This paper proposes a composite approach between

Transcription

1 Advance Reearch Journal o Multi-Diciplinary Dicoverie ISSN NO : ASSIVIY AND ISS FOR SINGULARLY ERURBED SYSEMS WIH EXOGENOUS DISURBANCE Original Reearch Article ISSN CODE: (Online) (ICV-AS/Impact Value): 3.8 (GIF) Impact Factor:.74 Copyright@IJF 7 Journal Code: ARJMD/AS/V-3./I-/C-3/MAY-7 Category : ALIED SCIENCE Volume : 3. / Chapter- III / Iue - (MAY) Webite: Received: Accepted: Date o ublication: age: -8 ABSRAC hi paper propoe a compoite approach between paivity and ISS or ingularly perturbed ytem (SS) with an eogenou input coupled with low and at dynamic. he inight link between paivity and ISS are preented eplicitly. We irt ind a linear matri inequality (LMI) condition under which the SS are trictly paive on an eogenou input or all uiciently mall ingular perturbation parameter and then etablih a relationhip between paivity and input-to-tate tability (ISS) or thi SS with an eogenou input and the problem o deigning a robut control law i addreed or uch SS with a reerence input uch that the cloed loop SS are robut paive and ISS on an eogenou input when the ingular perturbation parameter i uiciently mall. o achieve thi a matri inequality-baed criterion i propoed and proved. he criterion i then converted into a LMI eaibility iue or the purpoe o implementation in which the control gain matri i derived by olving uch a LMI. Key erm: Name o the Author: Dr. Boka Kuma Aitant proeor Department o Mathematic Wollega Univerity.O Bo 395 Nekemte EHIOIA kumab@yahoo.com Eogenou input Linear matri aivity Robut State tability Singular perturbed Citation o the Article Dr.Kuma B.(7). aivity and ISS or Singularly erturbed Sytem with Eogenou Diturbance. Advance Reearch Journal o Multidiciplinary Dicoverie.3.C-3(7):-8 ISSN An open acce journal o International Journal Foundation ( age I

2 Advance Reearch Journal o Multi-Diciplinary Dicoverie ISSN NO : I. INRODUCION he at and low dynamic phenomena occurring in multiple time-cale ytem i commonly encountered in real engineering application. It bring great challenge to control theory ince multiple time cale can caue problem o increaed order and tine o ytem. o overcome uch problem ingular perturbation method i introduced and ha been proven to be a powerul tool in ytem analyi and ynthei according to (. V. Kokotovic H. K. Khalil and J. O Reilly 986) (D. S. Naidu ) and recently (Lei M. and Cheniao C. 6) tability analyi and tabilization ynthei problem o the ingularly perturbed witched ytem (SSS) have been invetigated irt conidering tability o the SSS with table ubytem by uing an average dwell time approach with a ull-order piecewie Lyapunov unction. hen the reult i etended to the ituation that not all the ubytem are Hurwitz table. Furthermore tate eedback controller are deigned when all ubytem are tabilizable and the deign method i etended to the ituation that not all ubytem are tabilizable. (L. Moreau and D. Aeyel ) conidered the tability o ingularly perturbed ytem with time delay that i the boundary layer ytem i robutly globally aymptotically table and the reduced low ytem admit a Lyapunov-Razumikhin unction then the ingularly perturbed ytem ha certain practical and emi-global aymptotic tability property a the parameter o ingular perturbation reduce to zero and other many more reult have been orwarded. Motivated by the aorementioned work thi paper ocue on the robut control problem o ingularly perturbed ytem with eogenou input. In thi paper we will tudy the robut tability or ingularly perturbed ytem rom a paivity point o view. Firt uing LMI technique uicient condition or trict paivity o the ytem without control input are given. Meanwhile the relation between the paivity and input-to-tate tability (ISS) i urther etablihed. hen we addre the problem o deigning a tate eedback controller o that the cloed-loop ytem i trictly paive and ISS or uiciently mall perturbation parameter. he contribution o thi paper can be ummarized a below.. A more general cla o ytem i conidered