Design of Finite-Time Synchronization Controller and Its Application to Security Communication System

Transcription

1 Appl. Math. If. Sci. 8, No., 87-9 (4) 87 Applied Mathematics & Iformatio Scieces A Iteratioal Joural Desig of Fiite-Time Sychroizatio Cotroller ad Its Applicatio to Security Commuicatio System Tao Re, Zhi-liag Zhu ad Hai Yu Software College, Northeaster Uiversity, Sheyag Liaoig 4, P R Chia Received: 7 Jul., Revised: 7 Nov., Accepted: Nov. Published olie: Ja. 4 Abstract: I this paper, a fiite-time sychroizatio cotroller is proposed for the geeral chaos model. Compared to other results, the cotroller desiged is simpler ad idepedet o the oliear fuctio of chaos system. So the fiite-time cotrol scheme ca be used i a wide rage of chaos system. The simulatio results are give to illustrate the effectiveess of the fiite-time cotroller ad the applicability to security commuicatio system of this sychroizatio scheme. Keywords: Chaos system, sychroizatio, fiite-time cotrol, security commuicatio. Itroductio Chaos sychroizatio has bee a focused research topic durig the last decade due to its theoretical ad practical applicatios especially i security commuicatio. May schemes ad techiques for cotrollig chaos sychroizatio such as adaptive feedback cotrol [ 4], state feedback cotrol [5, 6] ad observer-based approach [7] etc have bee developed. Furthermore, some attractive results of sychroizatio ad applicatios i secure commuicatio for a class of chaotic systems have bee reported [ 6]. Based o the above results, a large umber of chaos sychroizatio schemes are aimed at asymptotical stability of error sychroizatio dyamics. From the practical poit of view, however, it is more valuable that the sychroizatio objective is realized i a fiite time. Especially i secure commuicatio systems for istace, the rage of time durig which chaotic systems are out of sychroy is equivalet to the rage of time i which the ecoded message/data caot be recovered or set. Therefore, the sychroizatio time optimizatio is essetial for satisfyig the requiremet of commuicatio ad this could be doe by meas of fiite-time cotrol-a very promisig techique by which we ca decide the settlig time flexibly. Some authors have ivestigated chaos sychroizatio based o fiite-time recetly [8 6]. However, most of the reported works are more specific to some systems, such as Lorez system ad Chua system. There are some limitatios for these cotributios to be applied to the other systems except the oe studied. The other problem for some works is the complex cotroller desig. Some cotrollers proposed cotai all or parts of the oliear fuctio, ad the the closed-loop error system ca be deduced to liear system. This method will reduce the difficulties of mathematical proof, meawhile make the cotroller too complex compared to systems beig cotrolled. These approaches are usually difficult to implemet i experimets whe possible. Motivated by the above discussio, i this paper, a simple sychroizatio cotroller is proposed for realizig sychroizatio i fiite time based o the geeral chaos system. The structure of the cotroller is idepedet o the chaos system, so it ca be implemeted i may chaos systems. The umerical simulatios are give to illustrate the effectiveess. The rest of the paper is orgaized as follows. I Sectio, the fiite-time cotroller is desiged to guaratee the stability of the error system. I Sectio, the simulatio example is give to illustrate the applicability for security commuicatio of the proposed approach. A coclusio is draw i the last sectio. Notatios: I deotes the idetity matrix of appropriate dimesios. x(t) meas the Euclidea vector orm at time t, while x(t) := x(t) dt. sg( ) is deoted Correspodig author chiaretao@6.com, zzl@mail.eu.edu.c c 4 NSP

2 88 T. Re et al : Desig of Fiite-Time Sychroizatio Cotroller... as sig fuctio. diag( ) represets a block-diagoal matrix. The Mai Results Cosider the followig master-slave sychroizatio system M : ẋ(t)=ax(t)+ f(x(t)) S : ẏ(t)=ay(t)+ f(y(t))+u(t) where M ad S deote the master system ad slave system with the state vectors x(t) R ad y(t) R respectively. A is the costat matrix with appropriate dimesios. u(t) = (u (t),,u (t)) is the fiite-time cotroller to be desiged. Defiig e(t) = y(t) x(t), the we have the error dyamic system as follows () ė(t)=ae(t)+ f(y(t)) f(x(t))+u(t) () The cotroller u(t) is proposed as: u i (t)= k e i (t) α sg(e i (t)) () where costat k is the cotroller gai ad the costat α (,). Fact Accordig to the features of chaos system, each state of chaos system ẋ(t) = f(x(t)) is bouded, furthermore the error of ay two states is bouded too. From Fact we ca get the coclusio that there exists a costat M, such that the state of error system () satisfies e M. Assumptio The oliear fuctio f : R R i () satisfies the followig Lipschitz coditio: f(x ) f(x ) γ x x where γ is a positive scalar. Lemma Suppose a,a,,a ad <q< are all real umbers, the the followig iequality holds: a q + a q + + a q ( a + a + +a ) q/. (4) Lemma [5] Cosider the system ẋ(t)= f(x(t)), f()=,x R (5) where f : D R is cotiuous o a ope eighborhood D R. Suppose there exists a cotiuous differetial positive-defiite fuctio V(x(t)) : D R, real umbers p>, <η <, such that V(x(t))+ pv η (x(t)), x(t) D (6) The, the origi of system (5) is a locally fiite-time stable equilibrium, ad the settlig time, depedig o the iitial state x()=x, satisfies T V η (x ) p( η) (7) I additio, if D = R ad V(x(t)) is also radially ubouded (i.e. V(x(t)) as x(t) ), the the origi is a globally fiite-time stable equilibrium of system (5). Theorem. The error system () ca achieve stability (i.e. the sychroizatio of the master ad slave system () is achieved) i fiite time T by the / cotroller (), if there exists costat k (d+ λm α ) >, the settlig time T is determied by T (V(e )) α d( α). (8) where λ = max{,λ max }, i which λ max is the maximum eigevalue of (A+γI) ad γ is Lipschitz costat of f( ), costat d >, α (,) ad M such that e M. Proof: Costruct a Lyapuov fuctio cadidate as V(e)=(e + e +...+e ) (9) Takig the time derivative of V(t) ad substitutig () ad () ito V(t) yield V(e) = (ė e + ė e +...ė e ) = e T Ae+ e T Ae+ k e i α+, ( f i (y) f i (x)+u i ) e i f i (y) f i (x) e i ) by virtue of Assumptio, we have V(e) e T Ae+e T γe k e i α+ = e T A + γie k ( e i ) α+ λ max e T e k ( e i ) α+, Due to the fact.5< α+ < ad Lemma, we get V(e) λ max e T e k(v(e)) α+ = (k λ max (V(e)) α )(V(e)) α+, By Theorem, we ca get k (d+ λm α ) the / >, V(e) (d+ λm α λ max (V(e)) α )(V(e)) α+ () Next, we will discuss the value of V(e) accordig to λ max. There are two possibilities for λ max : c 4 NSP

3 Appl. Math. If. Sci. 8, No., 87-9 (4) / d(v (e)) α + α )(V (e)) α , y V (e) (d λmax (V (e)) x () Whe λmax, the λ = due to the fact that λ = max {, λmax }, it yeilds 89 () Whe λmax >, the λ = λmax, oe ca get V (e) (d + λmax (M (V (e)) α ))(V (e)) α + z 5 α 4 5 Accordig to Fact, we have the followig iequality V (e) = ei = kek M, it yields V (e) d(v (e)) α + () Fig. : The three state trajectories of Lorez system Therefore, o matter what the value of λmax is, we ca α + coclude that V (e) d(v (e)), i which d >, α (, ). Accordig to Lemma, we ca get the coclusio that the error dyamic ca achieve stability i fiite time α ad the settlig time T (V (e )) d( α ). Hece 4 z the chaos system () ca sychroize before time T. This completes the proof. 6 Numerical Simulatio x 5 y Fig. : The chaotic attractors of Lorez system e e e I this sectio, the followig Lorez systems are cosidered as master ad slave system for the security commuicatio scheme based o chaos maskig scheme. x = a(x x ) Master system: x = cx x x x x = x x bx y = a(y y ) + u y = cy y y y + u Slave system: y = y y by + u where a =, b = 8/, c = 8. We set the iitial value for both chaos systems as (x (), x (), x ()) = (,,) ad (y (), y (), y ()) = (,,), the the iitial value of error system is (e (), e (), e ()) = (-,-,). As is show i Figure ad, the three states of Lorez system are bouded idepedet o the iitial value, i.e. x(t) (, ), y(t) (, ), z(t) (, 5), so are the error state values. Due to Fact kek M, we chose M = 77 for Lorez system. The Lipschitz costat is calculated as γ = 65. The we ca get λmax = 47 by calculatig the maximum eigevalue of (A + γ I). Hece, the rage of the cotroller gai is k 8, we chose k = 85 for simulatio. Meawhile, we set the parameters d = 4 ad α =.8, by Theorem the settlig time is calculated as T.99s. Figure shows that the states of error dyamic ca regulate to zero i short time, less tha the settlig time T. This implies that the master ad slave Fig. : The state trajectories of error system c 4 NSP

