THE phenomena of time delays are often encountered in

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1 0 American Control Conference on O'Farrell Street San Francisco CA USA June 9 - July 0 0 Robust stability criteria for uncertain systems with delay and its derivative varying within intervals Luis Felipe da Cruz Figueredo João Yoshiyuki Ishihara Geovany Araújo Borges and Adolfo Bauchspiess Abstract In this paper stability criteria are proposed for linear systems liable to model uncertainties and with the delay and its derivative varying within intervals. he results are an improvement over previous ones due to the development of a new Lyapunov-Krasovskii functional LKF. he analysis incorporates recent advances such as convex optimization technique and piecewise analysis method with new delay-intervaldepedent LKFs terms and a novel auxiliary delayed state. Stability conditions are provided for the cases when the delay derivative is upper and lower bounded when the lower bound is unknown and when no restrictions are cast upon the derivative. he analysis is enriched with numerical examples that illustrate the effectiveness of our criteria which outperform previous criteria in the literature for noal and uncertain delayed systems. I. INRODUCION HE phenomena of time delays are often encountered in various practical systems such as chemical engineering systems biological systems aircraft stabilization networked control systems etc. Nonetheless since time delays can degrade a system s performance and even cause system instability considerable attention has been devoted to the subject of stability analysis and design of systems with timevarying delays see e.g. 9. During the last decade the problem of time-delayed systems stability analysis have been deeply investigated under delay-dependent criteria for the exposure of the delay information leads to less conservative results. Various methods have been taken for deriving stability conditions using different Lyapunov Krasovskii functionals LKFs 6. Particularly the employment of Jensen s inequality instead of the cross-terms bounding 0 is a well-established approach that leads to less conservative results. However this still is a conservative analysis for the time-varying delay is bounded when considering terms in the LKF derivative containing not only the delay bounds but also the delay itself. Instead of bounding the time-varying delay the convex optimization technique incorporated with the Jensen s inequality proved to be effective in 3. Further impovements were obtained using similar technique with different LKFs see e.g Recently new Lyapunov functional candidates inspired on have enriched the stability analysis by extending the piecewise analysis method from to systems with timevarying delays see e.g Particularly 7 9 also explore the information about the delay derivative s lower All the authors are with the Automation and Robotics Laboratory LARA Department of Electrical Engineering University of Brasilia Brasilia Brazil. s: {figueredo ishihara gaborges bauchspiess}@lara.unb.br bound through the employment of delay-interval-dependent terms in the Lyapunov functional. Nevertheless in practice it is very difficult to obtain an exact mathematical model due to environmental noise or slowly varying parameters. Systems with time-varying delays almost inevitably present some uncertainties. However most recent advances in the analysis of systems with time-varying delays aren t fully exploited in most recent works concerning