European Journal of Scientific Research - EuroJournals
European Journal of Scientific Research - EuroJournals
European Journal of Scientific Research - EuroJournals
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
<strong>European</strong> <strong>Journal</strong> <strong>of</strong> <strong>Scientific</strong> <strong>Research</strong><br />
ISSN 1450-216X Vol.14 No.3 (2006), pp. 326-332<br />
© Euro<strong>Journal</strong>s Publishing, Inc. 2006<br />
http://www.eurojournals.com/ejsr.htm<br />
Model-Based Adaptive Chaos Control using Lyapunov<br />
Exponents<br />
Amin Yazdanpanah Goharrizi<br />
Department <strong>of</strong> Electrical Engineering<br />
K. N. Toosi University <strong>of</strong> Technology<br />
Tehran, Iran<br />
Mehdi Semin<br />
Department <strong>of</strong> Electrical Engineering<br />
Tabriz University, Tabriz, Iran<br />
E-mail: yazdanpanah@ee.kntu.ac.ir<br />
Abstract<br />
A model-based approach to adaptive control <strong>of</strong> chaos in non-linear chaotic discrete<br />
time systems is presented. In the case <strong>of</strong> unknown or time varying chaotic plants, the<br />
Lyapunov exponents may vary during the plant operation. In this paper, an effective<br />
adaptive strategy is proposed for on-line identification <strong>of</strong> Lyapunov exponents. The control<br />
aim is that the plant output changes in accordance with the output <strong>of</strong> the linear desired<br />
model. Also, a nonlinear observer for estimation <strong>of</strong> the states is proposed. Simulation<br />
results are provided to show the effectiveness <strong>of</strong> the proposed methodology.<br />
1. Introduction<br />
The analysis and control <strong>of</strong> chaotic behavior in dynamical systems has been widely investigated in<br />
recent years [1], [2], [3], [4] and [5].Also, Lyapunov exponents have been used to characterize and<br />
quantify the chaoticity <strong>of</strong> complex dynamical systems [6], and the computation <strong>of</strong> the Lyapunov<br />
exponents for nonlinear dynamical systems is an effective tool in this respect [7]. In [8], a model-based<br />
approach for anticontrol <strong>of</strong> some discrete-time systems is proposed and in this paper, a reverse method<br />
for control <strong>of</strong> chaotic systems is presented and also, a new method to adaptive control <strong>of</strong> chaos via<br />
adaptive calculation <strong>of</strong> Lyapunov exponents is introduced. The adaptive calculation <strong>of</strong> Lyapunov<br />
exponents proposed in [9-10-11], greatly facilities the design <strong>of</strong> adaptive chaos control. Thus, a<br />
generalized adaptive algorithm recursive least square for estimation <strong>of</strong> Lyapunov exponents is<br />
developed when the parameters <strong>of</strong> the system change abruptly. We use an efficient QR based method<br />
for the computation <strong>of</strong> Lyapunov exponents [12]. Then, if the maximum Lyapunov exponent becomes<br />
positive it's indicates the chaotic behavior and the control aim is that the plant output changes in<br />
accordance with the output <strong>of</strong> the linear desired model. So, the behavior <strong>of</strong> the closed-loop system<br />
depends on the linear model and it can be periodic or tends to zero after controlling. With the above<br />
strategy and adaptive calculation <strong>of</strong> Lyapunov exponents an efficient methodology for adaptive chaos<br />
control is presented. Also, a nonlinear observer is proposed when the sate <strong>of</strong> nonlinear chaotic plant are<br />
not available. Finally, simulation results for Henon map with time varying parameters are provided to<br />
show the effectiveness <strong>of</strong> the proposed methodology.