Aume tht, for given α >, there exit ymmetric poitive definite mtrice P R 3n 3n, Q,R,Z R n n tifying the following LMI Π +Π Π Π 3 < (8 Then ytem (7 i exponentilly tble with convergence rte σ = α/ Proof: Conider the following Lypunov-Krovkii functionl V(x t = x T P x+ + + h t+ h e α( t x T (Qx(d e α(u t ẋ T (urẋ(udud t+u e α(θ t ẋ T (θzẋ(θdud { where x = col x, x(d, } x(udud (9 It follow from (9 tht V(x t λ min (P x Tking 3

4 derivtive of V(x t long trjectorie of (7 we obtin V(x t +αv(x t = χ T ( Π +Π χ h e α( t ẋ T (Rẋ( t+ e α(u t ẋ T (uzẋ(udud ( where χ = col{x,x(, x(d, h t+ x(udud} By pplying Lemm nd Lemm 3 to the firt nd the econd term in (, repectively, we then obtin V(x t +αv(x t χ T (Π +Π Π Π 3 χ ( It follow from (8 nd ( tht V(xt + αv(x t, which yield V(x t V(φe αt Thi led to x(t,φ V(φ λ min(p e α/t The proof i completed Remrk 7 Note tht the exponentil trnformtion z = e σt x,σ >, in generl, i not pplicble to cce the exponentil tbility of ytem (7 becue it led to ytem with time-vrying coefficient When A =, uing the forementioned trnformtion, ytem (7 become ż = (A +σi n z+e σh A z( ( In mny work found in the literture, in order to get exponentil etimte for ytem (7 (with A =, it w trnformed to ( firt nd then ymptotic tbility condition for ( were propoed However, thi pproch uully produce conervtim in exponentil tbility condition due to the fct tht the exponentil tbility of (7 (with A = i jut equivlent to the boundedne of ( which i le retrictive then ymptotic tbility Differ from thoe, nd dicued in Section in thi pper, we here propoe n improved pproch ued in exponentil tbility nlyi for time-dely ytem by employing our new weighted inequlitie derived in Lemm nd Lemm 3 in thi pper B Sytem with intervl time-vrying dely Conider cl of liner ytem with intervl time-vrying dely of the form { ẋ = Ax+A d x(, t (3 x = φ, t [ h,] where A,A d R n n re given contnt mtrice, h i time-vrying dely tifying h h h, where h,h re known contnt involving the upper nd the lower bound of time-vrying dely Let e i = [ n (i n I n n (7 in ], i =,,,7 We denote A = Ae +A d e 3 nd x t h χ = col x( x(d x(, h h x(d, x( h h x(d Υ(h = col{e,h e 5,(h h e 6 +(h he 7 }, Υ = col{a,e e,e e 4 }, Υ = col{e e,h (e e 5 }, Υ = col{e e 3,e +e 3 e 6 }, Υ 3 = col{e 3 e 4,e 3 +e 4 e 7 }, = [Υ T ΥT 3 ]T, Ω (h = Υ(h T PΥ +Υ T PΥ(h+αΥ(hT PΥ(h, Ω = e T Q e e αh e T Q e +e αh e T Q e e αh e T 4Q e 4, Ω = A T ( h R +h eαh R A, Ω 3 = αh (e e T R (e e + αh Υ T γ ρ ( L T L R Υ, γ = e αh, γ = γ αh, L = [ α γ / γ ], [ γ (αh e αh], h = h h ρ = γ γ Theorem Aume tht there exit ymmetric poitive definite mtrice P R 3n 3n, Q i,r i R n n, i =,, nd mtrix X R n n uch tht the following LMI hold for h {h,h } ] [ R X Π =, (4 R Ω(h = Ω (h+ω +Ω Ω 3 e αh T Π <, (5 where R = dig{r,3r } Then ytem (3 i exponentilly tble with convergence rte α/ Firt, we need the following lemm Lemm 4 If Ω(h < nd Ω(h < then Ω(h <, h [h,h ] Proof: It i obviou tht d dh Ω(h = αγ T PΓ, where Γ = [ 7n n (e 6 e 7 T ] T Therefore, Ω(h i convex qudrtic function with repect to h Thi complete the proof Lemm 5 (Improved reciproclly convex combintion [4] For given ymmetric poitive definite mtrice R R n n, R[ R m m ], if there exit mtrix X R n ch tht R X then the inequlity R [ δ R ] [ ] R X δ R R hold for ll δ (, Proof: Inpired from [4], we now conider the following LKF V(x t = χ T Pχ + e α( t x T (Q x(d h + e α( t x T (Q x(d (6 +h h +h h h t+ t+ e α(u t ẋ T (ur ẋ(udud e α(u+h t ẋ T (ur ẋ(udud 4

