Improved Results on Stability of Time-delay Systems using Wirtinger-based Inequality

Transcription

1 Preprints of the 9th World Congress he Interntionl Federtion of Automtic Control Improved Results on Stbility of ime-dely Systems using Wirtinger-bsed Inequlity e H. Lee Ju H. Prk H.Y. Jung O.M. Kwon S.M. Lee Deprtment of Electricl Engineering Yeungnm University Kyongsn Republic of Kore e-mil: fesuse@ynu.c.kr; jessie@ynu.c.kr. School of Electricl Engineering Chungbuk University Cheongju Republic of Kore School of Electronics Engineering Degu University Gyeongsn Republic of Kore Abstrct: his pper concerns with the problem of dely-dependent stbility nlysis of time-dely systems. By help of Wirtinger bsed inequlity which gives very close estimting bound of Jensen s inequlity n extended inequlity is proposed. Using the new inequlity nd tuning prmeters generlized criterion for stbility of time-dely systems is estblished. wo numericl exmples re given to describe the less conservtism of the proposed methods. Keywords: Stbility ime-dely systems Lypunov method.. INRODUCION During the lst decde mny reserchers hve been devoted much ttention to time-dely systems becuse time-delys re nturlly encountered in mny dynmic systems such s chemicl or process control systems nd networked control systems nd often sources of poor performnce nd instbility of the system. Existing stbility criteri cn be clssified into two ctegories tht is dely-independent ones nd delydependent ones. It is well known tht dely-independent ones re usully more conservtive thn the dely-dependent ones so much ttention hs been pid in recent yers to the study of dely-dependent stbility conditions. he min issue of delydependent stbility nd stbiliztion is to reduce the conservtism of stbility nd stbiliztion criteri. One of the importnt index for checking the conservtism is to enlrge the fesible region of the criteri or to get mximum dely bounds for gurnteeing the system stbility. Recently to reduce the conservtism mny methods were dopted bsed on Lypunov- Krsovskii pproch or Lypunov-Rzumikhin pproch. A descriptor model trnsformtion method which is n equivlent model of the originl system is introduced in Fridmn nd Shked 2003 in which the method significntly reduces the conservtism of the results. By using free-weighting mtrices method bsed on the Newton-Leibniz formul the further results on time-dely introduced in He et l Very recently there re few works Seuret nd Gouisbut b to reduce Jensen s gp g R ẋ = b ẋsrẋsds b ẋsdsr b ẋsds/b where R > 0. Jensen s inequlity ws used s one of essentil techniques to reduce conservtism in deling with the time dely systems so it is widely utilized for estimting upper bound of time derivtive of constructed Lypunov functionl. herefore reducing the Jensen s gp gives more fesible region of stbility his reserch ws supported by Bsic Science Reserch Progrm through the Ntionl Reserch Foundtion of Kore NRF funded by the Ministry of Eduction 203RAA2A nd BK2+ Progrm. of time dely systems. In Seuret nd Gouisbut bsed on the Wirtinger s inequlities more close lower bound of Jensen s inequlity is proposed s g R ẋ π2 4 ν brν b where ν b = xb + x 2 b xsds nd the choice of prticulr signl which stisfies the necessry ssumptions to pply the Wirtinger inequlities hs been proved. As these b result n integrl form of sttes b xsds is tken s ugmented vectors to include further informtion of the timedely systems. Moreover the improved result which reduces more Jensen s inequlity gp s g R ẋ 3ν brν b insted of hs been presented in Seuret nd Gouisbut 203b. In this pper n extension result which covers the existing ones s specil cses is introduced to discuss the problem of stbility nlysis of time-dely systems. In ddition by introducing tuning prmeters the conservtism of stbility of the system is reduced without ny chnge of Lypunov functionl nd LMI liner mtrix inequlity condition. Numericl exmples re given to illustrte tht the proposed methods re effective nd led to less conservtive results. Nottions: R n is the n-dimensionl Eucliden spce X > 0 respectively X 0 mens tht the mtrix X is rel symmetric positive definite mtrix respectively positive semidefinite. in mtrix represents the elements below the min digonl of symmetric mtrix. For X R m n X denotes bsis for the null-spce of X. b 2. PRELIMINARIES In this pper the following time-dely systems re considered: ẋt = Axt + Bxt h xt = ϕt t h 0] 2 Copyright 204 IFAC 6826

