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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

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 [26].

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)),𝑡𝑡0𝑥𝑡𝑅,0=𝑥0,(2.1) where the state 𝑥(𝑡) take values in 𝑋, 𝑓(𝑡,𝑥)𝑅×𝑋𝑋 is a given nonlinear function and 𝑓(𝑡,0)=0, for all 𝑡𝑅. 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,𝑡0𝑅, 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 all𝑡𝑡0.

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 lim𝑡𝑥(𝑡)=0.
Consider the control system ̇𝑥(𝑡)=𝑓(𝑡,𝑥(𝑡),𝑢(𝑡)),𝑡0,(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))),𝑡0,(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,𝑦𝑅,𝑓,𝑔𝐶𝑟[𝑅×𝑅,𝑅],𝑢𝐶𝑟[𝑅×𝑅×𝑅,𝑅],𝑟1,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,  lim𝑡𝑥(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)̇𝑉0,(3)̇𝑉(𝑥,𝑡) is uniformly continuous in time,then lim𝑡̇𝑉(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𝜕𝑦𝑓𝑢,𝐷𝑤1𝐷𝑡=𝜌𝜕𝑝𝜕𝑧𝑔,(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=0𝜕𝑢+𝜕𝑥𝜕𝑣𝜕𝑦𝑑𝑧=𝜕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.,𝐷/𝐷𝑡=𝑤|surface). Thus 𝐷𝐷𝑡=𝜕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)=𝑧𝑔𝜌𝑑𝑧=(𝑧)𝜌𝑔.(3.6) Take the partial derivatives of 𝑝 (at the surface) with respect to 𝑥 and 𝑦, 𝜕𝑝=𝜕𝜕𝑥𝜕𝑥((𝑧)𝜌𝑔)=𝜌𝑔𝜕1𝜕𝑥𝜌𝜕𝑝𝜕𝑥=𝑔𝜕,𝜕𝑥𝜕𝑝=𝜕𝜕𝑦𝜕𝑦((𝑧)𝜌𝑔)=𝜌𝑔𝜕1𝜕𝑦𝜌𝜕𝑝𝜕𝑦=𝑔𝜕.𝜕𝑦(3.7) Taking (3.2)–(3.7) into (3.1), so the shallow water model in Cartesian coordinates is as follows: 𝐷𝑢𝐷𝑡=𝑔𝜕𝜕𝑥+𝑓𝑣,𝐷𝑣𝐷𝑡=𝑔𝜕𝜕𝑦𝑓𝑢,𝐷𝑤𝐷𝑡=𝜕𝑢+𝜕𝑥𝜕𝑣.𝜕𝑦(3.8) In the vector form, the shallow water model is as follows: ̇̇𝑉=𝑓𝑘×𝑉Φ,Φ=Φ𝑉,(3.9) where 𝐕=𝑢𝑖+𝑣𝑗 is the horizontal velocity, Φ=𝑔 is the geopotential height.