robust stability of delayed systems see e.g. 7. herefore the results from these works are usually more conservative than the results from criteria for delayed systems which do not consider the possibility of model uncertainties see e.g herefore in this paper we present a novel robust stability analysis for uncertain systems with delay and its derivative varying within intervals. New delay-interval-dependent LKF terms that are ignored in previous works are introduced to exploit all possible information about the delay derivative s lower and upper bounds. Moreover we introduce an auxiliary delayed state in order to make further use of the delay s lower bound value. hese methods considerably improve the stability analysis even for systems with no uncertainties. he resulting criteria can be applied for the case when the delay derivative is upper and lower bounded when the lower bound is unknown and when no restrictions are cast upon the derivative characteristics. Numerical examples illustrate the effectiveness of the proposed robust stability criteria which outperform previous criteria in the literature for time-delayed systems with and without uncertainties. II. PRELIMINARIES Consider the following continuous-time linear system with time-varying delay: ẋt Axt+A d xt dt t > 0 xt ρt t τ max 0 where xt R r x is the system s state ρt is a given function which describes the state s initial condition and the matrices A and A d are considered not exactly known but belonging to bounded sets: A A R r x r x and A d A d R r x r x. he continuous function dt denotes the time-varying delay that satisfies τ dt τ max where 0 τ τ max are constants. he time-varying delay is assumed to be either fast varying with no restrictions cast upon the delay derivative or //$ AACC 4884

2 differentiable with given bounds: d dt d max 3 where d d max are constants. Considering the parameter uncertainties equation can be rewritten as: ẋt A+ Axt+A d + A d xt dt 4 he uncertainties A e A d are time-varying matrices with appropriate dimensions which are defined as follows: A Ad DFt EA E Ad 5 where D E A and E Ad are known real constant matrices with appropriate dimensions and Ft represents an unknown time-varying matrix which is Lebesque measurable in t and satisfies Ft Ft I. hroughout this paper the following results will be useful to derive conditions for the establishment of new delaydepedent stability criteria for the uncertain system with timevarying delay 4. Lemma 8 For given scalars r r and matrix M R m m such that r r 0 and M>0 and any vectorial function x : r r R m we have: r r r r x r smxsds xsds r r M r xsds. Lemma 9 Given matrices M M R m m B R r m the following statement x Mx > 0 M + FB+B F >0 holds for some F R m r and any x R m \{0} such that Bx0. III. SABILIY ANALYSIS his section presents the main results of this paper. Firstly we shall similarly to 7 9 divide the delay range τ τ max. Here we will consider two equally spaced subintervals: τ τ and τ τ 3 where τ τ τ 3 τ max and τ τ max + τ. herefore the linear uncertain delayed system 4 can be rewritten as ẋt A+ Axt+ χ τ τ dta d + A d xt dt + χ τ τ dt A d + A d xt dt 6 where χ τ τ :R {0} is the characteristic function of τ τ : { if s τ τ χ τ τ s 0 otherwise. he proposed stability analysis for systems with timevarying delay and model uncertainties is based on the Lyapunov Krasovskii functional candidate Vt 6 i V it 7 where V t χ τ τ dtx dt τ t P + τ dt P xt τ τ τ τ + χ τ τ dt x dt τ t P 3 + τ 3 dt P xt τ 3 τ τ 3 τ τ V t x sq xsds t dt τ xs N N V 3 t xs t τ xs τ +τ N ds N xs τ +τ xs M M V 4 t xs t τ xs τ M M xs τ ds V 5 t τ 0 ẋ sz ẋsdsdβ + τ τ t ẋ s τ t+β τ t+β τ Z ẋsdsdβ +τ τ ẋ sz 3 ẋsdsdβ τ t+β τ +τ 3 τ ẋ sz 4 ẋsdsdβ τ 3 t+β dt V 6 tχ τ τ dt ẋ sr R 3 ẋsdsdβ τ t+β τ + χ τ τ dt ẋ sr 3 R ẋsdsdβ dt t+β 0 + ẋ sr +R ẋsdsdβ dt t+β dt + ẋ sr 3 +R 4 ẋsdsdβ. τ 3 t+β One can note that the if the conditions P P 3+P P >0 P 3 >0 Q 0 Z j >0 j {34} τ τ Z 3 +R R 3 >0 τ 3 τ Z 4 +R 3 R >0 R +R >0 N N R 3 +R 4 >0 N M M N N 0 and M M M 0 8 are satisfied we guarantee the positiveness of 7. Also it can be seen that Vt in 7 is continuous in t since lim V t x tp xt dt τ lim dt τ V 6 t 0 ẋ sr + R ẋtdsdβ τ t+β τ + ẋ sr 3 + R 4 ẋtdsdβ. τ 3 t+β In the following we propose novel robust stability criteria for linear systems with model uncertainties and delay and its derivative varying within intervals. heorem For given scalars τ τ max d and d max such that 0<τ τ max and d <d max the system 4 with time-varying delay dt satisfying -3 and uncertainties described by 5 is robust asymptotically stable if there exist scalars ε >0 and ε >0 and matrices P i i {3} Q Z j R j j {34} N and M with appropriate dimensions satisfying 8 and Z +U dt d max >0 Z +U dt d max >0 9 Z +U dt d >0 Z +U dt d >0 0 and free-weighting matrices H R 7r x 3r x and H R 7r x 3r x such that the following LMIs hold: Ω dt d Ω dt d max Ω dt d Ω dt d max Ω dt d Ω dt d max Ω dt d Ω dt d max <

3 where Ω k Ψ dt τk + H B +H B τ τ H Γ k H Γ 3 D ε E τ τ Λ k 0 0 ε I 0 ε I Ψ dt τk+ +H B +H B τ 3 τ H Γ k Ω k with k {} and H Γ 3 D ε E τ 3 τ Λ k 0 0 ε I 0 ε I Γ 0 I 0 Γ I 0 0 Γ3 0 0 I Λ τ τ Z 3 +R +R 4 Λ τ τ Z 3 +R + dt R + dtr 4 Λ τ 3 τ Z 4 +R 3 +R 4 Λ τ 3 τ Z 4 +R 3 + dt R + dtr 4 B 0 I 0 0 I I I 0 B 0 I I 0 0 I I A A d I A A d I E E A E Ad dt τ Ψ 0 τ τ P + τ dt τ τ P Ψ Ψ Ψ Ψ 33 dt+ψ Ψ 44 Ψ Ψ 55 N 0 0 N N U U N 0 U N dt τ Ψ 0 τ 3 τ P 3 + τ 3 dt τ 3 τ P Ψ Ψ Ψ Ψ 33 dt+ψ Ψ 44 Ψ Ψ 55 U 3 N +U 3 0 N N U 3 N N with U τ R + dt R + dtr 4 U τ 3 τ τ 3 τ Z 4 +R 3 +R 4 U 3 τ τ τ τ Z 3 +R + dt R + dtr 4 dt Ψ P P +M Z U τ τ Ψ dt Q τ Z Ψ 33 +Z +τ τ Z 3 +τ 3 τ Z 4 +τ R +τ 3 τ R 3 Ψ 33 dtτ dt τ τ dt 3 τ R 4 +τ dtr 4 + τ R + τ R τ τ τ τ Ψ 33 dtτ dt τ τ 3 dt 3 dtr 4 + τ 3 R + τ R τ 3 τ τ 3 τ Ψ 44 M M Z Z U Ψ 55 Q + N M Z U Ψ 4 Z +M +U Ψ 45 Z M +U. It is also interesting to consider two special cases of the previous result. he case when the lower bound of the timevarying delay derivative is unknown and the case when no restrictions are cast upon delay derivative. For the first case by fullfilling the restrictions P 3 > P and R > R 4 3 the following corollary arises directly from heorem. Corollary For given scalars τ τ max and d max such that 0<τ τ max the system 4 with the delay dt satisfying and dt d max and uncertainties described by 5 is robust asymptotically stable if there exist scalars ε >0 ε >0 and matrices P i i {3} Q Z j R j j {34} N and M with appropriate dimensions satisfying 8 9 and 3 and free-weighting matrices H R 7r x 3r x and H R 7r x 3r x such that the following LMIs with notations given in hold: Ω dt d max Ω dt d max Ω dt d max Ω dt d max < 0. We shall now consider the second case i.e. fast-varying delays. In this case as we have no information about the delay derivative by assug P P P 3 Q 0 and R R 4 4 we can eliate the terms with dt from. hen it is straightforward to obtain the following corollary. Corollary For given scalars τ and τ max such that 0<τ τ max the system 4 with time-varying delay dt satisfying and uncertainties described by 5 is robust asymptotically stable if