Implicit Euler numerical simulations of sliding mode systems
For researchers in control theory and numerical simulation, this method offers a stable, event-capturing approach to simulate sliding mode systems without the oscillations of explicit methods.
The paper shows that implicit Euler time-discretization of sliding mode systems eliminates spurious oscillations and handles Zeno phenomena and multiple switching surfaces, providing an alternative for chattering suppression.
In this report it is shown that the implicit Euler time-discretization of some classes of switching systems with sliding modes, yields a very good stabilization of the trajectory and of its derivative on the sliding surface. Therefore the spurious oscillations which are pointed out elsewhere when an explicit method is used, are avoided. Moreover the method (an {\em event-capturing}, or {\em time-stepping} algorithm) allows for accumulation of events (Zeno phenomena) and for multiple switching surfaces (i.e., a sliding surface of codimension $\geq 2$). The details of the implementation are given, and numerical examples illustrate the developments. This method may be an alternative method for chattering suppression, keeping the intrinsic discontinuous nature of the dynamics on the sliding surfaces. Links with discrete-time sliding mode controllers are studied.