5 where χ = col{x, x(d, h x(d} It follow from (6 tht V(x t λ min (P x In regrd to the fct χ = G (hχ nd d dt χ = G χ, the derivtive of (6 long trjectorie of (3 give V(x t +αv(x t = χ T (Ω (h+ω +Ω χ h h h By Lemm we hve h e α( t ẋ T (R ẋ(d e α(+h t ẋ T (R ẋ(d (7 e α( t ẋ T (R ẋ(d χ T Ω 3χ (8 Corollry Sytem (3 i ymptoticlly tble for ny dely h [h,h ] if there exit ymmetric poitive definite mtrice P R 3n 3n, Q i,r i R n n, i =,, nd mtrix X R n n tifying (4 nd the following LMI hold for h {h,h } where Ω (h+φ Φ T Π <, (3 Ω (h = Υ(h T PΥ +Υ T PΥ(h, Φ = e T Q e e T Q e +e T Q e e T 4Q e 4 +A T ( h R +h R A, Φ = Υ T 4 dig{r,3r }Υ 4, Υ 4 = col{e,e +e e 5 } Next, by plitting h e α(+h t ẋ T (R ẋ(d = h h e α(+h t ẋ T (R ẋ(d where χ = Υ χ nd χ 3 = Υ 3 χ Note tht the previou inequlity i till vlid when h tend to h or h Thi led to h h e α(+h t ẋ T (R ẋ(d e αh χ T T Π χ Combining (7-(9 we then obtin (9 V(x t +αv(x t χ T Ω(hχ (3 By Lemm 4, (5 implie tht Ω(h < for ll h [h,h ] Therefore, if (5 hold for h = h nd h = h then, from (3, V(xt + αv(x t which conclude the exponentil tbility of (3 with gurnteed decy rte σ = α/ The proof i completed Remrk 8 When α pproche zero, by Remrk 5 nd Theorem, we obtin the me ymptotic tbility condition for ytem (3 derived from improved Wirtinger inequlity [4] V EXAMPLES Exmple Conider the following ytem [6], [] [ ] [ ] t ẋ = x+ x(d (3 + e α(+h t ẋ T (R ẋ(d An eigenvlue nlyi [6] how tht (3 remin tble for contnt dely in the rnge h [, 4] By the the econd integrl term of (7 cn be bounded by (5 nd Wirtinger-bed inequlity pproch, Theorem 6 in [6] nd Lemm 4 follow Theorem in [] gurntee the ymptotic tbility of (3 h for h in the intervl [, 877] nd [, 954], repectively h e α(+h t ẋ T (R ẋ(d In Theorem we fix α t, it i found tht (8 i feible for h [, 9778] Thi clerly how reduction h e αh χ T h h R χ h e αh h h χt 3 R of conervtim of Theorem χ 3 [ ] [ ] Exmple Conider ytem (7 with the mtrice tken T h [χ ] χ = e αh h h R from the literture χ 3 h R [ ] [ ] [ ] h h χ 3 [ ] T ][ ] A =, A =, A = χ [ R X χ e αh χ 3 R χ 3 It i urpriing tht, for thi ytem, the exponentil tbility criteri propoed in [7] [9], epecilly [9], re not feible for ny h >, α > By uing the nottion ] given in [ [4], we hve M = A M + A = Thu, for ny 3 [ ] v = (v,v T R v, v >,v >, Mv = > 3v +v Thi how tht the tbility condition propoed in [4] nd [] re not pplicble to thi ytem Note lo tht, ince Re(eig(A + A = 5 >, the dely-free ytem i untble nd the reult to cce exponentil tbility of thi ytem re much more crce By employing the Wirtinger-bed integrl inequlity, ignificnt improvement of the ymptotic tbility condition w provided in [6] To compre with our pproch, we pply Theorem 6 in [6] to ( The obtined reult in Tble how tht, thnk to our new weighted integrl inequlitie propoed in Lemm nd Lemm 3, Theorem in thi pper give ignificntly better reult The imultion reult preented in Figure i tken with h = 6, σ = 45, incrementl tep = 4 nd initil condition φ = [ ] T It cn be een tht the tte trjectory z = e σt x i bounded, nd thu, 5