Consider the controlled system of (3.9) which has the forṁ𝑉=𝑓𝑘×𝑉Φ+𝑢1,̇Φ=Φ𝑉+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=𝑔𝑉𝑉,𝑢2=𝑘ΦΦ,(3.11) where 𝑘,𝑔 (estimate of 𝑘, 𝑔, resp.) are updated according to the following adaptive algorithm: ̇𝑔=𝜇𝑉𝑉2,̇𝑘=𝜌ΦΦ2,(3.12) where 𝜇,𝜌 is adaption gains. Then the controlled systems have the following form: ̇𝑉=𝑓𝑘×𝑉Φ𝑔𝑉𝑉̇,(3.13)Φ=Φ𝑉𝑘ΦΦ.(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=12𝑉𝑉2+ΦΦ2+1𝜇𝑔𝑔2+1𝜌𝑘𝑘2.(3.15) The time derivative of 𝑉 is ̇𝑉=𝑉𝑉̇𝑉+ΦΦ̇1Φ+𝜇𝑔𝑔1̇𝑔+𝜌𝑘𝑘̇𝑘.(3.16) By substituting (3.13)-(3.14) in (3.16), ̇𝑉=𝑉𝑉𝑓𝑘×𝑉Φ𝑔𝑉𝑉+ΦΦΦ𝑉𝑘ΦΦ+1𝜇𝑔𝑔𝜇𝑉𝑉2+1𝜌𝑘𝑘𝜌ΦΦ2.(3.17) Let 𝜂1=(𝑉𝑉),𝜂2=(ΦΦ). Since (𝑉,Φ) is an equilibrium point of the uncontrolled system (3.9), ̇𝑉 becomes ̇𝑉=𝜂1𝑓𝑘×𝑉Φ𝑔𝑉𝑉+𝜂2Φ𝑉𝑘ΦΦ+𝑔𝑔𝜂21+𝑘𝑘𝜂22=(𝑓𝑘×𝑉)𝜂1Φ𝜂1𝑔𝜂21ΦV𝜂2𝑘𝜂22+𝑔𝑔𝜂21+𝑘𝑘𝜂22.(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,𝑘𝐿. From ̇𝑉(𝑡)0, we can easily show that the square of 𝜂1,𝜂2 is integrable with respect to 𝑡, namely, 𝜂1,𝜂2𝐿2. From (3.13)-(3.14), for any initial conditions, we have ̇𝜂1,̇𝜂2𝐿. 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×𝑉1Φ1,̇Φ=Φ1𝑉1,̇𝑉=𝑓2𝑘2×𝑉2Φ2𝑢1,̇Φ=Φ2𝑉2𝑢2(4.1) where 𝑢=[𝑢1,𝑢2]𝑇 is the controller. We choose 𝑢1=𝑘1𝑒𝑉,𝑢2=𝑘2𝑒Φ,(4.2) where 𝑘1,𝑘20 and 𝑒𝑉,𝑒Φare the error states which are defined as follows 𝑒𝑉=𝑉2𝑉1,𝑒Φ=Φ2Φ1.(4.3)

Theorem 4.1. Let 𝑘1,𝑓1,𝑘1,𝑘20 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×𝑉2Φ2+𝑓1𝑘1×𝑉1+Φ1𝑢1,̇𝑒Φ=Φ2𝑉2+Φ1𝑉1𝑢2.(4.5) Let 𝑒𝑘𝑓=𝑘2𝑓2𝑘1𝑓1. 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=𝑒𝑉𝑓2𝑘2×𝑉2Φ2+𝑓1𝑘1×𝑉1+Φ1𝑢1+𝑒ΦΦ2𝑉2+Φ1𝑉1𝑢2=𝑒𝑉𝑓2𝑘2×𝑉2+Φ2𝑓1𝑘1×𝑉1Φ1+𝑢1𝑒ΦΦ2𝑉2Φ1𝑉1+𝑢2=𝑒𝑉𝑓2𝑘2×𝑉2𝑓1𝑘1×𝑉1+𝑓1𝑘1×𝑉2𝑓1𝑘1×𝑉2𝑒𝑉Φ2Φ1𝑒𝑉𝑢1𝑒ΦΦ2𝑉2Φ1𝑉1+Φ1𝑉2Φ1𝑉2𝑒Φ𝑢2=𝑒𝑉𝑒𝑘𝑓×𝑉2+𝑓1𝑘1𝑉2𝑉1𝑒𝑉Φ2Φ1𝑒𝑉𝑘1𝑒𝑉𝑒ΦΦ2Φ1𝑉2+Φ1𝑉2𝑉1𝑒Φ𝑘2𝑒Φ=𝑒𝑉𝑒𝑘𝑓×𝑉2+𝑓1𝑘1𝑒𝑉𝑒𝑉𝑒Φ𝑒2𝑉𝑘1𝑒Φ𝑒Φ𝑉2+Φ1𝑒𝑉𝑒2Φ𝑘2=𝑒𝑉𝑒𝑘𝑓×𝑉2+𝑓1𝑘1𝑒2𝑉𝑒𝑉𝑒Φ𝑒2𝑉𝑘1𝑒2Φ𝑉2𝑒ΦΦ1𝑒𝑉𝑒2Φ𝑘2𝑓1𝑘1𝑒2𝑉𝑒2𝑉𝑘1𝑒2Φ𝑘2𝑓1𝑘1+𝑘1𝑒2𝑉𝑘2𝑒2Φ=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,𝑘2𝐿. Froṁ𝑉(𝑡)𝑒𝑇𝑃𝑒, we can easily show that the square of 𝑒𝑉,𝑒Φ is integrable with respect to 𝑡, namely, 𝑒𝑉,𝑒Φ𝐿2. 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.

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