Composite Tracking Control for Generalized Practical Synchronization of Duffing-Holmes Systems with Parameter Mismatching, Unknown External Excitation, Plant Uncertainties, and Uncertain Deadzone Nonlinearities

Abstract

The generalized practical synchronization (GPS) of uncertain Duffing-Holmes chaotic systems with parameter mismatching, unknown external excitation, plant uncertainties, and uncertain deadzone nonlinearities is investigated. Based on the composite control approach, a tracking control is derived to realize the GPS for the uncertain Duffing-Holmes chaotic systems with parameter mismatching, unknown external excitation, plant uncertainties, and uncertain deadzone nonlinearities. Besides, the guaranteed exponential decay rate, convergence radius, and desired scaling factor can be prespecified. Finally, numerical simulations are provided to illustrate the feasibility and effectiveness of the proposed GPS scheme.

1. Introduction

Recently, synchronizations of various dynamic systems or chaotic systems have been intensively studied; see, for instance, [1–6] and the references therein. Occasionally, chaos in many systems is a source of the generation of oscillation. Chaos synchronizations exist in certain fields of application, such as, secure communication, ecological systems, and system identification.

It is well known that there inevitably exist nonlinearities in electric components, such as, operational amplifier, resistor, inductors, capacitor, and electromechanical actuator. Furthermore, the control schemes of controlled driver-response chaotic systems can be realized by various electric components. Input nonlinearities not only often appear in the controlled driver-response chaotic systems but also frequently cause undesirable behavior, such as, instabilities or spurious limit cycles. Over the past decades, researchers have been concerned with several input nonlinearities common in dynamical systems, such as, deadzones, saturation, hysteresis, relays, and others; see, for instance, [7–10]. Generally speaking, the synchronization of chaotic system with parameter mismatching, unknown external excitation, plant uncertainties, and uncertain deadzone nonlinearities is in general not as simple as that without any uncertainties and nonlinearities.

Over the past decades, Duffing systems have been received a great deal of interest due to theoretical interests and successful applications in numerous areas. In [11], the virtual stabilizability of Duffing-Holmes control systems has been studied and a tracking control has been proposed such that the states of Duffing-Holmes control system track the desired trajectories. In addition, a harmonic balance method in conjunction with the successive integration technique has been offered in [12] to solve the Duffing oscillator equation with damping and excitation. In this paper, the concept of the GPS for chaotic systems is presented and the GPS of uncertain Duffing-Holmes chaotic systems with parameter mismatching, unknown external excitation, plant uncertainties, and uncertain deadzone nonlinearities is explored. Using the composite control strategy, a tracking control is offered to realize the GPS for uncertain Duffing-Holmes chaotic systems, with any prespecified exponential decay rate, convergence radius, and desired scaling factor.

2. Problem Formulation and Main Result

Before presenting the problem formulation, let us introduce a definition as follows.

Definition 2.1. The deadzone nonlinearities 𝐷(𝑢,𝑑,𝑚), with 𝑑≥0 and 𝑚>0, is defined to be the collection of all functions Δ𝜙(𝑢)∶=ℜ→ℜ satisfying (𝑢)𝑚(Δ𝜙∶=𝑢−Δ𝑟),if𝑢≥Δ𝑟,0,if−Δ𝑟𝑚≤𝑢≤Δ𝑢+Δ𝑟if𝑢≤−Δ𝑟,𝑟, (2.1) for any Δ𝑟 and Δ𝑟 with 0≤Δ𝑟,Δ𝑟≤𝑑.

In this paper, we consider the following uncertain Duffing-Holmes chaotic systems with parameter mismatching, unknown external excitation, plant uncertainties, and uncertain deadzone nonlinearities described as

Driver system: 𝑟̈+𝑝₁𝑟+𝑝₂𝑟̇+𝑟³+𝑝₃cos(𝑤𝑡)+Δ𝑓₁()𝑟(0)(0)=𝑟𝑡,𝑟,𝑟̇=0,𝑡≥0,𝑟̇_𝑖1𝑟_𝑖2, (2.2)

