Delay-Dependent Stability Criteria for Linear Time-Delay System of Neutral Type

Transcription

1 World Academy of Science Engineering and Technology Vol:4 No:1 1 Delay-Dependent Stability Criteria for Linear Time-Delay Sytem of Neutral Type Myeongjin Park Ohmin Kwon Juhyun Park and Sangmoon Lee International Science Index Computer and Information Engineering Vol:4 No:1 1 waet.org/publication/798 Abtract Thi paper propoe improved delay-dependent tability condition of the linear time-delay ytem of neutral type. The propoed method employ a uitable Lyapunov-Kraovkii functional and a new form of the augmented ytem. New delay-dependent tability criteria for the ytem are etablihed in term of Linear matrix inequalitie (LMI which can be eaily olved by variou effective optimization algorithm. Numerical example howed that the propoed method i effective and can provide le conervative reult. Keyword Neutral ytem; Time-delay; Stability; Lyapunov method; LMI. I. INTRODUCTION Time-delay occur in variou phyical indutrial and engineering ytem uch a aircraft biological ytem neural network networked control ytem and o on. It ha been hown that the delay i a ource of ocillation poor performance or intability of control ytem. Therefore the tudy on tability analyi for time-delay ytem ha been widely invetigated. For more detail ee 1-. In general tability analyi for time-delay ytem can be claified into two type. One i the delay-dependent tability analyi which include the information on the ize of delay and another i the delay-independent tability analyi which do not. Generally peaking the former i le conervative then the latter particularly when the time-delay i mall. In delay-dependent tability analyi an important iue i to enlarge the feaibility region of tability criteria or to provide an upper bound of time delay for guaranteeing aymptotic tability of time-delay ytem. Therefore a great number of reult on time-delay ytem have been reported in the literature 5-1. Above all Ariba and Gouaibaut 1 propoed ome new tability criteria by an augmented model of time-varying delay ytem and preented one by a form of the Lyapunov-Kraovkii functional that include a triple-integral term. On the other hand ome practical ytem can be modeled by uing the model of time-delay ytem of the neutral type which have delay in both it tate and the derivative of the tate. Thi ytem refer to the time-delay ytem which are the amount of the pat tate variable affect the current tate variable. The example of time-delay ytem of M.J. Park and O.M. Kwon are with the School of Electrical Engineering Chungbuk National Univerity Cheongju Republic of Korea ( netgau; madwind@chungbuk.ac.kr. Ju H. Park i with the Department of Electrical Engineering Yeungnam Univerity Gyeongan Republic of Korea ( jeie@ynu.ac.kr. S.M. Lee i with the School of Electronic Engineering Daegu Univerity Gyeongan Republic of Korea ( moony@daegu.ac.kr. Correponding author. Tel.: ; Fax: neutral type include lole tranmiion line (LC circuit partial element equivalent circuit (PEEC 3 the control of contrained manipulator with delay meaurement 4 the ytem which need the information of the pat tate variable and o on. The variou approach to the delay-dependent tability analyi for time-delay ytem of neutral type have been invetigated in the literature 13- on account of theoretical and practical importance for time-delay ytem of neutral type. In thi paper we propoe improved delay-dependent tability criteria for linear time-delay ytem of neutral type. By contructing a uitable Lyapunov-Kraovkii functional and a new augmented ytem new delay-dependent criteria are derived in term of