Complete Quadratic Lyapunov functionals using Bessel-Legendre inequality

Transcription

1 Complete Quadratic Lyapunov functionals using Bessel-Legendre inequality Alexandre Seuret Frédéric Gouaisbaut To cite tis version: Alexandre Seuret Frédéric Gouaisbaut Complete Quadratic Lyapunov functionals using Bessel- Legendre inequality European Control Conference Jun 214 Strasbourg France pp6 214 <al v2> HAL Id: al ttps://alarcives-ouvertesfr/al v2 Submitted on 22 Sep 214 HAL is a multi-disciplinary open access arcive for te deposit and dissemination of scientific researc documents weter tey are publised or not Te documents may come from teacing and researc institutions in France or abroad or from public or private researc centers L arcive ouverte pluridisciplinaire HAL est destinée au dépôt et à la diffusion de documents scientifiques de niveau recerce publiés ou non émanant des établissements d enseignement et de recerce français ou étrangers des laboratoires publics ou privés

2 Complete Quadratic Lyapunov functionals using Bessel-Legendre inequality Alexandre Seuret ac and Frédéric Gouaisbaut ab Abstract Te article is concerned wit te stability analysis of time-delay systems using complete- Lyapunov functionals Tis class of functionals as been employed in te literature because of teir nice properties Indeed suc a functional can be built if a system wit a constant time delay is asymptotically stable Hence several articles aim at approximating teir parameters tanks to a discretization metod or polynomial modeling Te interest of suc approximation is te design of tractable sufficient stability conditions expressed on te Linear Matrix Inequality or te Sum of Squares setups In te present article we provide an alternative metod based on polynomial approximation wic takes advantages of te Legendre polynomials and teir properties Te resulting stability conditions are scalable wit respect to te degree of te Legendre polynomials and are expressed in terms of a tractable LMI I Introduction Delays are inerent of real-time closed loop systems Indeed multiplication of networked controlled systems requires te explicit consideration of te communication delays since delays drastically affect te performances of te overall system Hence tis practical problem motivates a large number of papers dedicated to te stability of linear time delay systems (see [6] and references terein) In te context of te stability analysis of suc time-delay systems te use of Lyapunov-Krasovskii functionals (LKF) as been very popular [4] [8] [11] [16] [18] All teses papers focus on te coice of a particular structure for te LKF wic is usually composed by te sum of several typical terms [6] including quadratic function of te instantaneous state x(t) and integral of quadratic functional of te entire delay state x t (θ) In general all teses functionals are particular case of te well-known complete Lyapunov-Krasovskii functionals (see Teorem 59 in [6]) given by V (x t ) = x T (t)px(t) + 2x T (t) Q(θ)x t(θ)dθ + x t(θ 1 )T(θ 1 θ 2 )x t (θ 2 )dθ 1 dθ 2 + xt t (θ)s(θ)x t (θ)dθ (1) were x t (θ) = x(t + θ) represents te state of te time-delay system and > te delay and were te a CNRS LAAS 7 avenue du Colonel Roce 3177 Toulouse France {aseuretfgouaisb}@laasfr b Univ de Toulouse UPS LAAS F-314 Toulouse France c Univ de Toulouse LAAS F-314 Toulouse France Tis work is supported in part by te ANR project LimICoS contract number 12 BS3 5 1 matrix P is a symmetric positive definite and te matrix functions Q S and T are differentiable (see Section 562 in [6] for more details) Teorem 59 from [6] ensures tat if te solutions of a time delay system is asymptotically stable tis functional is a LKF provided tat te different functions Q and T satisfy some partial differential equations wic is not an easy task especially for ig dimensional delay systems In practice numerically cecking te existence of suc functionals often requires an approximation of tis matrix functions in an appropriate manner In [6] a discretization metod was proposed were te functions Q T and S were cosen piecewise linear and te conditions are presented troug te LMI setup In [12] tese matrices were cosen as polynomials functions and te numerical test was performed