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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 529251, 9 pages
http://dx.doi.org/10.1155/2012/529251
Research Article

Adaptive Control and Synchronization of the Shallow Water Model

P. Sangapate

Department of Mathematics, Faculty of Science, Maejo University, Chiang Mai 50290, Thailand

Received 6 July 2011; Revised 24 September 2011; Accepted 26 September 2011

Academic Editor: Carlo Cattani

Copyright © 2012 P. Sangapate. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The shallow water model is one of the important models in dynamical systems. This paper investigates the adaptive chaos control and synchronization of the shallow water model. First, adaptive control laws are designed to stabilize the shallow water model. Then adaptive control laws are derived to chaos synchronization of the shallow water model. The sufficient conditions for the adaptive control and synchronization have been analyzed theoretically, and the results are proved using a Barbalat's Lemma.

1. Introduction

A dynamical system is a system that changes over time. Chaotic systems are dynamical systems that are highly sensitive to initial conditions. Chaos phenomena in weather models were first observed by Lorenz equation; a large number of chaos phenomena and chaos behavior have been discovered in physical, social, economical, biological, and electrical systems.

Atmosphere is a dynamical system. An atmospheric model is a set of equations that describes behavior of the atmosphere. The shallow water model is simple model for the atmosphere. Shallow water model is the set of the equations of motion that describes the evolution of a horizontal structure, hydrostatic homogeneous, and incompressible flow on the sphere [1].

The control of chaotic systems is to design state feedback control laws that stabilize the chaotic systems. Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The usual objective of control theory is to calculate solutions for the proper corrective action from the controller that result in system stability.

Synchronization of chaotic systems is phenomena that may occur when two or more chaotic oscillators are coupled or when a chaotic oscillator drives another chaotic oscillator, because of the butterfly effect, which causes the exponential divergence of the trajectories of two identical chaotic systems started with nearby the same initial conditions. Synchronizing two chaotic systems is seemingly a very challenging problem in chaos literature [2–6].

In 1990, Pecora and Caroll [7] introduced a method to synchronize two identical chaotic systems and showed that it was possible for some chaotic systems to be completely synchronized. From then on, chaos synchronization has been widely explored in variety of fields including physical system [8], chemical systems [9], ecological systems [10], secure communications [11, 12], and so forth.

In most of the chaos synchronization approaches, the drive-response formalism has been used. If a particular chaotic system is called the drive system and another chaotic system is called the response system, then the idea of synchronization is to use the output of the drive system to control the response system so that the output of the response system tracks the output of drive system asymptotically stable.

This paper is organized as follows. Section 2 gives notations and definitions of the stability in the chaotic system. Section 3 presents the adaptive control chaos of the shallow water model. Section 4 presents adaptive synchronization of the shallow water model. The conclusion discussion is in Section 5.

2. Notations and Definitions

𝑋 denotes an infinite dimensional Banach Space with the corresponding norm ‖‖, 𝑅 denotes the real line.

Consider a nonlinear nonautonomous differential equation of the general form ̇𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑡≥𝑡0𝑥𝑡∈𝑅,0=𝑥0,(2.1) where the state 𝑥(𝑡) take values in 𝑋, 𝑓(𝑡,𝑥)∶𝑅×𝑋→𝑋 is a given nonlinear function and 𝑓(𝑡,0)=0, for all 𝑡∈𝑅. The stability conditions were proposed and presented in [13].

Definition 2.1. The zero solution of (2.1) is said to be stable if for every 𝜀>0,𝑡0∈𝑅, there exists a number 𝛿>0 (depending upon𝜀 and𝑡0) such that for any solution 𝑥(𝑡) of (2.1) with ‖𝑥0‖<𝛿 implies ‖𝑥(𝑡)‖<𝜀, for all𝑡≥𝑡0.

Definition 2.2. The zero solution of (2.1) is said to be asymptotically stable if it is stable and there is a number 𝛿>0 such that any solution 𝑥(𝑡) with ‖𝑥0‖<𝛿 satisfies lim𝑡→∞‖𝑥(𝑡)‖=0.
Consider the control system ̇𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡),𝑢(𝑡)),𝑡≥0,(2.2) where 𝑢(𝑡) is the external control input. The adaptive control is the control method to design state feedback control laws that stabilize the chaotic systems.