LMI which can be olved efficiently by uing the interior-point algorithm 5. Numerical example are included to how the effectivene of the propoed method. The organization of the paper i a follow. In Section we formulate the olved problem and review the general lemma which are needed to derive new tability criteria. In Section 3 we deal with new tability criteria for time-delay ytem. In Section 4 baed on the reult in Section 3 three numerical example and PEEC model which i a practical example are given for a comparion of the previou reult. Finally in Section 5 we ummarize the reult in thi paper. Notation: R n i the n-dimenional Euclidean pace R m n denote the et of m n real matrix. C nh = C(h R n denote the Banach pace of continuou function mapping the interval h into R n with the topology of uniform convergence. X> (repectively X mean that the matrix X i a real ymmetric poitive definite matrix (repectively poitive emi-definite. I denote the identity matrix with appropriate dimenion. refer to the Euclidean vector norm and the induced matrix norm. diag{ } denote the block diagonal matrix. repreent the element below the main diagonal of a ymmetric matrix. II. PROBLEM STATEMENTS Conider the following linear ytem with time-delay ẋ(t Cẋ(t h = Ax(t+A d x(t h t > x( = φ( h h > (1 x(t R n i the tate vector A A d C R n n are known contant matrice with appropriate dimenion φ( C nh i a given continuou vector valued initial function h i a contant time-delay. International Scholarly and Scientific Reearch & Innovation 4( cholar.waet.org/ /798

2 World Academy of Science Engineering and Technology Vol:4 No:1 1 International Science Index Computer and Information Engineering Vol:4 No:1 1 waet.org/publication/798 In thi paper we aume C < 1 which implie that the ytem (1 atifie Lipchitz condition in ẋ(t h with a contant le than 1 (ee 1 pp.9-3 for detail. To derive tability analyi of the ytem (1 we introduce two equivalent ytem. One i to tranform the original ytem (1 to the following equivalent ytem by integrating both term of Eq.(1 = A ẋ(d C x(d + A d ẋ( hd and another i the differentiating ytem (1 x( hd ( ẍ(t Cẍ(t h = Aẋ(t+A d ẋ(t h. (3 Eq.(1-(3 can be re-arranged to give the following new augmented ytem: E = z(t = Eż(t Cż(t h = Az(t+A d z(t h (4 I I I A A A I x(d x(t ẋ(t C = A d = C C C A d A d A d I. (5 Remark 1: The augmented ytem (4 i a new one and not conidered in other literature. By differentiating and integrating ytem (1 three equation (1 ( and (3 can be repreented a one dynamic ytem which have more information than ytem (1. Alo by contructing a uitable Lyapunov-Kraovkii functional for augmented tate z(t a le conervative tability condition may be derived which provide an improved feaible region for ytem (1. The goal of thi paper i to invetigate the delay-dependent tability analyi of ytem (1 with time-invariant delay. Before deriving our main reult we give the following fact and lemma. Fact 1: (Schur Complement Given contant matrice Σ 1 Σ Σ 3 with Σ 1 = Σ T 1 and < Σ = Σ T then Σ 1 + Σ T 3 Σ 1 Σ 3 < if and only if Σ1 Σ T 3 Σ 3 Σ < or Σ1 Σ T 3 Σ 3 Σ <. (6 Lemma 1: 3 Let ζ R n Φ = Φ T R n n and B R m n uch that rank(b <n. The following tatement are equivalent: (i ζ T Φζ< Bζ =ζ (ii B T ΦB < B i a right orthogonal complement of B. Lemma : 4 For any contant matrix M R n n M = M T > a calar γ>and a vector function x :γ R n uch that the integration concerned are well defined then γ γ x T (Mx(d ( γ T ( γ x(d M x(d. (7 To derive a le conervative tability criterion we ue the following lemma to be utilized in deriving an upper bound of double-integral term. Lemma 3: For any calar h> and any contant matrix M = M T > the following inequality hold: h ( x T (umx(udud T ( x(udud M x(udud. Proof: From Lemma the following inequality hold (8 (t x T (umx(udu ( T ( x(udu M x(udu (9 t h t. By uing Fact 1 Eq.