using te SOSTOOLS In te present paper we aim at presenting anoter approximation metod wic is also based on te polynomial approximation of te functions Q and T but using te particular setup of te Legendre polynomials Indeed tese polynomials are frequently used in te approximation teory because of teir relevant properties wic are described in te following section Tanks to te introduction of tese polynomials we are able to provide a new integral inequality wose conservatism can be made arbitrarily small Tis is te core tool for developing a set of new sufficient conditions indexed by N te degree of te polynomials modeling te parameters of te complete LKF It is proved also tat tis set forms an ierarcy wit respect to te pair ( N) in te sense tat increasing N improves te result Finally two examples sow te effectiveness of te metod Notations: Trougout te paper R n denotes te n-dimensional Euclidean 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 and positive definite Te set S n + represents te set of symmetric positive definite matrices of R n n Te set of continuous functions from an interval I R to R n wic are square integrable is denoted as space L 2 (I R n ) Te symmetric matrix A B C stands A B for diag(a B) stands for te diagonal matrix B T ] C Moreover for any square matrix A R n n we [ A B define He(A) = A + A T Te matrix I represents te identity matrix of appropriate dimension Te notation nm stands for te matrix in R n m wose entries are

3 zero and wen no confusion is possible ) te subscript will be omitted Te notation refers to te binomial coefficients given by k! (k l)!l! ( k l II New integral inequalities A Legendre polynomials In tis article we aim at taking advantages of te Legendre polynomials to provide new integral inequalities and a new metod to construct complete Lyapunov- Krasovskii functionals Let first recall te definition and te basic properties of te Legendre polynomials Definition 1: Te Legendre polynomials considered over te interval [ ] are defined by k ( ) l u + k N L k (u) = ( 1) k p k l ) ( k + l l ) l= ( wit p k l = ( 1) l k l Anoter reason for employing tese polynomials comes from teir nice properties tat are summarized below wic will be useful in te latter developments Property 2: Te Legendre polynomials described in Definition 1 satisfy te following properties: P1 Ortogonality: (k l) N 2 { k l L k (u)l l (u)du = 2k+1 k = l (2) P2 P3 Boundary conditions: k N L k () = 1 L k () = ( 1) k Differentiation: d du L k(u) = if k = and d du L k(u) = k 1 i= (2i+1) (1 ( 1) k+i )L i (u) if k 1 Proofs of tese properties can be found in [3] B Bessel-Legendre inequalities Based on te Legendre polynomials and an application of Bessel s inequality [3] we obtain te following lemma Lemma 3: Let x L 2 (I R n ) and R S n + Te integral inequality x(u)rx(u)du 1 olds for all N N were Ω Ω N T R N Ω Ω N R N = diag(r 3R (2N + 1)R) Ω k = L k(u)x(u)du for all k N (3) Proof: Consider a function x in L 2 (I R n ) and a matrix R in S + n Define te function z by z(u) = x(u) N k= 2k + 1 Ω k L k (u) From its definition z is in L 2 (I R n ) and te integral zt (u)rz(u)du exists From te ortogonal property of te Legendre polynomials one as zt (u)rz(u)du = xt (u)rx(u)du 2 ( N ) k= 2k+1 L k(u)x(u)du RΩ k + N ) 2 k= L2 k (u)duωt k RΩ k ( 2k+1 Finally by noting tat Ω k = L k(u)x(u)du and ( 2k+1 ) 2 L2 2k+1 k (u)du = it yields zt (u)rz(u)du = xt (u)rx(u)du N k= 2k+1 ΩT k RΩ k Te proof is concluded by noting tat since R te left and side of te previous equation is positive definite wic directly implies te inequality (3) In te remainder of tis paper inequality (3) will be recalled as Bessel-Legendre (B-L) Inequality Remark 1: Te previous inequality encompasses te Jensen inequality [6] and te recent Wirtinger-based integral inequality [15] as te particular cases N = and N = 1 respectively Tus te set of inequalities provided in Lemma 3 represents more general formulation tan tese two inequalities Additionally te Parseval identity proves tat te inequalities (3) becomes non conservative as N goes to III Application to te stability analysis of systems wit a discrete delay Consider a linear time-delay system of te form: { ẋ(t) = Ax(t) + Ad x(t ) t x(t) = φ(t) t [ ] (4) were x(t) R