in which eogenou input depend on the at and low dynamic.. he relation between the paivity and ISS i etablihed it will be epecially helpul or u to analyze the tability o ytem. 3. A LMI baed uicient condition or paivity and ISS tability o ytem i provided. he control gain matri can be olved eiciently in which olving much more comple equation i not involved. he ret o the paper i organized a ollow. Section give the problem ormulation. he main reult are given in Section 3. Section 4 give two eample to how the eectivene o the propoed method. Finally the concluion i drawn in Section 5. II. ROBLEM FORMULAION Conider the ollowing nonlinear uncertain SS with eogenou diturbance given by ( t) ( A A ( t)) ( t) ( A A ( t)) ( t) B w( t) ( B B ( t)) u( t) () ( t) ( A A ( t)) ( t) ( A A ( t)) ( t) B w( t) ( B B ( t)) u( t) () where ( ) i the ytem tate i the control input; w n R and n R ( m R i the bounded diturbance input; n n n ) are the low tate and the at tate repectively; u R i a perturbation parameter which i mall and poitive but may be unknown repreenting the repone o the at dynamic. ( t ) and ( t ) are initial condition A ij and B ij ( i j ) are contant matrice with appropriate dimenion; A () t and B () t ( i j are vector-value time- varying uncertaintie which are aumed to be norm bounded i.e. ij ij q A ( t) A ( t) B ( t) H F () t E E E 3 A( t) A ( t) B( t) H (3) H H E E and E 3 are contant matrice with appropriate dimenion. Ft () I an unknown time-varying atiying Deine F ( t) F( t) I t [ ). (4) H H E I O A A O I A A A H E E E B A ( t) A ( t) A() t A ( t) A ( t) B B B B () t B () t. B () t B B hen ytem ()-() can be rewritten in a compact orm E ( t) ( A A( t)) B w( t) ( B B ( t)) u( t). (5) It eay to veriy that the uncertain item At () and B () t atiy A ( t) HF( t) E B ( t) HF( t) E3. (6) We now give ome baic material beore continuing our dicuion which will be ued in the derivation o the main reult. An open acce journal o International Journal Foundation ( age I

3 Advance Reearch Journal o Multi-Diciplinary Dicoverie ISSN NO : Deinition.: Sytem ()-() with u i aid to be a tandard ingularly perturbed ytem i the algebraic equation ( A A ( t)) ( t) ( A A ( t)) ( t) B w( t) (7) ha a unique iolate root ( t w) or any given ( w ). From the Deinition. we can ee that the eitence o the iolate root enure that the n -dimenional reduced model i well deined. hu it become a tandard requirement or mot ingularly perturbed control ytem according to (. V. Kokotovic H. K. Khalil and J. O Reilly 986) (D. S. Naidu ). Deinition.: According to (H. K. Khalil ) Conider the ytem: ( t w) (8) n m n m n tate t () i in R and control input wt () in R. :[ ) R R R i continuou and locally Lipchitz in and w. he input w i a bounded unction or all t. hen the ytem i aid to be input-to-tate table (ISS) i there eit a cla KL unction and a cla K unction uch that or any initial tate t ( ) the olution t () eit or all t t and atiie: ( t) ( ( t ) t t ) (up w( ) ). Remark he lat inequality guarantee that or any bounded input wt () the tate t () will be bounded and a t increae the tate t () will be ultimately bounded by a cla K unction o w. Furthermore the inequality alo how that i wt () converge to zero a t o doe t () which will be veriied in the net ection. Lemma.: n According to (H. K. Khalil ) Let V :[ ) R R be a continuouly dierentiable unction uch that are cla t t ( ) V( t ) ( ) V V ( t w ) W ( ) ( w ) t K unction i a cla K unction and W( ) i a continuou poitive deinite unction on ytem (8) i input-to-tate table with. n R. hen the Lemma.: According to (H. K. Khalil ) Let and be real matrice o appropriate dimenion. hen or any matri F atiying F F I and a calar F ( F ). III. RESULS AND DISCUSSION A. he Eitence o an Iolate Root Analyi In the ubequent two ubection we will conider ytem ()-() without control input. he ollowing reult preent a uicient condition in term o linear matri inequality to guarantee the eitence o an iolate root or the ytem ()-(). Lemma 3.: I there eit a calar matrice and uch that the ollowing linear matri inequality hold A A E E H * I (9).hen the uncertain ytem ()-() i a tandard ingularly perturbed ytem. roo: Let we obtain the reduced-order ytem rom (5) E diag{ I O}. Condition (9) implie n Noticing that EE i non-negative which mean that E ( A A( t)) B w( t) A A E E. A A. () Now making a partition or () give A A. So A i non-ingular which implie A i non-ingular too. An open acce journal o International Journal Foundation ( age I

