Control systems with stochastic feedback

In this paper we use the analogy of Parrondo's games to design a second ord er switched mode circuit which is unstable in either mode but is stable whe n 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 Lyap unov's second method to prove that the randomly switched system is stable w ith probability of one. Simulations show that the solution to the randomly switched system is very similar to the analytic solution for the time-avera ged system. This is consistent with the standard techniques for switched st ate-space systems with periodic switching. We perform state-space simulatio ns of our system, with a randomized discrete-time switching policy. We also examine the case where the control variable, the loop gain, is a continuou s Gaussian random variable. This gives rise to a matrix stochastic differen tial equation (SDE). We know that, for a one-dimensional SDE, the differenc e between solution for the time averaged system and any given sample path f or the SDE will be an appropriately scaled and conditioned version of Brown ian motion. The simulations show that this is approximately true for the ma trix 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. W e also perform some simulations, without analysis, for the same system with large amounts of noise. In this case, the solution is significantly shifte d 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 e rratic and "bursty" behavior. The stored energy in the system takes the for m a logarithmic random walk. The simulations of our curious circuit suggest that it is possible to implement a control algorithm that actively uses no ise, although too much noise eventually makes the system unusable. (C) 2001 American Institute of Physics.