(9 i equivalent to the following xt (umx(udu t xt (udu x(udu. (1 1 (t M Integration of (1 from t h to t yield ϕ(t d xt (udud x(udud (t M 1 (11 d ϕ(t = xt (umx(udu. Eq.(11 i equivalent to inequality (8 according to Fact 1. III. MAIN RESULTS In thi ection we propoe new tability criteria for timedelay ytem (1. Before introducing our main reult the notation of everal matrice are defined for implicity: B = A A d E C ζ T (t = z T (t z T (t h ż T (t ż T (t h. (1 Now we have the following theorem. International Scholarly and Scientific Reearch & Innovation 4( cholar.waet.org/ /798

3 World Academy of Science Engineering and Technology Vol:4 No:1 1 International Science Index Computer and Information Engineering Vol:4 No:1 1 waet.org/publication/798 Theorem 1: For a given poitive calar h the ytem (1 with time-invariant delay i aymptotically table if C < 1 and there exit poitive definite matrice P =P ij 3 3 Q 1 = Q 1ij 3 3 Q 3 =Q 3ij 3 3 R =R ij 3 3 and any matrix Q =Q ii 3 3 atifying the follow LMI: Φ = B T ΦB <. (13 Q 1 R R P+ Q Q 1 R Q Q 3 + h R Q 3. (14 Proof: Let u chooe the well-known Lyapunov- Kraovkii functional candidate a V 1 = z T (tpz(t z( V = ż( V 3 = h V = V 1 + V + V 3 (15 Q 3 ż( T Q1 Q z( d ż T (urż(udud. (16 Firt the time-derivative of V 1 can be calculated a V 1 = z T (tpż(t. (17 Second the time-derivative of V can be obtained a T z(t Q1 Q = z(t ż(t Q 3 ż(t T z(t h Q1 Q z(t h ż(t h Q 3 ż(t h Finally calculating the time-derivative of V 3 lead to V 3 = h ż T (trż(t h. (18 ż T (Rż(d. (19 By uing Lemma an upper bound of integral term of 3 can be obtained a = h ż T (Rż(d ( T ( ż(d R ż(d T z(t R R z(t z(t h R z(t h. ( From (16-( the time-derivative of V ha a new upper bound a V ζ T (tφζ(t (1 ζ(t and Φ are defined in (1 and (14 repectively. In addition the ytem (1 with time-invariant delay can be rewritten a Bζ(t = B i defined in (1. By Lemma 1 the inequality ζ(tφζ(t < i equivalent to the inequality B T ΦB <. Therefore if for all ζ(t uch that Bζ(t = the LMI (13 are atified then the ytem (1 with time-invariant delay i guaranteed to be aymptotically table. Thi complete our proof. Theorem 1 i derived by utilizing a well-known double integral form of Lyapunov-Kraovkii functional. If a triple-integral form of Lyapunov-Kraovkii functional are included an improved delay-dependent tability criterion which will be introduced in Theorem can be derived. Theorem : For a given poitive calar h the ytem (1 with time-invariant delay i aymptotic table if C < 1 and there exit poitive definite matrice P =P ij 3 3 Q 1 = Q 1ij 3 3 Q 3 =Q 3ij 3 3 R =R ij 3 3 S and any matrix Q =Q ii 3 3 atifying the follow LMI: ˆΦ = B T ˆΦB <. ( Q 1 R R P+ Q Q 1 R Q T Q 3 T = Q 3 + h R + S S hs S = h S. (3 (h / S Proof: Let u chooe the Lyapunov-Kraovkii functional candidate that contain a triple-integral term a V = V 1 + V + V 3 + V 4 (4 V 1 = z T (tpz(t T z( Q1 Q V = ż( Q 3 V 3 = h V 4 = h ż T (urż(udud u z( ż( d ż T (vπ T SΠż(vdvdud (5 and Π in V 4 i defined a Π= I. Firt with the imilar method of the proof of Theorem 1 the time-derivative of V 1 V and V 3 can be calculated a 1 = z T (tpż(t V = T z(t Q1 Q z(t ż(t Q 3 ż(t z(t h ż(t h Q 3 ż(t h T Q1 Q z(t h 3 h ż T (trż(t T z(t R R + z(t h R z(t z(t h. (6 International Scholarly and Scientific Reearch & Innovation 4( cholar.waet.org/ /798