Definition 2.3. The control system (2.2) is stabilizable if there exists feedback control 𝑢(𝑡)=𝑘(𝑥(𝑡)) such that the system ̇𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡),𝑘(𝑥(𝑡))),𝑡≥0,(2.3) is asymptotically stable.
Consider two nonlinear systems ̇𝑥=𝑓(𝑡,𝑥(𝑡)),(2.4)̇𝑦=𝑔(𝑡,𝑦(𝑡))+𝑢(𝑡,𝑥(𝑡),𝑦(𝑡)),(2.5) where 𝑥,𝑦∈𝑅,𝑓,𝑔∈𝐶𝑟[𝑅×𝑅,𝑅],𝑢∈𝐶𝑟[𝑅×𝑅×𝑅,𝑅],𝑟≥1,𝑅 is the set  of nonnegative real number. Assume that (2.4) is the drive system, (2.5) is the response system, and 𝑢(𝑡,𝑥(𝑡),𝑦(𝑡))is the control vector.

Definition 2.4. Response system and drive system are said to be synchronic if for any initial conditions𝑥(𝑡0),𝑦(𝑡0)∈𝑅,  lim𝑡→∞‖𝑥(𝑡)−𝑦(𝑡)‖=0.

Lemma 2.5 (Barbalat’s lemma as used in stability). For nonautonomous system, ̇𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡))(2.6) If there exists a scalar function 𝑉(𝑥,𝑡) such that (1)𝑉 has a lower bound,(2)̇𝑉≤0,(3)̇𝑉(𝑥,𝑡) is uniformly continuous in time,then lim𝑡→∞̇𝑉(𝑥,𝑡)=0 by applying the Barbalat’s Lemma to stabilize the chaotic systems.

3. Adaptive Control Chaos of the Shallow Water Model

A chaotic system has complex dynamical behaviors; those posses some special features, such as being extremely sensitive to tiny variations of initial conditions. In this section, adaptive control method is applied to control chaos shallow water model.

Shallow water model is the set of the equations of motion that describes the evolution of a horizontal structure, hydrostatic homogeneous, and incompressible flow on the sphere. Euler’s equations of motion of an ideal fluid are as follows: 𝐷𝑢1𝐷𝑡=−𝜌𝜕𝑝𝜕𝑥+𝑓𝑣,𝐷𝑣1𝐷𝑡=−𝜌𝜕𝑝𝜕𝑦−𝑓𝑢,𝐷𝑤1𝐷𝑡=−𝜌𝜕𝑝𝜕𝑧−𝑔,(3.1) where 𝜌 is the density of the fluid, 𝑝 is the pressure, 𝑔 is the gravity, and 𝑓 is coliolis parameter. Using the hydrostatic approximation, 𝜕𝑝𝜕𝑧=−𝜌𝑔.(3.2) This implies𝐷𝑤/𝐷𝑡=0. Assume the pressure 𝑝 of fluid is constant, this implies that 𝜕𝑝/𝜕𝑡=0 and consider the continuity equation (or the incompressibility condition), 𝜕𝑢+𝜕𝑥𝜕𝑣+𝜕𝑦𝜕𝑤𝜕𝑧=0.(3.3) By solving for 𝜕𝑤/𝜕𝑧 and integrating with respect to 𝑧, then 𝑤 can be expressed as 𝜕𝑤𝜕𝑧=−𝜕𝑢+𝜕𝑥𝜕𝑣,𝜕𝑦𝑤=ℎ0−𝜕𝑢+𝜕𝑥𝜕𝑣𝜕𝑦𝑑𝑧=−ℎ𝜕𝑢+𝜕𝑥𝜕𝑣.𝜕𝑦(3.4) The surface (of the fluid) boundary condition on 𝑤 is that the fluid particles follow the surface(i.e.,𝐷ℎ/𝐷𝑡=𝑤|surface). Thus 𝐷ℎ𝐷𝑡=−ℎ𝜕𝑢+𝜕𝑥𝜕𝑣𝜕𝑦.(3.5) To get an expression for the pressure in the fluid, integrate the hydrostatic equation (3.2) from 𝑝=0 at the top downward, 𝑝(𝑥,𝑦,𝑧)=𝑧ℎ−𝑔𝜌𝑑𝑧=(ℎ−𝑧)𝜌𝑔.(3.6) Take the partial derivatives of 𝑝 (at the surface) with respect to 𝑥 and 𝑦, 𝜕𝑝=𝜕𝜕𝑥𝜕𝑥((ℎ−𝑧)𝜌𝑔)=𝜌𝑔𝜕ℎ1𝜕𝑥⟹−𝜌𝜕𝑝𝜕𝑥=−𝑔𝜕ℎ,𝜕𝑥𝜕𝑝=𝜕𝜕𝑦𝜕𝑦((ℎ−𝑧)𝜌𝑔)=𝜌𝑔𝜕ℎ1𝜕𝑦⟹−𝜌𝜕𝑝𝜕𝑦=−𝑔𝜕ℎ.𝜕𝑦(3.7) Taking (3.2)–(3.7) into (3.1), so the shallow water model in Cartesian coordinates is as follows: 𝐷𝑢𝐷𝑡=−𝑔𝜕ℎ𝜕𝑥+𝑓𝑣,𝐷𝑣𝐷𝑡=−𝑔𝜕ℎ𝜕𝑦−𝑓𝑢,𝐷𝑤𝐷𝑡=−ℎ𝜕𝑢+𝜕𝑥𝜕𝑣.𝜕𝑦(3.8) In the vector form, the shallow water model is as follows: ̇̇𝑉=−𝑓𝑘×𝑉−∇Φ,Φ=−Φ∇𝑉,(3.9) where 𝐕=𝑢⇀𝑖+𝑣⇀𝑗 is the horizontal velocity, Φ=𝑔ℎ is the geopotential height.