4 Advance Reearch Journal o Multi-Diciplinary Dicoverie ISSN NO : n n According to (L. Dai 989) it i eay to obtain that the pair ( E A) i regular and impule ree thu there eit matrice M R n n M R n n N R n n N R uch that M ( M M ) and N ( N N ) are non-ingular upper- and lower triangular matrice repectively and the ollowing decompoition hold: ME N diag( I ) n MAN diag( A I ) n n n A R. Noticing that M HH M i poitive emi deinite thu one ha Q ( M HH M I ) i poitive deinite or any. Let diag( I Q ) n M M N N then we have and Q M HH M Q ME N diag( I ) n MAN diag( A I ) n ( M HH M I) ( M HH M ) ( M HH M I) ( M HH M I) ( M HH M I)( M HH M I) I Which implie that Q M H By the Schur complement lemma it i obtained rom LMI (9) that A A F F HH. () re- and pot-multiplying both ide o () with N and N repectively we obtain that Let ( MAN ) M N ( M ) N MAN N F FN ( M N) MHH M M N. hen it i eay to know that the tructure o () M N 3 4 M and N implie that By decompoing () and urther calculation we have that the block matri at the econd block row and the econd block column o the let-hand ide o () i negative deinite that i Q N F FN Q Q M HH M Q It implie that there eit a uiciently mall uch that that i hi i equivalent hen it implie that Q N F FN Q I 4 4 Q N F FN Q Q ( M HH M I ) Q 4 4 Q N F FN Q ( ) ( ) I I I Q N F FN Q 4 4. hu there eit a uiciently mall calar uch that FN Q. In order to how the eitence o the iolate root we introduce a change o coordinate n R R n N ( ) (3). hen the reduced-order ytem can be rewritten equivalently a ollow: t A M HF t E N N Q M B w (4) ( ) ( ) ( ) For any given R Q M HF( t) E( N N Q ) Q M B w. m (5) we have Q M HF( t) E( N N Q ) Q M HF( t) E( N N Q ) Q M HF( t) EN Q ( ) Q M H FN Q ( ) FN Q. (6) ( ). According to the ied-point principle there eit a unique olution ( t w) or any given ( w ) rom (6). hu the eitence o iolate root ( t w) i obtained by (3) that i the ytem ()-() i in the tandard orm which complete the proo. An open acce journal o International Journal Foundation ( age I 3

5 Advance Reearch Journal o Multi-Diciplinary Dicoverie ISSN NO : Remark : From the proo o Lemma 3. it noticed that the eitence o the iolate root or ytem ()-() i inherited rom the reduced-order ytem which i a key tep or the two-time cale decompoition technique. However it i hown in (Yang. Y. Robut 3) and (H. K. Khalil ) that the correponding problem ha not been invetigated. hu thi can be viewed a an etenion according to (Yang. Y. Robut 3) and (H. K. Khalil ) urthermore we can alo how that ( t w) i Lipchitz with repect to ( w ) that i there eit two calar and atiying the ollowing contraint ( t w) w. (7) And the mathematical derivation i imilar to that o (6) thu the detail i omitted here. Now according to Lemma 3. and the time-cale property o ingularly perturbed ytem the low ubytem o ytem ()-() can be decribed by etting ( A A ( t)) ( A A ( t)) B w (8) ( t ) ( A A ( t)) ( A A ( t)) B w.. (9) ( t w ) here can be viewed a an intermediate variable which will contribute to the analyi o ISS or (8)-(9) Deine ( ) then ytem (8)-(9) can be rewritten in a compact orm E ( A A ( t)) B w. () Moreover let w w w the at ubytem valid in a boundary layer d ( A A ( t)) B w ( ) d () Can be obtained by etting and changing time cale t t In ytem ()-() Since ()-() i only an approimate model o ytem ()-() the ISS property o reduced order ytem doe not imply the one o the original ytem. hu attention i now ocued on how the ISS property o the original ytem ()-() can be deduced rom the reduced order ytem in eparate time cale. B. Input-to-State Stability Analyi