there exist scalars ε >0 and ε >0 and matrices P i i {3} Q Z j R j j {34} N and M with appropriate dimensions satisfying 8 and 4 and free-weighting matrices H R 7r x 3r x and H R 7r x 3r x such that the following LMIs with notations given in hold: Ω Ω Ω Ω < 0. Remark Because of the term U the results are valid only for imum delay strictly greater than zero. However it is straightforward to extend these results to the case where τ 0 by considering U 0. heorem Corollaries and provide stability conditions for linear systems liable to model uncertainties and time-varying delays and are the main results of the paper. Compared with previous criteria the conservativeness of the stability analysis is considerably reduced. o improve the results we have introduced a new auxiliary delayed state x t τ in the Lyapunov functional which allows further exploitation of the delay s lower bound value. Moreover further improvements were obtained through the introduction of new delay-interval-dependent terms in 7. he employment of these terms yields different expression in the derivative of the Lyapunov functional when dt<τ and when τ <dt. he examples in the next section illustrate the effectiveness of our criteria. It is important to emphasize that the stability results are less conservative than previous published criteria not only for systems with uncertainties but also for noal time-delayed systems. IV. NUMERICAL EXAMPLES Example Consider the system 4 with no uncertainties and 0 0 A A d A0 A d

4 ABLE I ALLOWABLE τ max VALUE FOR d max 0. AND d 0. EX. Method τ He et al Sun et al. 0 { Fridman et al. thm thm heorem ABLE II ADMISSIBLE τ max VALUE FOR τ AND GIVEN d AND d max EX. Method unknown d d 0. d max 0.3 unkn d max d max 0.3 d max He et al Shao Orihuela et al { hm Fridman et al. 7 hm heorem Assug slow-varying delays 0. dt 0. the maximum values of τ max which maintain the system s asymptotical stability for various τ are listed in able I. It is clear that the obtained results are less conservative than those in Example Consider the following delayed system described by A A d A0 A d 0. For τ and various d and d max the results from various criteria in the literature are listed in able II. For unknown d and for fast-varying delays the results are obtained using Corollaries and respectively. From the table it can be seen that our criteria present superior results when compared to previous methods. Moreover one can note that τ max grows for d 0 and for d max 0. Example 3 Consider now the uncertain system 4 with 0 0 A A 0 d D E 0 A E Ad From Corollary we find that the uncertain delayed system is stable for τ 0 and various values for d max with admissible τ max given in able III. he obtained result represents an important improvement over those from previous robust criteria. Example 4 Consider the following uncertain system 4 with A A d DI E A E Ad In able IV we compare the results from Corollaries and with those in 5 7 for τ 0 unknown d and various d max. From the table it is clear ABLE III ALLOWABLE UPPER BOUND VALUE OF τ max FOR τ 0 UNKNOWN d AND VARIOUS d max EX. 3 Methods d max Wu et al Lien Yue & Han Qian et al Park & Ko Corollary ABLE IV MAX. τ max VALUE FOR τ 0 AND UNKNOWN d EX. 4 Methods d max Unknown Wu et al Jing et al He et al Qian et al Corollary Corollary that our results are considerably less conservative than those in previous criteria in the literature. V. CONCLUSIONS his work s main result concern the establishment of new stability criteria for time-delayed systems liable