Plenary lecture 1: new approach to continuous and discrete-time systems based on abstract state space energy

Almost in any field of science and technology some sort of stability problem can appear. Instability is certainly the most important phenomena, which should be investigated before any other aspect of reality will be attacked. The stability and energy of the system are closely related. Simple linear and nonlinear continuous and discrete systems are investigated from the energy point of view. As an alternative to the method of Lyapunov functions a conceptually different approach can be based on the idea that, in fact abstract state space energy can be measured as distance from the system equilibrium to the actual state, what is needed also for stability analysis. Thus, instead of the physical energy a metric function will be defined in a proper way. The idea is based on based on a generalization of the well known Tellegen's principle.

Presented study deals with energy, stability and related structural properties of a relatively broad class of finite dimensional strictly causal continuous and discrete systems, which can be described in the state-space representation form. Dissipativity, instability, asymptotic stability as well as stability in the sense of Lyapunov is analyzed by a new approach based on an abstract state energy concept. The resulting energy metric function is induced by the output signal power and determines both, the structure of a digital system representation as well as the corresponding system state space topology.

A special form of physically correct internal structure of an equivalent state space representation has been derived for discrete time signals as a natural consequence of strict causality, signal energy conservation, dissipativity and state minimality requirements. New discretization approach of discrete systems based on energy preservations is presented. Results of simulation examples are shown for illustration of fundamental ideas and basic attributes of the proposed method.