In thi ubection baed on the reduced technique we will give the uicient condition under which the ull-order ytem i ISS. Firt we have ollowing reult or the low ubytem ()-(). heorem 3.: Under the condition o Lemma 3. i the matri then the low ubytem ()-() i made ISS with repect to the diturbance w. roo: In order to how that the low ubytem ()-() i ISS we chooe the Lyapunov unction candidate a ollow: S ( )... () Clearly S ( ) or any Notice that hu the derivative o S along ytem () yield S ( ) E. S ( ) (( A A ( t)) B w ) (( A A ( t)) B w ) Furthermore or any calar uing the contraint (3) and Lemma. we have A A E E HH ( ) ( ( ) ) S A HF t E B w ( A HF ( t ) E B w ) w w B w By the Schur Complement Lemma we can ee that can be guaranteed by (9). Let a ( ) then it ollow rom () that min a. hu it yield Noticing the contraint (9) we urther obtain S a B w. ( ) w S a B B w ( ) ( ) w w a b w c w An open acce journal o International Journal Foundation ( age I 4

6 Advance Reearch Journal o Multi-Diciplinary Dicoverie ISSN NO : b ( ) B w and c B w b b 4ca a( ) w a. Hence the condition o Lemma. are atiied and we can conclude that there eit a cla KL unction and a cla K unction uch that or any initial tate the olution () t eit or all t t and atiie ( t) ( ( t ) t) (up w ( ) ). (3) t t hat i the low ubytem ()-() i ISS with repect to the diturbance w thi i complete proo. For the at ubytem () we have the ollowing reult. heorem 3.: Under the condition o Lemma 3. i the matri then the at ubytem () i made ISS with repect to the diturbance w roo: In order to how that the at ubytem () i ISS We make a partition or LMI (9) which i equivalent to () () ( ) * ( ) * * * * H H O I O H O I A A A A E E ( ) A A A E E ( ) A A E E. hi indicate that Let A A E E H * I S ( ) calculating the derivative o S along the trajectorie o ytem () we have. (4) S ( ) ( A A ( t) B w ) ( A A ( t) B w ) B w ( A A E E H H ) B w. hen the ret o the proo can ollow along the ame line o the proo o heorem 3.. he eact argument i omitted or brevity. Remark : According to (Yanbo Gao Binghua Sun and Guoping Lu ) ISS property and aymptotic tability o the low and at ubytem were preuppoed. Baed on thee and the aumption o the eitence o an iolate root a type o total tability or ISS property o ingularly perturbation wa etablihed. Compared with (Yanbo Gao Binghua Sun and Guoping Lu ) heorem 3. and 3. preent veriiable uicient condition or ISS o the ubytem via LMI which can be regarded a an advance although the conidered ytem here i le general than that in (Yanbo Gao Binghua Sun and Guoping Lu ). Baed on heorem 3. and 3. we are now in the poition to how the main reult o thi ubection which i tated in the ollowing theorem. heorem 3.3: I the condition o heorem 3. and 3. hold then there eit an uch that the ollowing reult hold: ) Sytem ()-() i a tandard ingularly perturbed ytem; ) Sytem ()-() i made ISS with repect to diturbance w or any given ( ]. roo: ) he proo o Lemma 3. ha hown that ytem ()-() i in tandard orm which complete the proo o part ) We now how the ISS property o ytem ()-(). Under the condition o heorem 3. and 3. it i hown that both and are poitive deinite matrice then there eit a uiciently mall calar uch that yield or all ( ]. By the Schur Complement Lemma it E E ( ]. Deine a Lyapunov unction candidate or ytem ()-() a ollow S( ) E (5) An open acce journal o International Journal Foundation ( age I 5

7 Advance Reearch Journal o Multi-Diciplinary Dicoverie ISSN NO : hen or any a contant the derivative o S( ) ( E ) E (( A A( t)) B w( t)) (( A A( t)) B w( t)) ( A A E E HH ) B w.. A A E E HH For the above inequality by the Schur complement Lemma we have that i equivalent to A A H O * O... (6) O O It ollow rom (9) that there eit a uiciently mall calar uch that or any given ( ]. hu or ( ] i guaranteed. Let a ( ) min then a or ( ]. Further let min{ } then we have E and Simultaneouly or any given ( ] thu we obtain S( ) a B w w ma B [ ] S( ) a B w a( ) w (7) a hence the condition o Lemma. are atiied and we conclude that ytem ()-() i