Response system:𝑞̈+𝑝₁𝑞+𝑝₂𝑞̇+Δ𝑓₂(𝑢)𝑞(0)(0)=𝑞𝑡,𝑞,𝑞̇+Δ𝜙=0,𝑡≥0,𝑞̇_𝑖1𝑞_𝑖2, (2.3) where 𝑟,𝑞∈ℜ,Δ𝑓₁andΔ𝑓₂, for all 𝑖∈{1,2}, represent the mixed uncertainties (parameter mismatching, unknown external excitation, and plant uncertainties), [𝑟_𝑖1𝑟_𝑖2] and [𝑞_𝑖1𝑞_𝑖2] represent the initial conditions, 𝑢∈ℜ is the control input, and Δ𝜙(𝑢)∈𝐷(𝑢,𝑑,𝑚) is the uncertain deadzone nonlinearity. For the existence of the solutions of (2.2), we assume that the uncertainties Δ𝑓₁ and Δ𝑓₂ are continuous. It is noted that the system (2.2) without any uncertainties (i.e., Δ𝑓₁=Δ𝑓₂=0) displays chaotic behavior for certain values of the parameters [13]. Letting 𝑥(𝑡)∶=𝑥₁(𝑡)𝑥₂(𝑡)^𝑇∶=𝑟(𝑡)𝑟̇(𝑡)^𝑇,𝑧(𝑡)∶=𝑧₁(𝑡)𝑧₂(𝑡)^𝑇∶=𝑞(𝑡)𝑞̇(𝑡)^𝑇, (2.4)the corresponding state-space equations of system (2.2) and system (2.3) are 𝑥̇₁=𝑥₂,𝑥̇₂=−𝑝₁𝑥₁−𝑝₂𝑥₂−𝑥³₁−𝑝₃cos(𝑤𝑡)−Δ𝑓₁𝑡,𝑥₁,𝑥₂𝑧̇,∀𝑡≥0,₁=𝑧₂,𝑧₂̇=−𝑝₁𝑧₁−𝑝₂𝑧₂−Δ𝑓₂𝑡,𝑧₁,𝑧₂−Δ𝜙(𝑢),∀𝑡≥0. (2.5)(2.6)

Throughout this paper, an assumption is made as follows.(A1) There exist two continuous functions 𝑔₁(𝑡,𝑟,𝑟)̇≥0 and 𝑔₂(𝑡,𝑞,𝑞)̇≥0 such that, for all arguments,Δ𝑓₁(𝑡,𝑟,𝑟)̇≤𝑔₁(𝑡,𝑟,𝑟)̇,Δ𝑓₂𝑡,𝑞,𝑞̇≤𝑔₂𝑡,𝑞,𝑞̇. (2.7) For brevity, let us define the synchronous error vector as 𝑒(𝑡)∶=𝑒₁(𝑡)𝑒₂(𝑡)^𝑇∶=𝑧(𝑡)−𝛽𝑥(𝑡), (2.8) where 𝛽∈ℜ is the scaling factor.

The purpose of this paper is to search a composite control 𝑢=𝑢₁+𝑢₂ such that the states 𝑧₁ and 𝑧₂ of the response system (2.6) track, respectively, the signals 𝛽𝑥₁ and 𝛽𝑥₂ of the driver system (2.5), with any desired scaling factor𝛽.

The precise definition of the GPS, in terms of error response, is given as follows.

Definition 2.2. The driver-response dynamical systems (2.5) and (2.6) are said to realize the GPS, provided that, for any 𝛼>0, 𝜀>0, and 𝛽∈ℜ, there exists a tracking control 𝑢(𝑡)∶=𝑢(𝛼,𝜀,𝛽) and positive number 𝑘 such that the synchronous error satisfies ‖𝑒(𝑡)‖≤𝑘⋅𝑒^−𝛼𝑡+𝜀,∀𝑡≥0. (2.9)

In this case, the positive number 𝜀 is called the convergence radius, the positive number 𝛼 is called the exponential decay rate, and 𝛽 is called the desired scaling factor. In other words, the GPS means that the states 𝑧₁ and 𝑧₂ of the response system can track, respectively, the signals 𝛽𝑥₁ and 𝛽𝑥₂ of the driver system, with any prespecified exponential decay rate, convergence radius, and desired scaling factor.