Consider the controlled system of (3.9) which has the forṁ𝑉=−𝑓𝑘×𝑉−∇Φ+𝑢1,̇Φ=−Φ∇𝑉+𝑢2,(3.10) where 𝑢1,𝑢2 is external control input which will drag the chaotic trajectory (𝑉,Φ) of the shallow water model to equilibrium point 𝐸=(𝑉,Φ) which is one of two steady states 𝐸0,𝐸1.

In this case the control law is 𝑢1=−𝑔𝑉−𝑉,𝑢2=−𝑘Φ−Φ,(3.11) where 𝑘,𝑔 (estimate of 𝑘∗, 𝑔∗, resp.) are updated according to the following adaptive algorithm: ̇𝑔=𝜇𝑉−𝑉2,̇𝑘=𝜌Φ−Φ2,(3.12) where 𝜇,𝜌 is adaption gains. Then the controlled systems have the following form: ̇𝑉=−𝑓𝑘×𝑉−∇Φ−𝑔𝑉−𝑉̇,(3.13)Φ=−Φ∇𝑉−𝑘Φ−Φ.(3.14)

Theorem 3.1. For 𝑔<𝑔∗,𝑘<𝑘∗, the equilibrium point 𝐸=(𝑉,Φ) of the system (3.13), (3.14) is asymptotically stable.

Proof. Let us consider the Lyapunov function 𝑉𝜉1,𝜉2,𝜉3=12𝑉−𝑉2+Φ−Φ2+1𝜇𝑔−𝑔∗2+1𝜌𝑘−𝑘∗2.(3.15) The time derivative of 𝑉 is ̇𝑉=𝑉−𝑉̇𝑉+Φ−Φ̇1Φ+𝜇𝑔−𝑔∗1̇𝑔+𝜌𝑘−𝑘∗̇𝑘.(3.16) By substituting (3.13)-(3.14) in (3.16), ̇𝑉=𝑉−𝑉−𝑓𝑘×𝑉−∇Φ−𝑔𝑉−𝑉+Φ−Φ−Φ∇𝑉−𝑘Φ−Φ+1𝜇𝑔−𝑔∗𝜇𝑉−𝑉2+1𝜌𝑘−𝑘∗𝜌Φ−Φ2.(3.17) Let 𝜂1=(𝑉−𝑉),𝜂2=(Φ−Φ). Since (𝑉,Φ) is an equilibrium point of the uncontrolled system (3.9), ̇𝑉 becomes ̇𝑉=𝜂1−𝑓𝑘×𝑉−∇Φ−𝑔𝑉−𝑉+𝜂2−Φ∇𝑉−𝑘Φ−Φ+𝑔−𝑔∗𝜂21+𝑘−𝑘∗𝜂22=(−𝑓𝑘×𝑉)𝜂1−∇Φ𝜂1−𝑔𝜂21−Φ∇V𝜂2−𝑘𝜂22+𝑔−𝑔∗𝜂21+𝑘−𝑘∗𝜂22.(3.18) It is clear that if we choose 𝑔<𝑔∗ and 𝑘<𝑘∗, then ̇𝑉 is negative semidefinite. Since 𝑉 is positive definite and ̇𝑉 is negative semidefinite, 𝜂1,𝜂2,𝑔,𝑘∈𝐿∞. From ̇𝑉(𝑡)≤0, we can easily show that the square of 𝜂1,𝜂2 is integrable with respect to 𝑡, namely, 𝜂1,𝜂2∈𝐿2. From (3.13)-(3.14), for any initial conditions, we have ̇𝜂1,̇𝜂2∈𝐿∞. By the well-known Barbalat’s Lemma, we conclude that 𝜂1,𝜂2→(0,0) as𝑡→∞. Therefore, the equilibrium point 𝐸=(𝑉,Φ) of the system (3.13)-(3.14) is asymptotically stable.