ISS with repect to the diturbance w or ( ]. hat i there eit a cla KL unction cla K unction uch that or any initial tate t ( ) the olution t () eit or all t t and atiie: So thi i complete the proo ( t) ( ( t ) t t ) (up w( ) ) ( ]. (8) t t Remark 3.3: heorem 3.3 preent a uniied uicient or the eitence o the iolate root and ISS property or ytem ()-() by the two-cale decompoition technique and compared with the eiting reult (Yang. Y. Robut 3) our uicient condition i independent o the coordinate tranormation and only dependent on the olution o LMI (). hu it i eaier to tet numerically. It i worth pointing out that the derived condition here i independent o thereore the numerically ti problem can alo be avoided. A a pecial cae when the diturbance w () t atiie the condition w( t) a t we have the ollowing reult. Corollary 3.: Under the condition o heorem 3.3 i the diturbance w( t) a t then ytem ()-() i aymptotically table or all ( ]. roo: In the cae o w( t) a t. For any given a calar ( ) can be ound uch that ( ) /. Since lim w( t) there eit a calar uch that wt ( ) or all t. ake t. For t t by the inequality (8) we have t ( t) ( ( t ) t t ) ( ) ( c ) / t t or ome c. Since ( c t t ) a t there eit a calar uch that ( c t t ) / or all t together with the lat inequality yield t ( ) t ma{ }.. hu thi he lat inequality how that lim t ( ) that i the ytem ()-() i aymptotically table or any ( ]. So thi i complete the proo. t An open acce journal o International Journal Foundation ( age I 6

8 Advance Reearch Journal o Multi-Diciplinary Dicoverie ISSN NO : C. he Input-to-State Stability o Cloed-loop Sytem From heorem 3.3 we know that the ytem ()-() i ISS with a poitive deinite Lyapunov unction require that ytem ()-() be table when the diturbance and input do not eit. However in many cae it might not be table. o remove thi retriction we include eedback a a mean to achieve ISS. Intead o being table the ytem i aumed to be tabilizable. hereore we in thi ubection need to ind a tate eedback tranormation u K K (9) K ( K K ) i a contant matri uch that the reulting cloed-loop ytem i ISS with repect to the diturbance wt () when eedback implementation. Subtituting the above control unction (9) into ()-() we obtain the cloed-loop ytem a ollow: E ( t) ( A B K A B ( t) K) ( t) B w( t) (3) Applying theorem 3.3 to the cloed-loop ytem (3) we have the ollowing reult. heorem 3.4: I there eit a contant matri Y and a lower triangular matri n n With X R and X n n R X X X X atiying the ollowing linear matri inequality then there eit an AX X A B Y Y B H H X E Y E 3 EX E Y I 3 (3) * uch that the reulting cloed-loop ytem (3) i in the tandard orm and ISS with repect to w or any ( ]. Moreover tate eedback gain matri can be choen a K YX. (3) roo: By ubtituting (3) into (3) and uing the Schur Complement Lemma we obtain that inequality (3) i equivalent to X ( A B K ) ( A B K ) X X ( E E K ) ( E E K ) X H u 3 3. (33) * I re- and pot-multiplying inequality (33) by diag ( X I) anddiag( X I) repectively let X Y K then (33) i equivalent to ( A B K) ( A B K ) ( E E K ) ( E E K ) H u u 3 3. (34) * I We chooe the Lyapunov unction candidate a ollow V ( ) E O O O O. hen or any a contant the derivative o V ( ) ( E ) E (( A B K A B ( t) K) ( t) B w( t)) (( A B K A B ( t) K) ( t) B w( t)) B w. ( ) ( ) ( ) ( ) A B K A B K E E K E E K HH 3 3 B w For the above inequality by the Schur complement Lemma we have that i equivalent to ( A B K) ( A B K) H * O O O O.. (35) An open acce journal o International Journal Foundation ( age I 7