to model uncertainties and with delay and its derivative varying within bounded intervals. he case when the derivative s lower bound is unknown is also considered as the case when no restrictions are cast upon the delay derivative. he conservativeness of the stability analysis is considerably reduced with the introduction of new delay-interval-dependent terms and a new auxiliary delayed state in the LKF. Although this paper deals mainly with uncertain delayed systems our criteria when applied to noal systems also yields less conservative results than previous criteria in the literature. hese analyses are ratified with numerical examples that illustrate the effectiveness of the proposed criteria. APPENDIX PROOF OF HEOREM Firstly we shall consider the case where dt < τ. aking the time derivative of the Lyapunov functional candidate 7 with χ yields V t dt<τ dtx t P P xt+ẋ dt τ t P + τ dt P xt τ τ τ τ τ τ V t x t τ Q xt τ dt x t dtq xt dt xt τ N N V 3 t xt τ xt τ N N xt τ N N xt τ xt τ xt τ 3 N N xt τ 3 xt M M V 4 t xt xt τ M M xt τ M M xt τ xt τ xt τ M M xt τ V 5 tẋ t τ Z +Z +τ τ Z 3 +τ 3 τ Z 4 ẋt τ ẋ sz ẋsds τ τ ẋ sz ẋsds τ τ t τ t τ τ τ ẋ sz 3 ẋsds τ 3 τ ẋ sz 4 ẋsds t τ t τ 3 V 6 t dt<τ ẋ tτ dtr R 3 +dtr +R +τ 3 dtr 3 +R 4 ẋt 4887

5 ẋ s R + dt ẋsds R + dtr 4 t dt dt dt ẋ sr R 3 ẋsds ẋ sr 3 +R 4 ẋsds. 5 t τ t τ 3 Suppose now we take V 5 t and V 6 t dt<τ in 5 and expand the integral terms using the fact that τ dt τ. hen defining γ d : τ ẋsds and γ d : dt ẋsds dt τ t dt τ dt t τ where lim dt τ γ d ẋt τ and lim dt τ γ d ẋt τ and applying Jensen s inequality Lemma we have the following inequalities V 5 t+ V 6 t dt<τ ẋ τ t Z +Z +τ τ Z 3 +τ 3 τ Z 4 +τ R dt τ τ dt +τ 3 τ R 3 +τ 3 τ R 4 +τ dtr 4 +τ R +τ R ẋt τ τ τ τ xt x t τ Z + R + dt R + dtr 4 xt x t τ τ x t τ xt τ Z + R + dt R + dtr 4 x t τ xt τ τ xt τ xt τ 3 Z 4 + R 3 + R 4 xt τ xt τ 3 τ 3 τ γ d dt τ τ τ Z 3 +R + dt R + dtr 4 γd γ d τ dtτ τ Z 3 + R + R 4 γ d. 6 hen from 5 and 6 after some manipulation one can conclude that Vt dt<τ ζ t Ω dt<τ ζ t 7 where Ψ 0 Ω dt<τ Λ Λ dt τ Λ 0 0 τ dtλ and Ψ Λ and Λ are defined in. Also we have defined ζ t: ζ x γ d γ d R 9r x where ζ x t : x t x t dt ẋ t x t τ x t τ x t τ x t τ 3 8 Suppose now we introduce B B B R 3r x 9r x and H H 0 R 9r x 3r x where H is a 7r 3r freeweighting matrix and 0 I 0 0 I 0 0 B 0 I I 0 B dt τ I 0 0 τ dti. A+ A A d + A d I It is interesting to note that B ζ t 0. hen a straightforward consequence of applying Finsler s lemma Lemma is that the right side of 7 is negative definite if Ξ < 0 holds where Ψ Ξ Ω dt<τ + H B + B H +H B + B H H B. Λ Here we shall consider the terms Ξ and Ξ that arise from Ξ when dt τ and dt τ respectively Ψ Ξ k dt τk +H B + B H τ τ H Γ k 9 τ τ Λ k where Γ and Γ are defined in and k {}. Note that we have deleted the zero row and column from Ξ and Ξ. Now it can be seen that ζ tξ ζ t τ dt τ τ ζ tξ ζ t+ dt τ τ τ ζ tξ ζ t where ζ t: ζ x γ d ζ t: ζ x γ d and ζx is defined in 8. hus ζ tξ ζ t is convex in dt and is negative definite only if the vertices Ξ and Ξ are. Furthermore to eliate the time-varying matrix Ft from 9 we use the definition of A and A d from 5 and rewrite B as B B + Γ 3 A Ad B + Γ 3 DFt E 0 where B Γ 3 and E are defined in. hen according to 0 Ξ k in 9 is rewritten as Ψ Ξ k dt τk +H B +B H τ τ H Γ k +αftβ+β Ft α Ț τ τ Λ k where α H Γ 3 D 0 and β E 0. hen it follows from applying Lemma 3 in that Ξ k < 0 holds if and only if there exists a scalar ε > 0 such that Ψ dt τk +H B +B H τ τ H Γ k + αα + ε β β<0 τ τ Λ k ε holds for k {}. Moreover