State Space Multiple Leaves Representations of Complex Systems: The basic properties of system theory.

The purpose of this paper is to provide a reasonable comprehensive transition from the linear time- invariant systems to the linear time-varying systems at a level comparable on the state space multiple leaves representation. In this regard, the preliminary works are dedicated to the on-line identification procedures of a class of complex system. This class of systems is represented by a mechatronic point of view, i.e. communication, calculation, monitoring, linking and state-plan trajectories mapping. For the case in which both calculation and monitoring processes are used, these identification procedures show which type of state-plan is more attractive on Kc , or KR parameters. To complete the on-line identification procedures with a controller tuning of the model parameters, it was necessary to extend the sliding mode´s theorem to the case of multiple leaves state-plan, allowing limitation of short-time instability analysis.