6 TABLE I DECAY RATE σ FOR VARIOUSh IN EXAMPLE 5 h NoDv [6] n +n Theorem n +3n NoDv: Number of Deciion vrible z = e 45 t x 3 e 45 t x e 45 t x Time (ec Fig Trjectory z = e 45t x with h = 6 x exponentilly converge to the origin with decy rte σ = 45 Exmple 3 Conider n ctive qurter-cr upenion ytem with control dely introduced in [5] { ẋ = Ax+Bu(, (33 y = Cx, where x R 4 i the tte, u i the control input, y i the output nd A = k m c, B = k m kt, C = m c T c m c+ct The following prmeter re tken from [5] TABLE II QUARTER-CAR MODEL PARAMETERS m k k t c c t 973kg 4kg 47N/m 5N/m 95N/m 46N/m A ttic output feedbck controller i propoed u = Ky, where K i the controller gin The cloed-loop ytem i then in the form of (3 with A d = BKC For illutrtive purpoe we conider K = It i noted firt tht, in thi ce, the exponentil tbility criteri propoed in [4], [] bed on poitive ytem pproch lo cnnot cce the exponentil tbility of the ytem In [8], ome integrl equlitie were ued to overcome the conervtive etimte However, when mnipulting the derivtive of the Lypunov-Krovkii functionl, ll the integrl term were bndoned (ee, Eq ( in [8] Thi led to the fct tht the propoed condition in [8] re very conervtive We pply the min theorem in [8] to thi exmple, the obtined reult for h = nd vriou h re lited in Tble 3 In [4], exponentil convergence rte of olution w derived by etimting the mximl Lypunov exponent By Theorem 3 in [4], the exponentil convergence rte σ (,λ ], where λ i the unique poitive olution of eqution λ + 87e λh = 77 We pply Theorem in thi pper for h = nd vriou h The obtined reult re expoed in Tble 3 Clerly ignificnt reduction of conervtim i delivered by Theorem Thi how the effectivene of our pproch The imultion reult preented in Figure i tken with h = + 5 in, σ = 546 nd incrementl tep = 4 which illutrte our theoreticl reult e 546t x Fig 3 e 546t x e 546t x e 546t x 3 e 546t x Time (ec Trjectory e 546t x with h = +5 in Exmple 4 Conider ytem (3 with A = [ ], A d = [ ] Thi exmple h been tken from [6] The obtined reult by Corollry lited in Tble 4 Thee reult gin how the effectivene of our pproch in propoed in thi pper 6

7 TABLE III DECAY RATE σ FORh = AND VARIOUSh h NoDv [8] n +65n [4] Theorem n +35n TABLE IV UPPER BOUND OF h FOR VARIOUSh = h 3 7 NoDv [8] n +65n [7] n +55n [6] n +6n Corollry n +35n VI CONCLUSION In thi pper, new weighted integrl inequlitie (WII hve been propoed By employing WII, new exponentil tbility criteri hve been derived for ome cle of time-dely ytem in the frmework of liner mtrix inequlitie Numerou exmple hve been provided to how the potentil of WII nd lrge improvement on the exponentil convergence rte over the exiting method REFERENCES [] K Gu, SI Niculecu, Survey on recent reult in the tbility nd control of time-dely ytem, J Dyn Syt, Me, nd Control 5 ( [] R Siphi, SI Niculecu, CT Abdllh, W Michiel, K Gu, Stbility