9 Advance Reearch Journal o Multi-Diciplinary Dicoverie ISSN NO : he equent proo i imilar to that o heorem 3.3 thu there eit a calar uch that the cloed-loop ytem (3) i in the tandard orm and ISS with repect to wt () or any ( ]. hi complete the proo. Similar to Corollary 3. in the cae o wt () the cloed-loop ytem (3) reduce to the ollowing orm in (H. K. Khalil ). E ( t) ( A B K A B ( t) K) ( t).. (36) he ollowing reult can be obtained immediately rom theorem 3.4. Corollary 3.3: Aume that the condition o heorem 3.4 hold then the reulting cloed-loop ytem (36) i in the tandard orm and aymptotically table or any ( ] IV. CONCLUSION hi paper ha invetigated the robut paive control or ingularly perturbed ytem. A proper tate eedback control law in term o LMI technique ha been preented under which the cloed-loop ytem i trictly paive. Moreover the relation between the paivity and ISS tability ha been etablihed. It worth mentioning that uch a relation in the contet o ingularly perturbed ytem ha eldom been reported in the literature. REFERENCES [] Chakrabortty and E. Scholtz ime-cale eparation deign or perormance recovery o power ytem with unknown parameter and ault IEEE ran. Control Sytem and echnology vol. 9 no. pp [] R. eel Luc Moreau and Dragan Neˇic A uniied ramework or input-to-tate tability in ytem with two time cale IEEE ran. Autumn. Control vol. 48 no. 9 pp [3] Y. Yang Q. L. Zhang J. Sun and. Y. Chai Lur e Lyapunov unction and abolute tability criterion or Lur e ingularly perturbed ytem IEEE ran. Autumn. Control vol. 56 no. pp [4] S. Naidu Singular perturbation and time cale in control theory and application: an overview. Dynamic o Continuou Dicrete and Impulive Sytem Serie B: Application & Algorithm vol. 9 no. pp [5] H. K. Khalil Feedback control o nontandard ingularly perturbed ytem IEEE ran. Autumn. Control AC-34 pp [6] H. K. Khalil Nonlinear Sytem 3 rd ed. Upper Saddle River N. J. rentice Hall. [7] H.. Liu F. C. Sun and Z. Q. Sun Stability analyi and ynthei o uzzy ingularly perturbed ytem IEEE ran. Fuzzy Sytem vol. 3 no. pp [8] Hongwang Yu Guoping Lu and Yuan Zheng On the model-baed networked control or ingularly perturbed ytem with nonlinear uncertaintie Sytem and Control Letter vol. 6 no. 9 pp [9] L. Moreau and D. Aeyel Aymptotic method in the tability analyi o parameterized homogeneou low Automatica vol. 36 no. 8 pp [] L. Moreau and D. Aeyel ractical tability and tabilization IEEE ran. Autumn. Control Vol. 45 pp [] L. Dai Singular Control Sytem Berlin Germany Springer 989 [] Lawrence. erko Dierential equation and dynamical ytem 3 rd Springer-Verlag New York [3] Lei M. and Cheniao C. Stability Analyi and Stabilization Synthei o Singularly perturbed witched ytem: An average dwell time approach Cogent-Engineering 6. [4] Lei Zhou and Guoping Lu Robut tability o ingularly perturbed decriptor ytem with nonlinear perturbation IEEE ran Autumn Control vol. 56 no. 4 pp [5] M. G. Dmitriev and G. A. Kurina Singular perturbation in control problem Autumn. Remote Control Vol. 67 no. pp [6]. D. Chritoide A. R. eel Singular perturbation and input-to-tate tability IEEE ran Autumn. Control Vol. 4 no. pp [7]. Shi S.. Shue and R.K. Agarwal Robut diturbance attenuation with tability or a cla o uncertain ingularly perturbed ytem International Journal o Control vol. 7 no. 6 pp [8]. V. Kokotovic Application o ingular perturbation technique to control problem SIAM Review vol. 4 pp [9]. V. Kokotovic H. K. Khalil and J. O Reilly Singular perturbation method in control: analyi and deign London academic 986. [] R. Amjadiard M.. H. Behehti and M. J. Yazdanpanah Robut tabilization or a cla o nonlinear ingularly perturbed ytem Journal o Dynamic Sytem Meaurement and Control. [] Singh R. H. Brown. D. S. Naidu and J. A. Heinen Robut tability o ingularly perturbed tate eedback ytem uing uniied approach IEE roc.-control heory Appl. vol. 48 pp [] W. H. Chen Y. Guo and W. X. Zheng Robut tability o ingularly perturbed impulive ytem under nonlinear perturbation IEEE ran Autumn Control vol. 58 no. pp [3] Wei Wang A. R. eel and Dragan Neic Analyi or a cla o ingularly perturbed hybrid ytem via averaging Automatica vol. 48 pp [4] Yanbo Gao Binghua Sun and Guoping Lu aivity-baed integral liding-mode control o uncertain ingularly perturbed ytem IEEE ran. Circuit and Sytem II: Epre Brie vol. 58 pp [5] Yang Y. Robut Stability o Singularly erturbed ytem with delay 5nd IEEE Conerence on Deciion and Control 3. ***** An open acce journal o International Journal Foundation ( age I 8