4 World Academy of Science Engineering and Technology Vol:4 No:1 1 International Science Index Computer and Information Engineering Vol:4 No:1 1 waet.org/publication/798 Calculating the time-derivative of V 4 lead to 4 = (h / ż T (tπ T SΠż(t (h / = (h / ẍ T (tsẍ(t (h / ż T (uπ T SΠż(udud ẍ T (usẍ(udud. (7 and Π i defined in (5. By uing Lemma 3 an upper bound of double-integral term of 4 can be obtained a = (h / ( ẍ T (usẍ(udud T ( ẍ(udud S T ẋ(t h S hs t ẋ(td S ẍ(udud ẋ(t ẋ(td. (8 Uing the imilar method hown in the proof of Theorem 1 the LMI ( can be eaily obtained. IV. NUMERICAL EXAMPLES In thi ection we provide four example to how the le conervativene of the propoed new tability criterion in thi paper. Example 1: Conider the neutral ytem (1 with the following parameter 1 A.9 d = 1 1 c C = c<1. (9 c Table I how the reult of the upper bound of time-delay with different c. It can be een that Theorem in thi paper provide larger delay bound than the previou reult given in Table I. The example how that Theorem 1 and obtain the le conervative reult tep by tep. Baed on the well-known Lyapunov-Kraovkii functional (16 with the current tate of new augmented ytem (4 Theorem 1 i propoed which i proved to le conervative than the reult in 7 16 and 14. In Theorem for further improved reult we add the triple-integral term of ẍ(t on (16 ince we conider the current tate x(d of the integrating ytem (. (t Thee reult in Theorem indicate that the preented tability condition relieve the contraint of the tability caued by time-delay. Example : Conider the neutral ytem in the form of (1 with A.1.9 d = C =. (3..1 TABLE I UPPER BOUNDS OF TIME-DELAY WITH DIFFERENT c (EXAMPLE 1. c Fridman et al Xu et al Wu et al Our (Theorem Our (Theorem TABLE II UPPER BOUNDS OF TIME-DELAY (EXAMPLE. Mathod Upper bound Zhao et al Kwon et al Nian et al Our (Theorem.54 TABLE III UPPER BOUNDS OF TIME-DELAY WITH DIFFERENT β (EXAMPLE 4. β Yue et al Kwon et al Our (Theorem In Table II the reult of the upper bound of time-delay for guaranteeing tability are compared with the previou reult. It alo can be hown that the propoed tability criterion for thi ytem improve the tability region. Example 3: Conider the ytem (1 with A d = C = (31 In 16 for the above ytem the obtained the upper bound of time-delay wa By Theorem 1 in 1 it wa hown that the upper bound of time-delay with the ame condition wa By applying Theorem it can be obtained that the upper bound of time-delay i.661 which i larger delay bound than one in 16 and 1. When C = the upper bound of time-delay obtained.693 in 16 and.7918 in 1. However by uing Theorem one can obtain the upper bound of time-delay i Thi reult of our criterion give a larger delay bound than one in 16 and 1. Example 4: Conider the following PEEC model: 1 β A d = C = (3 4 1 International Scholarly and Scientific Reearch & Innovation 4( cholar.waet.org/ /798

5 World Academy of Science Engineering and Technology Vol:4 No:1 1 International Science Index Computer and Information Engineering Vol:4 No:1 1 waet.org/publication/798 In Table III the reult for different condition of β are compared with the reult in 15 and. From Table III it can be hown that our reult for thi example give larger upper bound of time-delay than the one in 15 and. V. CONCLUSION In thi paper new delay-dependent tability criteria for linear time-delay ytem of neutral type i propoed. To obtain a le conervative reult an augmented Lyapunov- Kraovkii functional that include a triple-integral term i ued to improve the feaible region of tability criterion. Numerical example have been given to how the uperiority of the preented criteria and it improvement over the exiting reult. ACKNOWLEDGMENT Thi reearch wa upported by the MKE(The Minitry of Knowledge Economy Korea under the