From (2.5)–(2.8), we have the following error dynamical system: 𝑒̇=01−𝑝₁−𝑝₂0𝑒+𝐵−Δ𝜙−Δ𝑓₂+𝛽𝑥³₁+𝛽𝑝₃cos(𝑤𝑡)+𝛽Δ𝑓₁=01−𝑝₁−𝑝₂0𝑒−𝑚𝐵𝑢+𝐵Δ𝑓,∀𝑡≥0, (2.10)with 𝐵∶=⁰₁ and Δ𝑓∶=[−Δ𝜙(𝑢)+𝑚𝑢]+𝛽𝑥³₁+𝛽𝑝₃cos(𝑤𝑡)+𝛽Δ𝑓₁−Δ𝑓₂. First consider the case of the system (2.10) without any parameter mismatching, unknown external excitation, and plant uncertainties, that is, Δ𝑓=0. Thus the nominal system of (2.10) can represented as 𝑒̇=01−𝑝₁−𝑝₂𝑒−𝑚𝐵𝑢. (2.11)According to the linear system theory, it can be easily obtained that given any positive constant𝛼, the system (2.11) subjected to the linear control law:𝑢(𝑡)=𝐾𝑒(𝑥)=(𝛼+1)²−𝑝₁𝑚2(𝛼+1)−𝑝₂𝑚𝑒(𝑡), (2.12)is globally exponentially stable with guaranteed exponential decay rate𝛼. Nevertheless, the nominal control law (2.12) may not suffice to render the uncertain error system (2.10) to be globally exponentially stable. Hence a corrective control term must be added to overcome the uncertain part of (2.10). In the following, a composite control, consisting of a nominal control and a corrective control, is proposed such that the GPS between systems (2.5) and (2.6) can be guaranteed.

Now we present the main result for the GPS between system (2.5) and system (2.6).

Theorem 2.3. The GPS between system (2.5) and system (2.6) can be achieved under the composite control 𝑢(𝑡)∶=𝑢(𝛼,𝜀,𝛽) defined by 𝑢(𝑡)=𝑢₁(𝑡)+𝑢₂(𝑡), (2.13) where 𝑢₁(𝑡)=𝐾𝑒(𝑥)=(𝛼+1)²−𝑝₁𝑚2(𝛼+1)−𝑝₂𝑚𝑒(𝑡), (2.14)𝑢₂(𝑡)=ℎ²(𝑡)𝑚⋅ℎ(𝑡)⋅𝑝₂₁𝑒₁+𝑝₂₂𝑒₂+𝛼⋅𝑚⋅𝜀²⋅𝜆_min(𝑃)𝑝₂₁𝑒₁+𝑝₂₂𝑒₂, (2.15) where ℎ(𝑡)=𝑚𝑑+𝛽⋅𝑔₁+𝑔₂+𝛽𝑥³₁+𝛽𝑝₃, (2.16) and 𝑃∶=⌊𝑝_𝑖𝑗⌋_2×2>0 is the unique solution to the following Lyapunov equation: (𝐴+𝛼𝐼)^𝑇𝑃+𝑃(𝐴+𝛼𝐼)=−2𝐼, (2.17) with 𝐴∶=⁰¹_−(𝛼+1)²_−2(𝛼+1).