n is te state vector φ is a continuous functions wic represents te initial conditions and A and A d are constant matrices Te delay is assumed to be constant In tis section we will sow ow te previous inequalities can be applied to te stability analysis of time delay systems A Coice of a Lyapunov-Krasovskii functional In tis subsection we aim at coosing a new structure for te LKF based on te use of Legendre polynomials Considering te LKF of te form (1) we propose to model te different matrices P Q(θ) T(θ s) and S(θ) as polynomials wit respect to te variables θ and s Contrary to an SOS formulation [12] tese polynomials are expressed in terms of te Legendre basis as follows: Q(θ) = N Q i L i (θ) T(θ s) = N N L i (θ)l j (s)t ij i= i= j= were te matrices Q i T ij = Tji T i { N} j { N} ave to be optimized Te polynomial matrix S(θ) is cosen as a linear function wit respect to θ and is terefore simply expressed wit te canonical basis S(θ) = S + ( + θ)r

4 Hence te functional V N is rewritten as : V N (x t ) = x T N (t)p N x N (t) + t t xt (s)(s + ( t + s)r)x(s)ds were P N = P Q T Q T N Q Q N T T N T N T NN and were extended state defined as: x t () x N (t) = L (s)x t (s)ds N L N(s)x t (s)ds (5) collects te current state and te projections of te state function x t to te N first Legendre polynomials B Systems wit constant and known delay We present in tis sub-section a first stability result for time-delay systems wic is based on te proposed LKF (5) and te use of te B-L inequality developed in te previous section Based on te previous inequalities te following stability teorem is provided by te use of Lemma 3 wit an arbitrary N Teorem 4: For a given integer N 1 and a constant delay assume tat tere exist a matrix P N S + (N+1)n and two matrices S R S + n suc tat te following LMI is satisfied Φ + N () := P N + 1 diag( S 3S (2N 1)S) (6) and were Φ N () = Φ N() diag( R N ) (7) Φ N () = He ( G T N ()PH N) + SN () S N () = diag{s [ + R S (N+1)n } ] I G N () = n nn(n+1) n(n+1)n n(n+1)n I n(n+1) H N = [ FN T ΓT N () ΓT N (1) ΓT N (N) ] T and were F N = [ A A d nn(n+1) ] Γ N (k) = [ I ( 1) k+1 I γnk I γn Nk I ] { (2i + 1)(1 ( 1) γnk i = k+i ) if i k if i k + 1 Ten te time delay system (4) is asymptotically stable for te constant delay Proof: Consider te LKF (5) since S and following te procedure provided in [6] Lemma 3 can be applied to te second term of V N to give a lower bound of te functional In order to be consistent wit te definition of x N Lemma 3 is considered wit te order N It tus yields V N (x t ) x T N (t)φ+ N () x N(t) + t t ( t + s)xt (s)rx(s)ds Ten te positive definiteness of V N results from te condition S R and Φ + N Let us concentrate on te differentiation of te functionals along te trajectories of te system To do so let us first define te vector x t () x t () ξn(t) T 1 = L (s)x t (s)ds N L N(s)x t (s)ds 1 wic will be employed to expressed te derivative of te functional Te computation of V N leads to were V N (x t ) = 2 x T N (t)p N x N (t) + x T t ()(S + R)x t () x T t ()Sx t () xt t (s)rx t (s)ds ẋ t () x T N(t) = L (s)ẋ t (s)ds L N(s)ẋ t (s)ds (8) Te following setup consists of te expression of te vector x N (t) using te augmented vector ξ N (t) On te first and it is clear tat ẋ t () = A x t () + A 1 x t () = F N ξ N (t) On te oter and for any positive integer k N an integration by parts ensures tat L k(s)ẋ t (s)ds = L k ()x t () L k ()x t () L k (u)x t (u)du Tanks to properties P2 and P3 of te Legendre polynomials te following expression is derived L k(s)ẋ t (s)ds = x t () ( 1) k x t () k 1 i= γi Nk 1 L i(u)x(u)du = Γ N (k)ξ N (t) Ten by putting togeter all te components of x N (t) we obtain x N (t) = H N ()ξ N (t) Finally by noting tat x N (t) = G N ()ξ N (t) it yields V N (x t ẋ t ) = ξ T N(t)Φ N ()ξ N (t) x T t (s)rx t (s)ds Applying Lemma 3 to te order N ensures tat xt t (s)rx t (s)ds ξ T N (t)diag( R N)ξ N (t) (9) Reinjecting tis inequality into (9) leads to V N (x t ẋ t ) ξ T N (t)φ N ()ξ N(t) Hence if te LMI (7) is satisfied te delay system (4) is asymptotically stable for te constant delay Remark 2: It is interesting to notice tat starting from a particular coice for a complete LKF te proposed