4 9 T. Re et al : Desig of Fiite-Time Sychroizatio Cotroller... chaos systems ca sychroize by the fiite-time cotrol scheme proposed very quickly. The iitial Sie wave sigal ca be masked well by the master system show i Figure 4, the trasmitted sigal looks like oise sigal ad impossibly attracts attetio of itruder. The recovered sigal is show i the bottom half of Figure 5, which is almost the same as the iitial Sie wave sigal show i the top half of the same Figure. 4 Coclusio I this paper, cotrol theory is used to formalize fiite-time chaos sychroizatio cotroller. The stability criteria is preseted for error system based o Lyapuov method. Compared to other results, the proposed cotroller is simpler. The umerical simulatio results for Lorez system show the effectiveess of our fiite-time cotrol scheme ad its applicability to security commuicatio system. The scheme provides a safe way to mask useful iformatio carryig sigal. trasmitted sigal si sigal recovered si sigal Fig. 4: The trasmitted/masked sigal Fig. 5: The Sie sigal (iitial sigal is i the top half; recovered sigal is i the bottom half) So the security commuicatio scheme based o chaos sychroizatio ca ot oly protect the message from attackig by the itruder but provide a good way to recover the useful sigal by the receiver. Ackowledgemet The authors ackowledge the fiacial support by the Natioal Natural Sciece Foudatio of Chia (Grat No. 6474), the Chia Postdoctoral Sciece Foudatio (Grat No. T694, 4746), the Fudametal Research Fuds for the Cetral Uiversities of Miistry of Educatio of Chia (Grat No. N7, N647, N474, N475, N67) ad the Scietific Research Foudatio for Doctor of Liaoig Provice, Chia (Grat No. ). The author is grateful to the aoymous referee for a careful checkig of the details ad for helpful commets that improved this paper. Refereces [] J. H. Park, S. M. Lee, O. M. Kwo, Phys. Lett., A7, 6 (7). [] M. F. Hu ad Z. Y. Xu, Chiese Physics, 6, (7). [] J. H. Park, It. J. Noliear Sci. Numer. Simul., 6, (5). [4] H. Salarieh ad A. Alasty, Commu Noliear Sci Numer Simul., 4, 58 (9). [5] H. H. Che, G. J. Sheu, Y. L. Li ad C. S. Che, Noliear Aalysis: Theory, Methods ad Applicatios, 7, 49 (9). [6] H. H. Che, Physics Letters, A7, 84 (8). [7] J. Meg ad X. Y. Wag, Physics Letters, A69, 94 (7). [8] S. H. Li ad Y. P. Tia, Chaos, Solitos ad Fractals, 5, (). [9] N. Cai, Y. W. Jig ad S. Y. Zhag, Commu Noliear Sci Numer Simulat, 5, 6 (). [] P. Wilfrid, F. Thierry ad M. Emmauel, IEEE Trasactios o Automatic Cotrol, 5, 56 (8). [] H. Wag, Z. Z. Ha, Q. Y. Xie ad W. Zhag, Commu Noliear Sci Numer Simulat, 4, 9 (9). [] H. Wag, Z. Z. Ha, Q. Y. Xie ad W. Zhag, Commu Noliear Sic Numei Simulat, 4, 78 (9). [] H. Wag, Z. Z. Ha, Q. Y. Xie ad W. Zhag, Noliear Aalysis: Real World Applicatios,, 84 (9). [4] U. E. Vicet ad R. Guo, Physics Letters, A75, (). [5] M. P. Aghababa, S. Khamohammadi ad G. Alizadeh, Applied Mathematical Modellig, 5, 8 (). [6] W. G. Yu, Physics Letters, A74, (). c 4 NSP

5 Appl. Math. If. Sci. 8, No., 87-9 (4) / 9 Tao Re was bor i 98 i Chia ad received the PhD degree from Northeaster Uiversity i cotrol theory ad cotrol egieerig i 7. He is curretly a associate professor i software college, Northeaster Uiversity. His mai research iterests iclude chaos sychroizatio cotrol, chaos security commuicatio, cogestio cotrol of high speed computer etworks. He has published more tha papers ad oe book. He is ow resposible for 7 academic research projects i Chia. Zhi-liag Zhu received the PhD degree i computer sciece from Northeaster Uiversity i. His mai research iterests iclude iformatio itegrate, complexity software system, etwork codig ad commuicatio security, chaos-based digital commuicatios, applicatios of complex-etwork theories, ad cryptography. By far, he has authored ad co-authored over iteratioal joural papers ad coferece papers. Additioally, he published 5 books, icludig Itroductio to Commuicatio ad Program Desigig of Visual Basic.NET. He is also the recipiet of 9 academic awards at the atioal, miisterial ad provicial level. Hai Yu received the PhD degree i Computer Software ad Theory i 6 at the Northeaster Uiversity. From May 9 to August 9, he worked shortly as a Seior Research Assistat to the Cetre for Chaos ad Complex Networks, Departmet of Electroic Egieerig, City Uiversity of Hog Kog, uder the supervisio of Prof. Guarog Che. His research iterests have focused maily o multimedia security ad secure chaos-based commuicatios, ecompassig Digital Watermarkig, Iformatio Hidig, Video Codig, Digital Chaotic Cipher, Network Codig, Chael Codig ad applicatios of complex-etwork theories to commuicatios. c 4 NSP