2 where xt R n is the stte vector h > 0 is the constnt time dely ϕt is comptible vector vlued initil function nd A B re known rel constnt mtrices with pproprite dimensions. he following lemm which plys key role in the derivtion of the min result is proposed: Lemm. For given mtrix R = R > 0 sclrs 2 m stisfying 2 m nd ll continuously differentible function x in m ] R n the following inequlity holds: m i+ i β i i+ Mβ i i+ 3 where β 2 = x 2 x 2 2 xsds] nd ] 4R 2R 6R M = 4R 6R. 2R Proof. It is cler tht m 2 = m 3 2. By pplying the following reltionship proposed in Seuret nd Gouisbut 203b to ech subintervl integrl terms of bove eqution: 2 x 2 x Rx 2 x 2 +3ν 2 Rν 2 where ν 2 = x 2 +x 2 to get xsds it is esy And the following lemm is used in this pper: Lemm 2. Finsler lemm Skelton et l. 997 Let vector ζ R n symmetric mtrix M R n n nd Γ R m n such tht rnkγ < n. hen the following sttements re equivlent: ζ Mζ < 0 Γζ = 0 ζ 0 2 Γ MΓ < MAIN RESULS Before stting further results for the ske of simplicity on mtrix representtion e i R 2m+2n n i = m+2 re defined s block entry mtrices; e.g. e 2 = 0 I 0. {{.. 0 ]. 2m heorem 3. For given constnt dely h nd positive integer m the time-dely system 2 is stble if there exist positive definite mtrices P Q i i =... m R such tht where Γ ΣΓ < 0 4 Σ = Π P Π 2 + Π 2 P Π + he 2 Re 2 Π 3 MΠ 3 Π 4 imi + Π 4 i + Φ Γ = A I {{ B 0. {{.. 0 m Π = e e 3+m e 4+m e 5+m... e 2m+2 ] Π 2 = e 2 e e 3 e 3 e 4 e 4 e 5... e +m e 2+m ] Π 3 = e e 3 e 3+m ] Π 4 i = e 2+i e 3+i e 3+m+i ] m Φ = e Q e e 2+i Q i e 2+i + e 2+i Q i+ e 2+i 6 4R 2R R h i h i 6 Mi = 4R R h i h i h i h i 2 h i h i 2 R h i = h m i. Proof. Define Lypunov functionl cndidte for system 2 s: V t = η tp η t + m hi th i x sq i xsds 0 + x srxsdsdθ 5 h m t+θ where η t = x t th x sds h th 2 x sds... h th m x sds]. hen the time-derivtive of V t long the solution of the system 2 gives V t = 2η tp η 2 t + m x t h i Q i xt h i x t h i Q i xt h i + hẋ trẋt th m where η 2 t = ẋ t x t h 2... x t x t h x t h x t h x t h m ]. 6827

3 Applying Lemm nd defining vector tht ζt = ẋ t β t β 2 t] with β t = x t x t h... x t h m ] nd β2 t = th x sds... h th m x sds] the following new upper bound of V t is obtined: V t ζ tσζt 6 nd ccording to Lemm 2 with 0 = Γζt Eq. 6 is equivlent to Γ ΣΓ. herefore if 4 is holds then the system 2 is stble. his completes the proof. Remrk. It is noted tht when m = the Lypunov functionl 5 cn be V t = + + th xt xsds th 0 h P x sq xsds t+θ th xt xsds x srxsdsdθ. 7 ] P Q By letting P = Q Z = S Lypunov functionl 7 reduces to the Lypunov functionl of Seuret nd Gouisbut 202. herefore the proposed method extend the results from Seuret nd Gouisbut Remrk 2. he considered Lypunov functionl 5 is very simple nd fundmentlly used in numerous ppers. Until now there re mny ttempt in constructing Lypunov functionl to reduce conservtism of stbility of the systems such s terms of double nd triple integrl of sttes nd time derivtive of sttes terms of fourth integrl of sttes nd so on. It should be pointed out tht our result my be improved esily when bove commented Lypunov functionl technique is pplied to our method. In heorem time-dely is evenly divided into m periods. On the contrry when uneven dividing bounds re considered the following theorem is obtined. Corollry 4. For given constnt dely h nd positive integer m positive sclrs α i < i =... m the liner time-dely system 2 is stble if there exist positive definite mtrices P Q i i =... m R such tht where Γ ΣΓ < 0 8 Σ = Π P Π 2 + Π 2 P Π + he 2 Re 2 Π 3 MΠ 3 Π 4 i Mi + Π 4 i + Φ 6 4R 2R h i h R i 6 Mi = h i h 4R i h i h R i 2 h i h i R 2 h i = α i h h i + h i i =... m h 0 = 0 h m = h nd other nottions re defined in heorem. Proof. he proof of Corollry 4 is sme to heorem 3 when h i re replced to h i so it is omitted. Remrk 3. In heorem 3 the rnge of the time dely h is divided into m subintervls evenly i.e. h i = h mi i =... m. On the other hnd by introducing tuning prmeters α i 0 i =... m uneven m subintervls of h i.e. h i = α i h h i + h i i =... m re considered in Corollry 4. It should be noted tht heorem 3 is specil cse of Corollry 4 e.g. when m = 3 α = 3 nd α 2 = 2 Corollry 4 is sme to heorem 3 with m = 3. he dvntge of this pproch is tht the fesible region of stbility criterion cn be enhnced thnks to djustble tuning prmeters nd it will be shown through numericl exmples. 4. NUMERICAL EXAMPLES In this section two numericl exmples re given to show less conservtive results of proposed methods thn the existing ones. 4. Exmple Consider the most well known model of time-dely system 2: A = ] ] B = his system is known tht its mximum nlytic dely bound is h = 6.72 nd it cn be esily computed by dely sweeping techniques. Our results from heorem 3 nd Corollry 4 nd recent results re shown in ble in which the mximum bound on dely using Corollry 4 is listed with best cse of tuning prmeter vlues. From ble it cn be seen tht heorem 3 improves the fesible region of stbility criteri compred to ll remrkble existing works. Furthermore Corollry 4 lso verifies the effectiveness in improvement of fesible region. It should be noticed tht heorem 3 with m = 6 nd Corollry 4 with m = 5 nd mny cses of α α 2 α 3 α 4 ] give the mximum upper bound of the dely of the system s 6.72 which is theoreticl bound to ensure the stbility of the system. In order to show the effectiveness of tuning prmeters by pplying Corollry 4 the mximum upper bounds on dely of the system with vrious tuning prmeters in the cses m = 2 nd m = 3 re presented in ble 2 nd Fig. respectively. 6828