5 stability criterion is equivalent to consider te stability of te following system: ẋ(t) = A x(t) + A 1 x(t ) L (s)ẋ t (s)ds = x(t) x(t ) L N(s)ẋ t (s)ds = x(t) ( 1) k x(t ) k 1 i= γi Nk L i(u)x(u)du studied wit te basic LKF V N (x t ) = x T N (t)p N x N (t) + t t xt (s)(s + ( t + s)rx(s)ds (1) Hence te main tool wic allows to deal wit suc a simplified LKF is te Bessel inequality wic connects te different states togeter Indeed tis inequality links te L 2 [ ] norm of te original delay state x t (θ) wit te L 2 [ ] norms of its projection onto te set of polynomial of degree less tan N C Delay range stability In Teorems 4 te delay is supposed to be perfectly known Terefore it ensures stability of te delay system only for te pointwise delay Te following subsection extends tis first result by considering tat te delay is unknown but belongs to a prescribed interval [ 1 2 ] We aim terefore at providing a criterion wic ensures stability for all constant delays in tis pocket Teorem 5: For a given integer N and an uncertain constant delay [ 1 2 ] assume tat tere exist a matrix P N = P T N R (N+1)n (N+1)n and two matrices S R S + n suc tat te LMIs Φ + N ( 2) and Ψ N ( 2 ) = Ψ N ( 2 ) 1 2 diag( R N ) (11) old for = { 1 2 } were Ψ N ( 2 ) = He ( G T N ()PH N) + SN ( 2 ) and were S N F N G N H N and Γ N (k) are defined in Teorem 4 Ten te time delay system (4) is asymptotically stable for any constant delay in te interval [ 1 2 ] Proof: Consider te LKF given by Ṽ N (x t ẋ t ) = x T N (t)p N x N (t) + t t xt (s)sx(s)ds + t t 2 ( 2 t + s)x T (s)rx(s)ds (12) were x N (t) as te same definition as in te proof of Teorem 4 Te only difference wit respect to te constant and known delay appears in te definition of te last term of V N wic is defined wit te delay 2 instead of Following te proof of Teorem 4 it yields Ṽ N (x t ẋ t ) = ξn T (t)ψ N( 2 )ξ N (t) 2 x T (t + s)rx(t + s)ds ξn T (t)ψ N( 2 )ξ N (t) xt (t + s)rx(t + s)ds (13) Applying Lemma 3 to te order N leads to Ṽ N (x t ẋ t ) ξ T N(t)Ψ N ( 2 )ξ N (t) By noting tat Ψ N ( 2 ) is affine in it is easy to see tat Ψ N ( 2 ) = 1 Ψ N ( 2 2 ) + 2 Ψ N ( 1 2 ) Hence it suffices to ensure Ψ N ( 1 2 ) and Ψ N ( 2 2 ) to guarantee tat te system is asymptotically stable for all constant delay [ 1 2 ] IV Hierarcy of LMI stability conditions Tis section aims at proving tat te previous stability conditions form a ierarcy of LMI conditions Tis is formulated in te following teorem based on te stability conditions of Teorem 4 Teorem 6: For any time delay system (4) define te set H N by { > : (PN S(N) R(N)) S H N := (N+1)n (S n + ) 2 st Φ + N () Φ N () } Ten H N H N+1 olds for all N Proof: Let N N If H N is empty te inclusion is trivial Assume tat H N is not empty and consider an element H N From te definition of H N tere exist P N = PN T S(N) and R(N) suc tat Φ+ N () and Φ N () Taking advantages of te construction of te LKF (5) we suggest te matrices PN P N+1 = ǫi S(N + 1) = S(N) = S R(N + 1) = R(N) = R were ǫ > is a scalar to be cosen Clearly tis coice of matrices ensures tat te functional V N+1 is positive definite According to te construction of te matrices G N H N F N and S N te following relation olds HN H N+1 = Nnn [ Γ N+1 (N + 1) ] GN () Nnn G N+1 () = [ nnn I SN S N+1 = Nnn nnn n F N+1 = F N n From tese expressions te matrix Φ N+1 () can be expressed using te matrix Φ + N () as follows Φ + N+1 () = ΦN () { (2N + 3)R } +ǫhe Γ T N+1 (N + 1) I Ten since Φ + N () R te first term of te previous expression is negative definite It implies tat tere exists a sufficiently small ǫ for wic Φ + N+1 () ]

6 wic proves tat belongs to H N+1 wic allows to conclude tat H N H N+1 Since Teorem 4 only provides sufficient stability condition te sequence of sets {H N } N N is an increasing sequence of set representing an inner approximation of te stability pockets However te previous teorem does not prove tat te conditions of Teorem 4 will converge to te analytical bounds of te delay An analogous teorem sowing tat Teorem 5 also forms a ierarcy of stability conditions can be obtained V Examples Te purpose of te following section is to illustrate our propositions on an academic and also a non trivial example Because of space limitations only te results provided by Teorem 4 are presented A Example 1 First consider te