taking the Schur s complement we have Ω k as described in. herefore Ξ is negative definite if and only if Ω and Ω are. Furthermore given 3 the following expressions hold Ω d max dt dt d Ω d max d dt d + Ω d max d dt d max Ω d max dt dt d Ω d max d dt d + Ω d max d dt d max. herefore Ω and Ω are convex in dt d d max. We will now consider the case where τ < dt τ 3. We shall prove that analogous results can be derived using exactly the same arguments of the former case. aking the time derivative of the Lyapunov functional candidate 7 with χ 0 yields xt+ẋ dt τ t τ 3 τ τ 3 τ V 6 t dt>τ ẋ tdt τ R 3 R +dtr +R +τ 3 dtr 3 +R 4 ẋt V t dt>τ x t dt P 3 P P 3 + τ 3 dt P xt τ 3 τ ẋ s R + dt ẋsds R + dtr 4 t dt τ dt ẋ sr 3 R ẋsds ẋ sr 3 +R 4 ẋsds. t dt t τ 3 and V t to V 5 t are defined in 5. hen similarly to 6 we apply Jensen s inequality Lemma to V 5 t and V 6 t dt>τ : V 5 t+ V 6 t dt<τ ẋ t τ Z +Z +τ τ Z 3 +τ 3 τ Z 4 dt τ τ 3 dt +τ R +τ 3 τ R 3 +τ 3 dtr 4 + τ 3 R + τ ẋt xt x x t τ t τ xt τ R τ 3 τ τ 3 τ Z + R + dt R + dtr 4 xt x t τ τ Z + R + dt R + dtr 4 x xt τ τ t τ xt τ xt τ Z 3 + τ τ R + dt R + dtr 4 xt τ xt τ γ d dt τ τ 3 τ Z 4 + dt R +R 3 + dtr 4 γd γd3 τ 3 dtτ 3 τ Z 4 + R 3 + R 4 γ d3 where γ d and γ d3 are defined by γ d : τ ẋsds and γ d3 : dt ẋsds dt τ t dt τ 3 dt t τ 3 with lim dt τ γ d ẋt τ and lim dt τ3 γ d3 ẋt τ 3. We denote ζ t: ζ x γ d γ d3 R 9r x where ζ x is defined in 8. hen taking V t V 3 t and V 4 t from

6 V t dt>τ and one concludes that Vt dt>τ ζ t Ω dt>τ ζ t 3 where Ψ 0 Ω dt>τ Λ Λ dt τ Λ 0 0 τ 3 dtλ and Ψ Λ and Λ are defined in. Suppose now that analogously to the case where χ we define a matrix B B B + Γ3 DFt E 0 such that B ζ t 0 where B is defined in and B is defined in an analogous fashion to B. hen the condition that arises from applying Finsler s lemma Lemma to the right side of 3 is that ζ t Ω dt>τ ζ t is negative definite if there exists a matrix H H 0 R 9r x 3r x such that Ξ < 0 holds where H R 7r x 3r x is a free-weighting matrix and Ξ Ω dt>τ + H B + B H. 4 Similarly to 9 we consider the terms Ξ and Ξ that arise from Ξ when dt τ and dt τ 3 respectively After some manipulation it can be seen that ζ tξ ζ t τ 3 dt τ 3 τ ζ tξ ζ t+ dt τ τ 3 τ ζ tξ ζ t where ζ t: ζx γd3 ζ t: ζx γd and ζx is defined in 8. hen from the convexity of ζ tξ ζ t it is sufficient to verify the feasibility for Ξ and for Ξ. hen it follows from applying Lemma 3 in that Ξ k < 0 holds if and only if there exists a scalar ε > 0 such that Ψ dt τk+ +H B +B H τ 3 τ H Γ k + αα + ε β β<0 τ 3 τ Λ k ε holds for k {}. Moreover taking the Schur s complement we have Ω k as described in. herefore Ξ is negative definite if and only if Ω and Ω are. Moreover given 3 the expressions Ω d max dt dt d Ω d max d dt d + Ω d max d dt d max Ω d max dt dt d Ω d max d dt d + Ω d max d dt d max hold. hus Ω and Ω are convex in dt d d max. We are now ready to complete the proof by establishing conditions that guarantee the negativeness of the Lyapunov functional s derivative. For the first case where dt τ it is easy to check that Vt dt τ χ τ τ dtζ tω ζ t+ χ τ τ dt ζ tω ζ t. For the second case where dt τ using exactly the same arguments of 7 and 9 we conclude that Vt dtτ max{ζ tω ζ t ζ tω ζ t}. herefore it is straightforward to conclude that if the conditions in are fullfilled then we guarantee that Vt<0 which concludes the proof. ACKNOWLEDGMENS he authors are supported by research grants funded by the National Council of echnological and Scientific Development - CNPq Brazil. he authors would also like to thank the support of the Scientific and echnological Developments Foundation FINAEC Brazil and the Dean of Research and Graduate Studies DPP University of Brasilia UnB Brazil. REFERENCES J.