4. Adaptive Synchronization of the Shallow Water Model

In this section, the adaptive synchronization is introduced to make two of the shallow water model. The sufficient condition for the synchronization has been analyzed theoretically, and the result is proved using a Barbalat’s Lemma. Assume that there are two shallow water models such that the drive system is to control the response system. The drive and response system are given as ̇𝑉=−𝑓1𝑘1×𝑉1−∇Φ1,̇Φ=−Φ1∇𝑉1,̇𝑉=−𝑓2𝑘2×𝑉2−∇Φ2−𝑢1,̇Φ=−Φ2∇𝑉2−𝑢2(4.1) where 𝑢=[𝑢1,𝑢2]𝑇 is the controller. We choose 𝑢1=𝑘′1𝑒𝑉,𝑢2=𝑘2𝑒Φ,(4.2) where 𝑘1,𝑘2≥0 and 𝑒𝑉,𝑒Φare the error states which are defined as follows 𝑒𝑉=𝑉2−𝑉1,𝑒Φ=Φ2−Φ1.(4.3)

Theorem 4.1. Let 𝑘1,𝑓1,𝑘1,𝑘2≥0 be property chosen so that the following matrix inequalities holds: ⎛⎜⎜⎝𝑘𝑃=1𝑓1+𝑘100𝑘2⎞⎟⎟⎠>0,(4.4) then the two shallow water models (4.1) can be synchronized under the adaptive control (4.2).

Proof. It is easy to see from (4.1) that the error system is ̇𝑒𝑉=−𝑓2𝑘2×𝑉2−∇Φ2+𝑓1𝑘1×𝑉1+∇Φ1−𝑢1,̇𝑒Φ=−Φ2∇𝑉2+Φ1∇𝑉1−𝑢2.(4.5) Let 𝑒𝑘𝑓=𝑘2𝑓2−𝑘1𝑓1. Choose the Lyapunov function as follows: 1𝑉(𝑡)=2𝑒2𝑉+𝑒2Φ.(4.6) Then the differentiation of 𝑉 along trajectories of (4.6) is ̇𝑉(𝑡)=𝑒𝑉̇𝑒𝑉+𝑒Φ̇𝑒Φ=𝑒𝑉−𝑓2𝑘2×𝑉2−∇Φ2+𝑓1𝑘1×𝑉1+∇Φ1−𝑢1+𝑒Φ−Φ2∇𝑉2+Φ1∇𝑉1−𝑢2=−𝑒𝑉𝑓2𝑘2×𝑉2+∇Φ2−𝑓1𝑘1×𝑉1−∇Φ1+𝑢1−𝑒ΦΦ2∇𝑉2−Φ1∇𝑉1+𝑢2=−𝑒𝑉𝑓2𝑘2×𝑉2−𝑓1𝑘1×𝑉1+𝑓1𝑘1×𝑉2−𝑓1𝑘1×𝑉2−𝑒𝑉∇Φ2−∇Φ1−𝑒𝑉𝑢1−𝑒ΦΦ2∇𝑉2−Φ1∇𝑉1+Φ1∇𝑉2−Φ1∇𝑉2−𝑒Φ𝑢2=−𝑒𝑉𝑒𝑘𝑓×𝑉2+𝑓1𝑘1𝑉2−𝑉1−𝑒𝑉∇Φ2−Φ1−𝑒𝑉𝑘1𝑒𝑉−𝑒ΦΦ2−Φ1∇𝑉2+Φ1∇𝑉2−𝑉1−𝑒Φ𝑘2𝑒Φ=−𝑒𝑉𝑒𝑘𝑓×𝑉2+𝑓1𝑘1𝑒𝑉−𝑒𝑉∇𝑒Φ−𝑒2𝑉𝑘1−𝑒Φ𝑒Φ∇𝑉2+Φ1∇𝑒𝑉−𝑒2Φ𝑘2=−𝑒𝑉𝑒𝑘𝑓×𝑉2+𝑓1𝑘1𝑒2𝑉−𝑒𝑉∇𝑒Φ−𝑒2𝑉𝑘1−𝑒2Φ∇𝑉2−𝑒ΦΦ1∇𝑒𝑉−𝑒2Φ𝑘2≤−𝑓1𝑘1𝑒2𝑉−𝑒2𝑉𝑘1−𝑒2Φ𝑘2𝑓≤−1𝑘1+𝑘1𝑒2𝑉−𝑘2𝑒2Φ=−𝑒𝑇𝑃𝑒,(4.7) where 𝑃 is as in (4.4). Since 𝑉(𝑡) is positive definite and ̇𝑉(𝑡) is negative semidefinite, it follows that 𝑒𝑉,𝑒Φ,𝑘1,𝑓1,𝑘1,𝑘2∈𝐿∞. Froṁ𝑉(𝑡)≤−𝑒𝑇𝑃𝑒, we can easily show that the square of 𝑒𝑉,𝑒Φ is integrable with respect to 𝑡, namely, 𝑒𝑉,𝑒Φ∈𝐿2. From (4.5), for any initial conditions, we have ̇𝑒𝑉(𝑡),̇𝑒Φ(𝑡)∈𝐿∞. By the well-known Barbalat’s Lemma, we conclude that (𝑒𝑉,𝑒Φ)→(0,0) as 𝑡→∞. Therefore, in the closed-loop system, 𝑉2(𝑡)→𝑉1(𝑡),Φ2(𝑡)→Φ1(𝑡)as 𝑡→∞. This implies that the two shallow water models have synchronized under the adaptive controls (4.2).

