TY - JOUR

T1 - Optimal control in a noisy system

AU - Asenjo, F.

AU - Toledo, B. A.

AU - Muoz, V.

AU - Rogan, J.

AU - Valdivia, J. A.

N1 - Funding Information:
This project has been financially supported by FONDECyT under Contract Nos. 1070854 (J.A.V.), 1070131 (J.A.V.), 1070080 (J.R.), 1071062 (J.R.), 1060830 (V.M.), 1080658 (V.M.), and 3060029 (B.T.). F.A. is grateful for the financial support from the doctoral fellowship of the Programa MECE Educación Superior.

PY - 2008

Y1 - 2008

N2 - We describe a simple method to control a known unstable periodic orbit (UPO) in the presence of noise. The strategy is based on regarding the control method as an optimization problem, which allows us to calculate a control matrix A. We illustrate the idea with the Rossler system, the Lorenz system, and a hyperchaotic system that has two exponents with positive real parts. Initially, a UPO and the corresponding control matrix are found in the absence of noise in these systems. It is shown that the strategy is useful even if noise is added as control is applied. For low noise, it is enough to find a control matrix such that the maximum Lyapunov exponent λmax <0, and with a single non-null entry. If noise is increased, however, this is not the case, and the full control matrix A may be required to keep the UPO under control. Besides the Lyapunov spectrum, a characterization of the control strategies is given in terms of the average distance to the UPO and the control effort required to keep the orbit under control. Finally, particular attention is given to the problem of handling noise, which can affect considerably the estimation of the UPO itself and its exponents, and a cleaning strategy based on singular value decomposition was developed. This strategy gives a consistent manner to approach noisy systems, and may be easily adapted as a parametric control strategy, and to experimental situations, where noise is unavoidable.

AB - We describe a simple method to control a known unstable periodic orbit (UPO) in the presence of noise. The strategy is based on regarding the control method as an optimization problem, which allows us to calculate a control matrix A. We illustrate the idea with the Rossler system, the Lorenz system, and a hyperchaotic system that has two exponents with positive real parts. Initially, a UPO and the corresponding control matrix are found in the absence of noise in these systems. It is shown that the strategy is useful even if noise is added as control is applied. For low noise, it is enough to find a control matrix such that the maximum Lyapunov exponent λmax <0, and with a single non-null entry. If noise is increased, however, this is not the case, and the full control matrix A may be required to keep the UPO under control. Besides the Lyapunov spectrum, a characterization of the control strategies is given in terms of the average distance to the UPO and the control effort required to keep the orbit under control. Finally, particular attention is given to the problem of handling noise, which can affect considerably the estimation of the UPO itself and its exponents, and a cleaning strategy based on singular value decomposition was developed. This strategy gives a consistent manner to approach noisy systems, and may be easily adapted as a parametric control strategy, and to experimental situations, where noise is unavoidable.

UR - http://www.scopus.com/inward/record.url?scp=54749087962&partnerID=8YFLogxK

U2 - 10.1063/1.2956981

DO - 10.1063/1.2956981

M3 - Article

AN - SCOPUS:54749087962

SN - 1054-1500

VL - 18

JO - Chaos

JF - Chaos

IS - 3

M1 - 033106

ER -