well known academic example of te form (4) wit te matrices 2 1 A = A 9 d = (14) 1 1 It can be proved by direct inspection of te caracteristic equation tat tis delay system is stable for all delays belonging to [ 61725] To illustrate te main teorem Table I compares te upperbound calculated by Teorem 4 wit tose found in te literature All papers except [9] use Lyapunov teory in order to derive stability criteria Some results based on Jensen lemma bounding tecnique gives nearly te same results [4] [8] [16] [18] Oter metods wic employ an augmented Lyapunov functional (wit triple integral term) can go furter but wit a numerically increasing burden Te partitioning approac proposed by [7] based on te discrete delay decomposition of a simple LKF is very efficient and goes along wit an important numerical complexity Two tecniques [5][12] are based on te structure of te complete quadratic functional Using an SOS optimisation setup [12] approximates te matrices (wic constitute te complete LKF (1)) as polynomials of a prescribed upperbound [5] proposes a partionning complete Quadratic LKF along wit a linear modeling of te matrices As expected all te proposed metod delivers very good results results wit a low numerical burden 1) Example 2: Tis example is taken from te dynamics modeling of macining catter [2] [17] and as been barely studied using te Lyapunov-Krasovskii Teorem and LMI conditions We consider: { ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) wit 1 1 A = B = 1 C = T A delayed static output feedback controller is proposed: u(t) = Ky(t) + Ky(t ) were K is te gain of te controller and is an unknown constant delay Te resulting dynamics is tus modeled by a time-delay system: ẋ(t) = A x(t) + A 1 x(t ) wit A = A BCK and A 1 = BCK Considering K = 1 and = 3 we are interesting firstly weter te classical metods of te literature could prove te stability and secondly if so to determine teir computational complexity Surprisingly te results based on Jensen lemma do not prove te stability as also te partitioning metod of [7] Te discretized Complete LKF of [5] is not able to guarantee stability wit a discretization step lower tan N = 12 (we did not look for iger levels because of te computation time) Te SOS optimization of [12] requires a relaxation of order N = 1 wic corresponds to consider for te matrices R Q S as polynomials of order 1 Our proposal gives a positive answer only for N = 6 VI Discussions Inspecting Example 1 te stability conditions provided in Teorem 4 is very competitive in terms of complexity wit respect to te most efficient stability conditions (Discretization and Sum of Squares metods) In addition our metod is very effective on te second example wen assessing stability is not trivial Te metod based on Legendre polynomials as a relevant feature Indeed Teorem 6 teoretically and elegantly proves tat increasing te degree N of te polynomials always elps in reducing te conservatism To te best of our knowledge tis type of result as not been addressed frequently in te literature In order to improve te efficiency of our metod te following problems ave to be addressed in future works a) A metod to reduce te conservatism to be more competitive wit existing stability conditions must be provided One direction would be to sligtly modify te Lyapunov-Krasovskii functional Indeed it is well-known tat te following integral quadratic term V (ẋ t ) = t t ( t + s)ẋ T (s)rẋ(s)ds significantly elps to reduce te conservatism Tis is te core of te work presented in [14] [15] wic indeed sows an impressive reduction of te conservative wit respect to te stability conditions of tis article b) Anoter key issue is also to extend te proposed results to delay wic are not constant Indeed it is not clear at least for te autors weter our metod based on te Legendre polynomials could be easily adapted to te situation wen te delay is time-varying