4 ble. Comprison of mximum bound of h. Methods h Number of vribles He et l Sho Ko nd Rntzer Arib et l Sun et l Kim Gu et l Arib nd Gouisbut Seuret nd Gouisbut Seuret nd Gouisbut 203b heorem 3 m = heorem 3 m = heorem 3 m = heorem 3 m = heorem 3 m = Corollry 4 m = 2 α = Corollry 4 m = 3 α α 2 ] = ] Corollry 4 m = 4 α α 2 α 3 ] = ] h ble 2. he mximum bound of h with m = 2 nd vrious α α h α h Best cse α α 2 ]= ] h= α 2 Fig.. he llowble mximum h with m = 3 nd vrious α nd α Exmple 2 Let us consider the liner time dely system 2 with the following mtrices: A = ] ] B =. 0 By heorem 3 nd Corollry 4 the improvement results of this pper re shown in ble 3. It is noticed tht the best cse of α i i =... m of exmple 2 is evenly divided 0.4 α subintervl cse i.e. α i = mi+ i =... m it mens heorem 3 is the best cse of Corollry 4. In ddition the results by Corollry 4 with m = 2 is presented in ble 4 s vrious α cses. ble 3. Comprison of mximum bound of h. Methods h Number of vribles Wu et l Prk nd Ko Kim heorem 3 m = heorem 3 m = heorem 3 m = Corollry 4 m = 2 α = Corollry 4 m = 3 α α 2 ] = 3 0.5] Corollry 4 m = 4 α α 2 α 3 ] = ] ble 4. he mximum bound of h with m = 2 nd vrious α. α h α h CONCLUSIONS In this pper the problem of dely-dependent stbility nlysis of time-dely systems hs been discussed. A new lemm hs been proposed bsed on Wirtinger inequlity which reduce Jensen s inequlity gp nd gives less conservtism of stbility of the system. In ddition by introducing tuning prmeters more generl criterion hs been derived. wo numericl exmples hve been given to illustrte the less conservtism of the proposed method. REFERENCES Y. Arib nd F. Gouisbut. An ugmented model for robust stbility nlysis of time-vrying dely systems. Internltionl Journl of Control 82: Y. Arib F. Gouisbut nd K.H. Johnsson. Stbility intervl for time-vrying dely systems. In Proc. of IEEE Conference on Decision nd Control E. Fridmn nd U. Shked. Dely-dependent stbility nd H control: constnt nd time-vrying delys. Interntionl Journl of Control 76: K. Gu V.-L. Khritonov nd J. Chen. Stbility of time-dely systems. Birkhuser Y. He M. Wu J.H. She nd G.P. Liu. Dely-dependent robust stbility criteri for uncertin neutrl systems with mixed delys. Systems nd Control Letters 5: Y. He Q. G. Wng L. Xie nd C. Lin. Further improvement of free-weighting mtrices technique for systems with timevrying dely. IEEE rns. on Automt. Control 52: C.Y. Ko nd A. Rntzer. Stbility nlysis of systems with uncertin time-vrying delys. Automtic 43: J.H. Kim. Note on stbility of liner systems with time vrying dely. Automtic 47:

5 P. Prk nd J.W. Ko. Stbility nd robust stbility for systems with timevrying dely. Automtic 43: A. Seuret nd F. Gouisbut. On the use of the wirtinger s inequlities for time-dely systems. In Proc. of the IFAC workshop on time dely systems A. Seuret nd F. Gouisbut. Jensen s nd Wirtinger s inequlities for time-dely systems. In Proc. of the IFAC workshop on time dely systems A. Seuret nd F. Gouisbut. Wirtinger-bsed intergrl inequlity: Appliction to time-dely systems. Automtic 49: b. H. Sho. New dely-dependent stbility criteri for systems with intervl dely. Automtic 45: R.E. Skelton. Iwski nd K.M. Grigordis. A Unified Algebric Approchto Liner Control Design. New York: ylor nd Frncis 997. J. Sun G.P. G.P. Liu J. Chen nd D. Rees. Improved delyrnge-dependent stbility criteri for liner systems with time-vrying delys. Automtic 46: M. Wu Y. He J.-H. She nd H.-P. Liu. Dely-dependent criteri for robust stbility of time-vrying dely systems. Automtic 40: