The generalized Rössler hyperchaotic systems are presented, and the state observation problem of such systems is investigated. Based on the differential inequality with Lyapunov methodology (DIL methodology), a nonlinear observer design for the generalized Rössler hyperchaotic systems is developed to guarantee the global exponential stability of the resulting error system. Meanwhile, the guaranteed exponential decay rate can be accurately estimated. Finally, numerical simulations are provided to illustrate the feasibility and effectiveness of proposed approach.
1. Introduction
In recent decades, several kinds of chaotic systems have been widely explored; see, for instance, [1–11] and the references therein. This is due to theoretical interests as well as to an efficient tool for chaos synchronization and chaos control design. As a rule, chaos in many systems is a source of the generation of oscillation and a source of instability. Chaotic systems frequently exist in various fields of application, such as system identification, master-slave chaotic systems, secure communication, and ecological systems.
Form practical considerations, it is either impossible or inappropriate to measure all the elements of the state vector. The state observer has come to take its pride of place in system identification, filter theory, and control design. As we know, the tasks of observer-based control systems (with or without chaos) can be divided into two categories: tracking (or synchronization) and observer-based stabilization (or regulation). The state observer can be skillfully applied in observer-based stabilization, synchronization of master-slave chaotic systems, and secure communication. For more detailed knowledge, one can refer to [1, 2, 7–9, 11–14]. However, the state observer design of dynamic systems with chaos is in general not as easy as that without chaos. Motivated by the above reasons, the observer design of chaotic systems is actually crucial and meaningful. On the other hand, a variety of methods have been proposed for the observer design of systems, such as Chebyshev neural network (CNN), sliding-mode approach, passivation of error dynamics, separation principle, and frequency domain analysis; see, for instance, [15–20] and the references therein.
In this paper, the nonlinear state reconstructor of the generalized Rössler hyperchaotic systems is investigated. Using the DIL methodology, a nonlinear observer for such systems is provided to guarantee the global exponential stability of the resulting error system. Furthermore, the guaranteed exponential decay rate can be correctly estimated. Finally, numerical simulations are given to verify the effectiveness of proposed approach.
2. Problem Formulation and Main Result
In this paper, we consider the generalized Rössler hyperchaotic systems as follows:ẋ1(t)=α1x1+α2x2+g1(x3,x4),ẋ2(t)=α3x1+α4x2+g2(x4),ẋ3(t)=r1x1x3+g3(x4),ẋ4(t)=r2x3+g4(x4),y(t)=x4,
where x(t):=[x1(t)x2(t)x3(t)x4(t)]T∈ℜ4is the state vector, y(t)∈ℜis the system output, r1, r2, and αi,foralli∈{1,2,3,4} are the system parameters with r1r2≠0. For the existence and uniqueness of system (2.1), we assume that all the functions gi(·),foralli∈{1,2,3,4}, are sufficiently smooth.
The following assumption is made on system (2.1) throughout this paper.
There exists a constant h1 such that
h1>α4,h1α4<-α2α3.
Remark 2.1.
It is noted that the Rössler hyperchaotic system [21] is the special cases of system (2.1) with
α1=0,α2=-1,α3=1,α4=0.25,r1=1,r2=-0.5,g1(x3,x4)=-x3,g2(x4)=x4,g3(x4)=3,g4(x4)=0.05x4.
The objective of this paper is to search a nonlinear observer for system (2.1) such that the global exponential stability of the resulting error systems can be guaranteed. Before presenting the main result, let us introduce a definition which will be used in the main theorem.
Definition 2.2.
System (2.1) is exponentially state reconstructible if there exist an observer Ex̂̇(t)=g(x̂(t),y(t))and positive numbers k and α such that
‖e(t)‖:=‖x(t)-x̂(t)‖≤kexp(-αt),∀t≥0,
where x̂(t) expresses the reconstructed state of system (2.1). In this case, the positive number α is called the exponential decay rate.
Now we present the main result for the state observer of system (2.1).
Theorem 2.3.
System (2.1) with (A1) is exponentially state reconstructible. Besides, a suitable nonlinear observer is given by
r1x̂3x̂̇1=(α1+h1)[x̂̇3-g3(y)]-h1r1x̂1x̂3+r1x̂3[α2x̂2+g1(x̂3,y)],x̂̇2=α3x̂1+α4x̂2+g2(y),x̂3=1r2[ẏ-g4(y)],x̂4=y.
In this case, the guaranteed exponential decay rate is given by α:=1/λmax(P), where P>0 is the unique solution to the following Lyapunov equation:
[-h1α2α3α4]TP+P[-h1α2α3α4]=[-200-2].
Proof.
From (2.1), (2.5) with
ei(t):=xi(t)-x̂i(t),∀i∈{1,2,3,4},
it can be readily obtained that
e4(t)=x4(t)-x̂4(t)=0,∀t≥0,e3(t)=x3(t)-x̂3(t)=x3(t)-1r2[ẏ-g4(y)]=x3(t)-1r2[ẋ4-g4(x4)]=x3(t)-1r2[r2x3(t)+g4(x4)-g4(x4)]=0,∀t≥0,ė2(t)=ẋ2(t)-ẋ̂2(t)=α3x1+α4x2+g2(x4)-α3x̂1-α4x̂2-g2(x̂4)=α3(x1-x̂1)+α4(x2-x̂2)+[g2(x4)-g2(x̂4)]=α3e1+α4e2+[g2(y)-g2(y)]=α3e1(t)+α4e2(t),∀t≥0,ė1(t)=ẋ1(t)-ẋ̂1(t)=α1x1+α2x2+g1(x3,x4)-(α1+h1)[x̂̇3-g3(y)]r1x̂3+h1x̂1-α2x̂2-g1(x̂3,y)=α1x1+α2e2+g1(x3,x4)-(α1+h1)[ẋ3-g3(x4)]r1x3+h1x̂1-g1(x3,x4)=α1x1+α2e2-(α1+h1)[g3(x4)+r1x1x3-g3(x4)]r1x3+h1x̂1=α1x1+α2e2-(α1+h1)x1+h1x̂1=h1e1(t)+α2e2(t),∀t≥0.
This implies that
[ė1(t)ė2(t)]=[-h1α2α3α4][e1(t)e2(t)],[e3(t)e4(t)]=0,∀t≥0,
with σ([-h1α2α3α4])⊆C-, in view of (A1):
Let
W(t):=[e1(t)e2(t)]P[e1(t)e2(t)].
The time derivative of W(t) along the trajectories of dynamical error system, with (2.9), (2.10), and (2.6), is given by
Ẇ(t)=[ė1ė2]P[e1e2]+[e1e2]P[ė1ė2]=[e1e2][-h1α2α3α4]TP[e1e2]+[e1e2]P[-h1α2α3α4][e1e2]=[e1e2]{[-h1α2α3α4]TP+P[-h1α2α3α4]}[e1e2]=-2[e1e2][e1e2]≤-2λmax(P)W(t)=-2αW(t),∀t≥0.
Thus, one has
e2αt⋅Ẇ+e2αt⋅2αW=ddt[e2αt⋅W]≤0,∀t≥0.
It follows that
∫0tddτ[e2ατ⋅W(τ)]dτ=e2αt⋅W(t)-W(0)≤∫0t0dτ=0.
Consequently, we conclude that
‖e(t)‖=e12(t)+e22(t)+e32(t)+e42(t)=e12(t)+e22(t)≤W(t)λmin(P)≤e-2αtW(0)λmin(P)=W(0)λmin(P)⋅e-αt,∀t≥0,
in view of (2.8), (2.10), and (2.13). This completes the proof.
3. Numerical Simulations
Consider the generalized hyperchaotic system:
ẋ1(t)=-x2-x3,ẋ2(t)=ax1+bx2+x4,ẋ3(t)=x1x3+3,ẋ4(t)=-0.5x3+0.05x4,y(t)=x4.
Case 1 ((a=1,b=0.25) or, equivalently, the Rössler hyperchaotic system).
It can be verified that condition (A1) is satisfied with h1=1.2. By Theorem 2.3, we conclude that system (3.1) with a=1 and b=0.25 is exponentially state reconstructible by the nonlinear observer:
x̂3x̂̇1=1.2(x̂̇3-3)-1.2x̂1x̂3-x̂2x̂3-x̂32,x̂̇2=x̂1+0.25x̂2+y,x̂3=0.1y-2ẏ,x̂4=y,
with the guaranteed exponential decay rate α=0.164.
Case 2 (a=-20,b=-50).
It can be verified that condition (A1) is satisfied with h1=10. By Theorem 2.3, we conclude that system (3.1) with a=-20 and b=-50 is exponentially state reconstructible by the nonlinear observer:
x̂3x̂̇1=10(x̂̇3-3)-10x̂1x̂3-x̂2x̂3-x̂32,x̂̇2=-20x̂1-50x̂2+y,x̂3=0.1y-2ẏ,x̂4=y,
with the guaranteed exponential decay rate α=8.47.
Case 3 (a=30,b=-40).
It can be verified that condition (A1) is satisfied with h1=5. By Theorem 2.3, we conclude that system (3.1) with a=30 and b=-40 is exponentially state reconstructible by the nonlinear observer:
x̂3x̂̇1=5(x̂̇3-3)-5x̂1x̂3-x̂2x̂3-x̂32,x̂̇2=30x̂1-40x̂2+y,x̂3=0.1y-2ẏ,x̂4=y,
with the guaranteed exponential decay rate α=3.79.
The time response of error states for system (3.1) with Case 1–Case 3 is depicted in Figures 1, 2, and 3, respectively. From the foregoing simulations results, it is seen that system (3.1) with Case 1–Case 3, regardless of chaotic system or nonchaotic system, is exponentially state reconstructible by the nonlinear observers (3.2)–(3.4), respectively.
The time response of error states, with a=1 and b=0.25.
The time response of error states, with a=-20 and b=-50.
The time response of error states, with a=30 and b=-40.
4. Conclusion
In this paper, the generalized Rössler hyperchaotic systems have been presented, and the state observation problem of such systems has been investigated. Based on the DIL methodology, a nonlinear state reconstructor of the generalized Rössler hyperchaotic systems has been developed to guarantee the global exponential stability of the resulting error system. Besides, the guaranteed exponential decay rate can be accurately estimated. However, the state observation design for more general uncertain hyperchaotic system still remains unanswered. This constitutes an interesting future research problem.
Nomenclatureℜn:
The n-dimensional real space
C-:
The set of {a+bj∣a<0,b∈ℜ}
|a|:
The modulus of a real number a
∥x∥:
The Euclidean norm of the vector x∈ℜn
∥A∥:
The induced Euclidean norm of the matrix A
AT:
The transpose of the matrix A
σ(A):
The set of all eigenvalues of the matrix A
P>0:
The symmetric matrix P is positive definite
λmax(P):
The maximum eigenvalue of the symmetric matrix P with real eigenvalues
λmin(P):
The minimum eigenvalue of the symmetric matrix P with real eigenvalues.
Acknowledgments
The author thanks the National Science Council of Republic of China for supporting this work under Grant NSC-100-2221-E-214-015. The author also wishes to thank the anonymous reviewers for providing constructive suggestions.