Proof. From (2.10), (2.13), and (2.14), the dynamical error system can be performed 𝑒̇=01−𝑝₁−𝑝₂𝑒−𝑚𝐵𝑢₁+𝑢₂0+𝐵Δ𝑓=𝐴𝑒−𝑚𝐵𝑢₂0+𝐵Δ𝑓,∀𝑡≥0, (2.18) with 𝐵∶=⁰₁ and Δ𝑓∶=[𝑚𝑢−Δ𝜙(𝑢)]+𝛽𝑥³₁+𝛽𝑝₃cos(𝑤𝑡)+𝛽Δ𝑓₁−Δ𝑓₂.
Obviously, one has 𝜎(𝐴)={−𝛼−1}, 𝜎(𝐴+𝛼𝐼)={−1}, and |Δ𝑓|≤ℎ(𝑡). Let 𝑊(𝑒(𝑡))=𝑒^𝑇(𝑡)𝑃𝑒(𝑡). (2.19)
The time derivative of 𝑊(𝑒(𝑡)) along the trajectories of dynamical error system, with (2.11)–(2.19) and (A1), is given by 𝑊̇(𝑒(𝑡))=𝑒^𝑇̇𝑃𝑒+𝑒^𝑇=𝑃𝑒̇𝐴𝑒+𝐵−𝑚𝑢₂+Δ𝑓^𝑇𝑃𝑒+𝑒^𝑇𝑃𝐴𝑒+𝐵−𝑚𝑢₂+Δ𝑓=𝑒^𝑇𝐴^𝑇𝑃+𝑃𝐴𝑒−2𝑚⋅𝑒^𝑇𝑃𝐵𝑢₂+2𝑒^𝑇𝑃𝐵⋅Δ𝑓=𝑒^𝑇(−2𝛼𝑃−2𝐼)𝑒−2𝑚⋅𝑒^𝑇𝑃𝐵𝑢₂+2𝑒^𝑇𝑃𝐵⋅Δ𝑓≤−2𝛼𝑊−2𝑚⋅𝑒^𝑇𝑃𝐵𝑢₂+2𝑒^𝑇⋅ℎ𝑃𝐵Δ𝑓≤−2𝛼𝑊−2²(𝑡)ℎ⋅𝐵^𝑇𝑃𝑒(𝑡)+𝛼⋅𝜀²⋅𝜆_min(𝑃)𝑒^𝑇𝑃𝐵²+2𝑒^𝑇𝑃𝐵⋅ℎ=−2𝛼𝑊+2𝛼⋅𝜀²⋅𝜆_min(𝑃)⋅ℎ⋅𝑒^𝑇𝑃𝐵ℎ⋅𝐵^𝑇𝑃𝑒(𝑡)+𝛼⋅𝜀²⋅𝜆_min(𝑃)=−2𝛼𝑊+2𝛼⋅𝜀²⋅𝜆_min(𝑃),∀𝑡≥0, (2.20) in view of |Δ𝑓(𝑡)|≤ℎ(𝑡) and 𝑝₂₁𝑒₁+𝑝₂₂𝑒₂=𝐵^𝑇𝑃𝑒=𝑒^𝑇𝑃𝐵. Thus, one has 𝑒^2𝛼1𝑡⋅𝑊̇+𝑒^2𝛼𝑡𝑑⋅2𝛼𝑊=𝑒𝑑𝑡^2𝛼𝑡⋅𝑊≤𝑒^2𝛼𝑡⋅2𝛼⋅𝜀²⋅𝜆_min(𝑃),∀𝑡≥0 (2.21) It follows that ^𝑡₀𝑑𝑒𝑑𝑡^2𝛼𝑡⋅𝑊(𝑒(𝑡))𝑑𝑡=𝑒^2𝛼𝑡(𝑒(𝑡(𝑒(0≤⋅𝑊))−𝑊))_𝑡0𝑒^2𝛼𝑡⋅2𝛼⋅𝜀²⋅𝜆_min(𝑃)𝑑𝑡=𝜀²⋅𝜆_min(𝑃)𝑒^2𝛼𝑡−1,∀𝑡≥0. (2.22) This implies 𝑊(𝑒(𝑡))≤𝑊(𝑒(0))−𝜀²⋅𝜆_min(𝑃)⋅𝑒^−2𝛼𝑡+𝜀²⋅𝜆_min(𝑃),∀𝑡≥0. (2.23) Thus, from (2.19) and (2.23), it can be readily obtained that 𝜆_min(𝑃)‖𝑒(𝑡)‖²≤𝑊(𝑒(𝑡))≤𝑒^−2𝛼𝑡𝑊(𝑒(0))−𝜀²𝜆_min(𝑃)+𝜀²𝜆_min(𝑃),∀𝑡≥0. (2.24) Consequently, we conclude that ‖𝑒(𝑡)‖≤𝑒^−2𝛼𝑡𝑊(𝑒(0))+𝜀²𝜆_min(𝑃)𝜆_min(𝑃)+𝜀²≤𝑒^−2𝛼𝑡𝑊(𝑒(0))+𝜀²𝜆_min(𝑃)𝜆_min(𝑃)+√𝜀²=𝑊(𝑒(0))+𝜀²𝜆_min(𝑃)𝜆_min(𝑃)⋅𝑒^−𝛼𝑡+𝜀,∀𝑡≥0. (2.25)This completes the proof.

Remark 2.4. In what follows, we present an algorithm to find a tracking control law of (2.13) stated in Theorem 2.3.
INPUT: the driver-response systems (2.5) and (2.6), the prespecified exponential decay rate𝛼>0, and the prespecified convergence radius 𝜀>0.OUPUT: tracking control of (2.13).Step 1. Choose 𝑔₁(𝑡,𝑟,𝑟)̇ and 𝑔₂(𝑡,𝑞,𝑞)̇ such that (A1) is satisfied.Step 2. Calculate 𝑚 and 𝑑, from Definition 2.1.Step 3. Calculate 𝑃, 𝜆_min(𝑃), 𝑝₂₁, and 𝑝₂₂, from (2.17).Step 4. Form 𝑒₁ and 𝑒₂ from (2.8).Step 5. Form 𝑢₁ from (2.14).Step 6. Form ℎ(𝑡) from (2.16).Step 7. Form 𝑢₂ from (2.15).Step 8. OUPUT 𝑢(𝑡)=𝑢₁(𝑡)+𝑢₂(𝑡).

3. Illustrative Example

In what follows, we provide an example to illustrate the main result.