7 Teorems max number of variables Teorems max number of variables [4] [8] [16] [18] n n [5] D d = n n [9] n 2 + 9n + 9 [5] D d = n n [2] 512 7n 2 + 4n [5] D d = n n [18] 52 18n n [5] D d = n n [1] n 2 + 5n [12] D p = n 2 + 3n [1] n n [12] D p = n n [19] 53 85n n [12] D p = n 2 + 6n [7]D d = n 2 + 2n T4 N = n 2 + 2n [7]D d = n n T4 N = n n [7]D d = n 2 + 3n T4 N = n 2 + 3n [13] 591 3n 2 + 2n T4 N = n n TABLE I: Results for Example (14) for constant delay Te notations D d and D p stands for te te degree of discretization and for te degree of te polynomial respectively c) Te set of polynomials is dense in te set of continuous functions defined on compact set wic implies x T 2k + 1 (u)rx(u)du = Ω T k RΩ k k= Ten since te only conservatism in te proof of Teorem 4 appears wen employing te integral inequality provided in Lemma 3 tere is some ope to prove tat te conservatism can be arbitrarily reduced Tis would mean tat our proposal could lead to an asymptotically necessary and sufficient stability condition Of course tis is still a result to be proven VII Conclusions In tis paper we provide a novel metod to construct complete Quadratic Lyapunov-Krasovskii functionals proposed originally by [5] Te matrices wic constitute te functional are cosen to be polynomials and are expressed wit te elp of te Legendre polynomials basis Tis new functional can also be viewed as a new simple Lyapunov-Krasovskii functional for an extended state composed by te instantaneous state x t () and te projection of te state x t (θ) onto te polynomial Legendre set wit respect to a well defined inner product An extensive use of Bessel inequality allows to develop efficient criteria at least on examples but wit a large numerical complexity Tis set of stability conditions forms a ierarcy of LMI indexed by te polynomial degree N in te sense tat increasing N reduces te conservatism of te proposed metod Future works will include te study of te asymptotic necessity of tis approac References [1] Y Ariba and F Gouaisbaut An augmented model for robust stability analysis of time-varying delay systems Int J Control 82: [2] Y Ariba F Gouaisbaut and KH Joansson Stability interval for time-varying delay systems In Decision and Control (CDC) 21 49t IEEE Conference on pages [3] W Gautsci Ortogonal Polynomials Computation and Approximation Oxford Science 24 [4] F Gouaisbaut and D Peaucelle A note on stability of time delay systems In 5 t IFAC Symposium on Robust Control Design (ROCOND 6) Toulouse France 26 [5] K Gu A furter refinement of discretized Lyapunov functional metod for te stability of time-delay systems Int Journal of Control 74(1): [6] K Gu V L Karitonov and J Cen Stability of Time-Delay Systems Birkäuser Boston 23 Control engineering [7] Q-L Han A discrete delay decomposition approac to stability of linear retarded and neutral systems Automatica 45(2): [8] Y He Q G Wang L Xie and C Lin Furter improvement of free-weigting matrices tecnique for systems wit timevarying delay IEEE Trans on Automat Control 52(2): [9] CY Kao and A Rantzer Stability analysis of systems wit uncertain time-varying delays Automatica 43(6): [1] JH Kim Note on stability of linear systems wit time-varying delay Automatica 47(9): [11] S-I Niculescu Delay Effects on Stability A Robust Control Approac Springer-Verlag 21 [12] MM Peet A Papacristodoulou and S Lall Positive forms and stability of linear time-delay systems SIAM Journal on Control and Optimization 47(6): [13] A Seuret and F Gouaisbaut On te use of te Wirtinger s inequalities for time-delay systems In 1 t IFAC Worksop on Time Delay Systems (IFAC TDS 12) Boston MA USA 212 [14] A Seuret and F Gouaisbaut Hierarcy of LMI-conditions for te stability of time delay systems Submitted to Automatica 214 [15] A Seuret and F Gouaisbaut Wirtinger-based integral inequality: Application to time-delay systems Automatica 49(9): [16] H Sao New delay-dependent stability criteria for systems wit interval delay Automatica 45(3): [17] R Sipai S Niculescu CT Abdalla W Miciels and K Gu Stability and stabilization of systems wit time delay Control Systems IEEE 31(1): [18] J Sun GP Liu J Cen and D Rees Improved delayrange-dependent stability criteria for linear systems wit timevarying delays Automatica 46(2): [19] J Sun GP Liu and J Cen Delay-dependent stability and stabilization of neutral time-delay systems International Journal of Robust and Nonlinear Control 19(12): [2] J Zang C R Knopse and P Tsiotras Stability of timedelay systems: Equivalence between Lyapunov and scaled small-gain conditions IEEE Trans on Automat Control 46(3):