BaekJ.LeeH.KimS.ParkM.Adaptive fuzzy bilinear observer based synchronization design for generalized Lorenz system200937347436843752-s2.0-7124910339610.1016/j.physleta.2009.09.064ZBL1234.34012GhoshD.SahaP.ChowdhuryA. R.Linear observer based projective synchronization in delay Rössler system2010156164016472-s2.0-7214908573910.1016/j.cnsns.2009.06.019ZBL1221.34138JovicB.UnsworthC. P.Fast synchronisation of chaotic maps for secure chaotic communications201046149502-s2.0-7604911995010.1049/el.2010.1532LamH. K.Chaotic synchronisation using output/full state-feedback polynomial controller20104112285229210.1049/iet-cta.2009.0328LiJ.LiW.LiQ.Sliding mode control for uncertain chaotic systems with input nonlinearity20121713413482-s2.0-7996107709510.1016/j.cnsns.2011.04.018LiY.Some new less conservative criteria for impulsive synchronization of a hyperchaotic Lorenz system based on small impulsive signals20101127137192-s2.0-7044963012910.1016/j.nonrwa.2009.01.017ZBL1180.37040LinJ. S.LiaoT. L.YanJ. J.YauH. T.Synchronization of unidirectional coupled chaotic systems with unknown channel time-delay: adaptive robust observer-based approach20052639719782-s2.0-1884438627710.1016/j.chaos.2005.02.005ZBL1093.93535SunY. J.A simple observer design of the generalized Lorenz chaotic systems201037479339372-s2.0-7374908598810.1016/j.physleta.2009.12.019TingC. S.An adaptive fuzzy observer-based approach for chaotic synchronization2005391971142-s2.0-1384429821310.1016/j.ijar.2004.10.011ZBL1107.93023YauH. T.LinJ. S.YanJ. J.Synchronization control for a class of chaotic systems with uncertainties2005157223522462-s2.0-2414449384510.1142/S0218127405013204ZBL1092.93594ZhuF.Observer-based synchronization of uncertain chaotic system and its application to secure communications2009405238423912-s2.0-6734916014510.1016/j.chaos.2007.10.052ZBL1198.94170ChenB.LamJ.WangZ.Observer design and stabilization for linear neutral delay systems200544135422-s2.0-12844285676LiZ. G.WenC. Y.SohY. C.Observer-based stabilization of switching linear systems20033935175242-s2.0-003736490610.1016/S0005-1098(02)00267-4ZBL1013.93045TorresL. A.Ibarra-JunqueraV.Escalante-MinakataP.RosuH. C.High-gain nonlinear observer for simple genetic regulation process20073801-22352402-s2.0-3424786094210.1016/j.physa.2007.02.105EngelR.KreisselmeierG.Nonlinear approximate observers for feedback control20075632302352-s2.0-3384631043910.1016/j.sysconle.2006.10.003ZBL1114.93036HajatipourM.FarrokhiM.Chattering free with noise reduction in sliding-mode observers using frequency domain analysis20102089129212-s2.0-7795551889010.1016/j.jprocont.2010.06.015ShaikF. A.PurwarS.PratapB.Real-time implementation of Chebyshev neural network observer for twin rotor control system2011381013043130492-s2.0-7995801237410.1016/j.eswa.2011.04.107ShimH.SeoJ. H.TeelA. R.Nonlinear observer design via passivation of error dynamics20033958858922-s2.0-003740148310.1016/S0005-1098(03)00023-2ZBL1103.93313TrinhH.AldeenM.NahavandiS.An observer design procedure for a class of nonlinear time-delay systems200430161712-s2.0-014188286510.1016/S0045-7906(03)00037-5ZBL1053.93011WangY.LynchA. F.Observer design using a generalized time-scaled block triangular observer form20095853463522-s2.0-6144911183010.1016/j.sysconle.2008.12.005ZBL1159.93327Al-Sawalha MossaM.NooraniM. S. M.Application of the differential transformation method for the solution of the hyperchaotic Rössler system2009144150915142-s2.0-5554914050910.1016/j.cnsns.2008.02.002