5. Conclusions

In this paper, we applied adaptive control theory for the chaos control and synchronization of the shallow water model. First, we designed adaptive control laws to stabilize the shallow water model based on the adaptive control theory and stability theory. Then, we derived adaptive synchronization to the shallow water model. The sufficient conditions for the adaptive control and synchronization of the shallow water model have been analyzed theoretically, and the results are proved using a Barbalat’s Lemma.

References

  1. J. Tribbia, An Introduction to Three-Dimensional Climate Modeling, Oxford University, New York, NY, USA, 1992.
  2. E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Physical Review Letters, vol. 64, no. 11, pp. 1196–1199, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. J. H. Park and O. M. Kwon, “A novel criterion for delayed feedback control of time-delay chaotic systems,” Chaos, Solitons and Fractals, vol. 23, no. 2, pp. 495–501, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. T. L. Liao and S. H. Tsai, “Adaptive synchronization of chaotic systems and its application to secure communications,” Chaos, Solitons and Fractals, vol. 11, no. 9, pp. 1387–1396, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. T. Yang and L. O. Chua, “Generalized synchronization of chaos via linear transformations,” International Journal of Bifurcation and Chaos, vol. 9, no. 1, pp. 215–219, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Y. G. Yu and S. Zhang, “Adaptive backstepping synchronization of uncertain chaotic system,” Chaos, Solitons and Fractals, vol. 27, no. 3, pp. 643–649, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. M. Lakshmanan and K. Murali, Chaos in Nonlinear Oscillators, World Scientific Publishing, River Edge, NJ, USA, 1996.
  9. S. K. Han, C. Kerrer, and K. Kuramoto, “Dephasing and brusting in coupled neutral oscillators,” Physical Review Letters, vol. 75, pp. 3190–3193, 1995. View at Publisher · View at Google Scholar
  10. B. Blasius, A. Huppert, and L. Stone, “Complex dynamics and phase synchronization in spatially extended ecological systems,” Nature, vol. 399, no. 6734, pp. 354–359, 1999. View at Publisher · View at Google Scholar · View at PubMed
  11. K. Murali and M. Lakshmanan, “Secure communication using a compound signal from generalized synchronizable chaotic systems,” Physical Review Letters, vol. 241, no. 6, pp. 303–310, 1998. View at Google Scholar · View at Zentralblatt MATH
  12. M. Feki, “An adaptive chaos synchronization scheme applied to secure communication,” Chaos, Solitons and Fractals, vol. 18, no. 1, pp. 141–148, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, New York, NY, USA, 1989.