-P. Richard ime-delay systems: an overview of some recent advances and open problems Automatica vol. 39 no. 0 pp F. Gouaisbaut and D. Peaucelle Delay-dependent stability of time delay systems in 5th IFAC Symp. Robust Control Design Jul P. Park and J. W. Ko Stability and robust stability for systems with a time-varying delay Automatica vol. 43 no. 0 pp Y. He Q.-G. Wang C. Lin and M. Wu Delay-range-dependent stability for systems with time-varying delay Automatica vol. 43 no. pp X. Jiang and Q.-L. Han New stability criteria for linear systems with interval time-varying delay Automatica vol. 44 no. 0 pp H. Shao New delay-dependent stability criteria for systems with interval delay Automatica vol. 45 no. 3 pp E. Fridman U. Shaked and K. Liu New conditions for delayderivative-dependent stability Automatica vol. 45 no. pp L. Orihuela P. Millan C. Vivas and F. R. Rubio Delay-dependent robust stability analysis for systems with interval delays in American Control Conference Baltimore USA Jun. 00 pp L. F. C. Figueredo J. Y. Ishihara G. A. Borges and A. Bauchspiess New delay-and-delay-derivative-dependent stability criteria for systems with time-varying delay in Proc. of the 49th IEEE Conf. on Decision and Control Atlanta USA Dec Y. S. Moon P. Park W. H. Kwon and Y. S. Lee Delay-dependent robust stabilization of uncertain state-delayed systems International Journal of Control vol. 74 no. 4 pp M. Wu Y. He J.-H. She and G.-P. Liu Delay-dependent criteria for robust stability of time-varying delay systems Automatica vol. 40 no. 8 pp X. J. Jing D. L. an and Y. C. Wang An LMI approach to stability of systems with severe time-delay IEEE rans. Autom. Control vol. 49 no. 7 July C. Lien Delay-dependent stability criteria for uncertain neutral systems with multiple time-varying delays via lmi approach IEE Proceedings Control heory and Applications vol. 5 no. 6 pp X. Jiang and Q. Han Delay-dependent robust stability for uncertain linear systems with interval time-varying delay Automatica vol. 4 no. 6 pp Y. He Q.-G. Wang L. Xie and C. Lin Further improvement of freeweighting matrices technique for systems with time-varying delay IEEE rans. on Automatic Control vol. 5 no. pp C. Peng and Y.-C. ian Delay-dependent robust stability criteria for uncertain systems with interval time-varying delay Journal of Comput. and Appl. Math. vol. 4 no. pp W. Qian S. Cong Y. Sun and S. Fei Novel robust stability criteria for uncertain systems with time-varying delay Applied Mathematics and Computation vol. 5 no. pp Sept K. Gu V. Kharitonov and J. Chen Stability of ime-delay Systems. Boston: Birkhauser S. Boyd L. El Ghaoui E. Feron and V. Balakrishnan Linear Matrix Inequalities in System and Control heory ser. Studies in Applied Mathematics. Philadelphia PA: SIAM 994 vol J. Sun G. Liu J. Chen and D. Rees Improved stability criteria for linear systems with time-varying delay IE Control heory & Applications vol. 4 no. 4 p D. Yue Q.-L. Han and C. Peng State feedback controller design of networked control systems IEEE ransactions on Circuits and Systems II: Express Briefs vol. 5 no. pp Nov L. F. C. Figueredo P. H. R. Q. A. Santana E. S. Alves J. Y. Ishihara G. A. Borges and A. Bauchspiess Robust stability of networked control systems in Proc. Intl. Conf. on Control and Automation. Christchurch New Zeland: IEEE Dec. 009 pp