nd tbiliztion of ytem with time dely, IEEE Control Syt 3 ( [3] Y Li, K Gu, J Zhou, S Xu, Etimting tble dely intervl with dicretized Lypunov-Krovkii functionl formultion, Automtic 5 ( [4] H R Feyzmhdvin, T Chrlmbou, M Johnon, Exponentil tbility of homogeneou poitive ytem of degree one with timevrying dely, IEEE Trn Autom Control 59 ( [5] LV Hien, HM Trinh, A new pproch to tte bounding for liner time-vrying ytem with dely nd bounded diturbnce, Automtic 5 ( [6] A Seuret, F Gouibut, Wirtinger-bed integrl inequlity: Appliction to time-dely ytem, Automtic 49 ( [7] S Mondié, V Khritonov, Exponentil etimte for retrded time-dely ytem: An LMI pproch, IEEE Trn Autom Control 5 ( [8] S Xu, J Lm, M Zhong, New exponentil etimte for time-dely ytem, IEEE Trn Autom Control, 5 ( [9] PL Liu, Exponentil tbility for liner time-dely ytem with dely dependence, J Frnklin Int 34 ( [] F Ren, J Co, Novel α-tbility of liner ytem with multiple time dely, Appl Mth Comput 8 (6 8 9 [] OM Kwon, JH Prk, Exponentil tbility of uncertin dynmic ytem including tte dely, Appl Mth Lett 9 ( [] PT Nm, An improved criterion for exponentil tbility of liner ytem with multiple time dely, Appl Mth Comput ( [3] LV Hien, VN Pht, Exponentil tbility nd tbiliztion of cl of uncertin liner time-dely ytem, J Frnklin Int 346 ( [4] AA Zevin, MA Pinky, Shrp bound for Lypunov exponent nd tbility condition for uncertin ytem with dely, IEEE Trn Autom Control 55 ( [5] LV Hien, VN Pht, New exponentil etimte for robut tbility of nonliner neutrl time-dely ytem with convex polytopic uncertintie, J Nonl Conv Anl ( [6] T Botmrt, P Nimup, VN Pht, Dely-dependent exponentil tbiliztion for uncertin liner ytem with intervl non-differentible time-vrying dely, Appl Mth Comput 7 ( [7] VN Pht, Y Khongthm, K Rtchgit, LMI pproch to exponentil tbility of liner ytem with intervl time-vrying dely, Liner Alg Appl 436 ( 43 5 [8] L Guo, H Gu, J Xing, X He, Aymptotic nd exponentil tbility of uncertin ytem with intervl dely, Appl Mth Comput 8 ( [9] J Co, Improved dely-dependent exponentil tbility criteri for timedely ytem, J Frnklin Int, 35 ( [] PHA Ngoc, Stbility of poitive differentil ytem with dely, IEEE Trn Autom Control 58 (3 3 9 [] K Gu, VL Khritonov, J Chen, Stbility of Time-Dely Sytem, Birkhäuer, Bel, 3 [] MJ Prk, OM Kwon, JH Prk, SM Lee, EJ Ch, Stbility of timedely ytem vi Wirtinger-bed double integrl inequlity, Automtic 55 (5 4 8 [3] WH Steeb, Y Hrdy, Mtrix Clculu nd Kronecker Product: A Prcticl Approch to Liner nd Multiliner Algebr, World Scientific, [4] A Seuret, F Gouibut, E Fridmn, Stbility of ytem with ftvrying dely uing improved wirtinger inequlity, IEEE Conference on Deciion nd Control, Florence, Itly, 3, pp [5] H Li, X Jing, HR Krimi, Output-feedbck-bed H control for vehicle upenion ytem with control dely, IEEE Trn Ind Electron 6 ( [6] Auxiliry function-bed integrl inequlitie for qudrtic function nd their ppliction to time-dely ytem, J Frnklin Int 35 ( [7] WI Lee, SY Lee, P Prk, Improved criteri on robut tbility nd H performnce for liner ytem with intervl time-vrying dely vi new triple integrl functionl, Appl Mth Comput 43 (