ITRC(Information Technology Reearch Center upport program upervied by the NIPA(National IT Indutry Promotion Agency (NIPA- 9-(C REFERENCES 1 J. Hale and S. M. V. Lunel Introduction to Functional Differential Equation. New York: Springer-Verlag J.P. Richard Time-delay ytem: an overview of ome recent advance and open problem Automatica vol.39 pp M.C. de Oliveira Invetigating duality on tability condition Syt. Control Lett. vol.5 pp K. Gu An integral inequality in the tability problem of time-delay ytem in Proc. IEEE Conf. Deciion Control Sydney Autralia Dec. pp P.G. Park A Delay-Dependent Stability Criterion for Sytem with Uncertain Time-Invariant Delay IEEE Tran. Autom. Control vol.44 pp E. Fridman and U. Shaked An Improved Stabilization Method for Linear Time-Delay Sytem IEEE Tran. Autom. Control vol.47 pp E. Fridman and U. Shaked Delay-dependent tability and H control: contant and time-varying delay Int. J. Control vol.76 pp S. Xu J. Lam and Y. Zou Simplified decriptor ytem approach to delay-dependent tability and performance analye for time-delay ytem IEE Proc.-Control Theory Appl. vol.15 pp S. Xu and J. Lam Improved Delay-Dependent Stability Criteria for Time-Delay Sytem IEEE Tran. Autom. Control vol.5 pp O.M. Kwon and Ju H. Park On Improved Delay-Dependent Robut Control for Uncertain Time-Delay Sytem IEEE Tran. Autom. Control vol.49 pp P.G. Park and J.W. Ko Stability and robut tability for ytem with a time-varying delay Automatica vol.43 pp Y. Ariba and F. Gouaibaut An augmented model for robut tability analyi of time-varying delay ytem Int. J. Control vol.8 pp J.H. Park and S. Won Aymptotic Stability of Neutral Sytem with Multiple Delay J. Optim. Theory Appl. vol.13 pp M. Wu Y. He and J.-H. She New Delay-Dependent Stability Criteria and Stabilizing Method for Neutral Sytem IEEE Tran. Autom. Control vol.49 pp D. Yue and Q.-L. Han A Delay-Dependent Stability Criterion of Neutral Sytem and it Application to a Partial Element Equivalent Circuit Model IEEE Tran. Circuit Syt. II-Expre Brief vol.51 pp S. Xu J. Lam and Y. Zou Further reult on delay-dependent robut tability condition of uncertain neutral ytem Int. J. Robut Nonlinear Control vol.15 pp Ju H. Park and O. Kwon On new tability criterion for delaydifferential ytem of neutral type Appl. Math. Comput. vol.16 pp Z. Zhao W. Wang and B. Yang Delay and it time-derivative dependent robut tability of neutral control ytem Appl. Math. Comput. vol.187 pp O.M. Kwon Ju H. Park and S.M. Lee On tability criteria for uncertain delay-differential ytem of neutral type with time-varying delay Appl. Math. Comput. vol.197 pp O.M. Kwon and Ju H. Park Augmented Lyapunov functional approach to tability of uncertain neutral ytem with time-varying delay Appl. Math. Comput. vol.7 pp M.N.A. Parlakci Extenively augmented Lyapunov functional approach for the tability of neutral time-delay ytem IET Contr. Theory Appl. vol. pp X. Nian H. Pang W. Gui and H. Wang New tability analyi for linear neutral ytem via tate matrix decompoition Appl. Math. Comput. vol.15 pp A. Bellen N. Guglielmi and A.E. Ruehli Method for Linear Sytem of Circuit Delay Differential Equation of Neutral Type IEEE Tran. Circuit Syt. I-Regul. Pap. vol.46 pp S.I. Niculecu and B. Brogliato Force meaurement time-delay and contact intability phenomenon Eur. J. Control vol.5 pp S. Boyd L. El Ghaoui E. Feron and V. Balakrihnan Linear Matrix Inequalitie in Sytem and Control Theory. Philadelphia: SIAM P. Gahinet A. Nemirovkii A. Laub and M. Chilali LMI Control Toolbox Uer Guide. Natick Maachuett: The MathWork Inc International Scholarly and Scientific Reearch & Innovation 4( cholar.waet.org/ /798