Micro-chaos in Relay Feedback Systems with Bang-Bang Control and Digital Sampling

摘要

We investigate a class of linear relay feedback systems with bang-bang control and with the control input applied at discrete time instances. Using a third order system as a representative example we show that stable oscillations with so-called sliding motion, with sliding present in continuous time system, loose the sliding segment of evolution, but do not loose their stability if the open loop system is stable. We then carry on our investigations and consider a situation when stable self-sustained oscillations are generated with the unstable open loop system. In the latter case a transition from a stable limit cycle to micro-chaotic oscillations occurs. The presence of micro-chaotic oscillations is shown by considering a linearised map that maps a small neighbourhood of initial conditions back to itself. Using this map the presence of the positive Lyapunov exponent is shown. The largest Lyapunov exponent is then calculated numerically for an open set of sampling times, and it is shown that it is positive. The boundedness of the attractor is ensured for sufficiently small sampling times; with the sampling time tending to zero these switchings become faster and they turn into sliding motion. It is the presence of the underlying sliding evolution that ensures the boundedness of the chaotic attractor. Our finding implies that what may be considered as noise in systems with digital control should actually be termed as micro-chaotic behaviour. This information may be helpful in designing digital control systems where any element contributing to what appears as noise should be suppressed.