Sampling and round-off, as sources of chaos in PD-controlled systems

It is well-known that nonlinear terms in the governing equations of dynamical systems may lead to chaotic behaviour. With this fact in mind, a well-trained engineer must be able to decide which system of equations can be linearized without a significant change in the solution. However, if the linearized dynamical system in question is part of a digital control loop, the interaction between the original mechanical or electrical system and the control system may still lead to un expected behaviour due to the so-called digital effects. Our goal is to analyze the problem of computer-controlled stabilization of unstable equilibria, with the application of the PD control scheme. We consider the problem of the inverted pendulum, with linearized equations of motion. As a consequence of the digital effects, i.e., the sampling and the round-off error, the solutions of the system can be described by a two dimensional piecewise linear map. We show that this system may perform chaotic behaviour. Although the amplitude of the evolving oscillations is usually very small, several disconnected strange attractors may coexist in certain parameter domains, rather far from the desired equilibrium position. We claim that since the amplitude is small the nonlinearity of the digital control system is the primary source of the stochastic-like vibrations of the inverted pendulum, instead of the nonlinearity of the mechanical system.