Control systems with stochastic feedback.

*(English)*Zbl 0977.37049Summary: We use the analogy of Parrondo’s games to design a second order switched mode circuit which is unstable in either mode but is stable when switched. We do not require any sophisticated control law. The circuit is stable, even if it is switched at random. We use a stochastic form of Lyapunov’s second method to prove that the randomly switched system is stable with probability of one. Simulations show that the solution to the randomly switched system is very similar to the analytic solution for the time-averaged system. This is consistent with the standard techniques for switched state-space systems with periodic switching. We perform state-space simulations of our system, with a randomized discrete-time switching policy. We also examine the case where the control variable, the loop gain, is a continuous Gaussian random variable. This gives rise to a matrix stochastic differential equation (SDE). We know that, for a one-dimensional SDE, the difference between solution for the time averaged system and any given sample path for the SDE will be an appropriately scaled and conditioned version of Brownian motion. The simulations show that this is approximately true for the matrix SDE. We examine some numerical solutions to the matrix SDE in the time and frequency domains, for the case where the noise power is very small. We also perform some simulations, without analysis, for the same system with large amounts of noise. In this case, the solution is significantly shifted away from the solution for the time-averaged system. The Brownian motion terms dominate all other aspects of the solution. This gives rise to very erratic and “bursty” behavior. The stored energy in the system takes the form a logarithmic random walk. The simulations of our curious circuit suggest that it is possible to implement a control algorithm that actively uses noise, although too much noise eventually makes the system unusable.

##### MSC:

37N35 | Dynamical systems in control |

34C29 | Averaging method for ordinary differential equations |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

93E99 | Stochastic systems and control |

93B52 | Feedback control |

PDF
BibTeX
XML
Cite

\textit{A. Allison} and \textit{D. Abbott}, Chaos 11, No. 3, 715--724 (2001; Zbl 0977.37049)

Full Text:
DOI

##### References:

[1] | DOI: 10.1214/ss/1009212247 · Zbl 1059.60503 · doi:10.1214/ss/1009212247 |

[2] | DOI: 10.1038/scientificamerican0701-56 · doi:10.1038/scientificamerican0701-56 |

[3] | DOI: 10.1038/47220 · doi:10.1038/47220 |

[4] | DOI: 10.1098/rspa.2000.0516 · Zbl 1054.91514 · doi:10.1098/rspa.2000.0516 |

[5] | DOI: 10.1088/0305-4470/33/23/101 · Zbl 1032.91554 · doi:10.1088/0305-4470/33/23/101 |

[6] | DOI: 10.1049/el:19970525 · doi:10.1049/el:19970525 |

[7] | DOI: 10.1109/63.641492 · doi:10.1109/63.641492 |

[8] | Wu M. K. W., J. Electr. Electron. Eng., Aust. 16 pp 193– (1996) |

[9] | DOI: 10.1016/S0005-1098(98)00167-8 · Zbl 0949.93014 · doi:10.1016/S0005-1098(98)00167-8 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.