Driver system: 𝑥̇₁=𝑥₂,𝑥̇₂=−𝑝₁𝑥₁−𝑝₂𝑥₂−𝑥³₁−𝑝₃cos(𝑤𝑡)−Δ𝑎(𝑡)𝑥₁𝑥₂,∀𝑡≥0. (3.1)

Response system: 𝑧₁̇=𝑧₂,𝑧₂̇=−𝑝₁𝑧₁−𝑝₂𝑧₂−Δ𝑏(𝑡)sin(𝑡)−Δ𝜙(𝑢),∀𝑡≥0, (3.2) where𝑝₁=−1,𝑝₂=0.25𝑝₃(𝑡)(𝑡)(𝑢)().=0.3,𝑤=1,−0.1≤Δ𝑎≤0.1,−0.1≤Δ𝑏≤0.1,Δ𝜙∈𝐷𝑢,1,2 (3.3)Our objective, in this example, is to design a tracking control such that the systems (3.1) and (3.2) can realize the GPS, with the exponential decay rate 𝛼=2 and convergence radius 𝜀=0.1. Comparison of (3.1)-(3.2) with (2.5)-(2.6), it can be obtained that 𝑑=1,𝑚=2,Δ𝑓₁𝑡,𝑥₁,𝑥₂=Δ𝑎(𝑡)⋅𝑥₁𝑥₂,Δ𝑓₂𝑡,𝑧₁,𝑧₂=Δ𝑏(𝑡)sin(𝑡). (3.4)The condition (A1) is evidently satisfied if we let𝑔₁𝑡,𝑥₁,𝑥₂=0.1⋅𝑥₁𝑥₂,𝑔₂𝑡,𝑧₁,𝑧₂=0.1⋅|sin(𝑡)|. (3.5)

Solving the Lyapunov equation (2.17) yields𝑃=4911113,𝜆_min(𝑃)𝑝=0.5049,₂₁=11,𝑝₂₂=3. (3.6)

Finally, owing to (2.13)–(2.16), we obtain the design controller 𝑢(𝑡)=5𝑒₁+2.875𝑒₂+ℎ²(𝑡)2⋅ℎ(𝑡)⋅11𝑒₁+3𝑒₂+0.0211𝑒₁+3𝑒₂, (3.7)withℎ(𝑡)=2+0.1⋅𝛽⋅𝑥₁𝑥₂+|sin(𝑡)|+𝛽𝑥³₁+0.3𝛽. (3.8)Consequently, by Theorem 2.3, we conclude that the GPS between the driver system (3.1) and the response system (3.2) can be achieved under the control law (3.7), with the guaranteed exponential decay rate 𝛼=2 and convergence radius 𝜀=0.1.

With, for instance,Δ𝑎(𝑡)=Δ𝑏(𝑡)=0.1, (3.9)the typical state trajectories of uncertain Duffing-Holmes chaotic system (3.1) are depicted in Figure 1. In addition, the synchronization errors of systems (3.1)–(3.7) with 𝛽=1,−1,5, and 0, are depicted in Figures 2, 3, 4 and 5, respectively. From the foregoing simulations results, it is seen that the controlled drive-response systems (3.1) and (3.2) achieve the GPS under the control law (3.7).

Figure 1 

Typical state trajectories of the uncertain Duffing-Holmes system for (3.1).

Figure 2 

Synchronization errors of systems (3.1)–(3.7) with 𝛽=1.

Figure 3 

Synchronization errors of systems (3.1)–(3.7) with 𝛽=−1.

Figure 4 

Synchronization errors of systems (3.1)–(3.7) with 𝛽=5.

Figure 5 

Synchronization errors of systems (3.1)–(3.7) with 𝛽=0.

4. Conclusion

In this paper, the GPS of uncertain Duffing-Holmes chaotic systems with parameter mismatching, unknown external excitation, plant uncertainties, and uncertain deadzone nonlinearities has been investigated. Based on the composite control methodology, a tracking control has been proposed to realize the GPS for such uncertain Duffing-Holmes chaotic systems, with prespecified exponential decay rate, convergence radius, and desired scaling factor. Finally, numerical simulations have also been given to verify the feasibility and effectiveness of the proposed GPS scheme.

Nomenclature

ℜ𝑛:	The 𝑛-dimensional real space
|𝑎|:	The modulus of a real number 𝑎
‖𝑥‖:	The Euclidean norm of the vector𝑥∈ℜ𝑛
𝐼:	The unit matrix
𝐴𝑇:	The transport of the matrix 𝐴
𝜆min(𝑃):	The minimum eigenvalue of the matrix 𝑃 with real eigenvalues
𝜎(𝐴):	The spectrum of the matrix 𝐴
𝑃>0:	The matrix 𝑃 is a symmetric positive definite matrix.

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 I-Shou University for supporting this work under Grant ISU100-01-01.