Physical Correctness of Systems, State Space Representations, Minimality and Dissipativity JOSEF HRUSAK 1 , MILAN STORK 1 , DANIEL MAYER 2 Department of Applied Electronics and Telecommunications 1 Theory of Electrical Engineering 2 University of West Bohemia P.O. Box 314, 30614 Plzen Czech Republic hrusak@kae.zcu.cz , stork@kae.zcu.cz , mayer@kte.zcu.cz Abstract: - This paper deals with internal stability and related structural properties of a relatively broad class of finite dimensional strictly causal systems, which can be described in the state-space representation form. Dissipativity, instability, asymptotic stability as well as stability in the sense of Lyapunov are 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 proper 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 both the continuous- and discrete-time signals as a natural consequence of strict causality, signal energy conservation, dissipativity and state minimality requirements. Several typical problems are solved in detail, and results of simulation examples are shown for illustration of fundamental ideas and basic attributes of the proposed method. Key-Words: Signal power, signal energy, structure, state space velocity, state space metric, nonlinear system, internal, external, representation. 1 Introduction Recall that from general point of view any collection of trajectories constitutes a dynamical system that, in principle, can be described either by external behavior, or by an internal structure. In the input-to-output framework the external behavior of a continuous-time causal system can be seen as a collection of all input- output trajectories satisfying the relation: ( ) ( ) (, , ,..., , , ,... ) 0, n m F tyy y uu u m n = ≤ (1) The input signals u(.) and output signals y(.), explicitly reflect a signal orientation property of causality relation and determine an external causality structure, which is important for external stability. Formally, we can write for an external stability property: {System S is stable} ⇔ { () ut <δ ⇒ () yt <ε} (2) In the present paper mainly the concepts of dissipativity and conservativity and their relationships with the internal stability problems will be examined. In such a case of the state-to-state framework, mainly an internal causality structure, reflecting a time orientation property of the causality relation and describing a collection of all state trajectories in, which no external signals are explicitly introduced seems to be appropriate: [ ] 0 () (), ( ) n x t f xt xt X R = ∈ ⊂ (3) Basic stability results concerning general internal stability problems are due to original work of A.M. Lyapunov. The main advantage of the Lyapunov’s approach is its generality. It applies for time-varying, linear and nonlinear systems as well. Notice that the resulting stability conditions based on the direct Lyapunov’s method are sufficient but not necessary in general. From practical point of view, main drawback of the direct method is lack of any systematic and universally applicable technique for generation of axiomatically defined Lyapunov functions V(x) having required properties [1 - 10]. Certainly, any abstract realizable system has to fulfill some causality and energy conservation requirements. Recall that existence of an abstract state space representation is necessary and sufficient for a system to be causal. On the other hand causality does not imply energy conservation. Consequently not every state space representation may be a’priori considered as physically correct in the sense of a signal/state energy conservation law. This is one of crucial points of the proposed approach. Some theoretical and practical consequences of the new concepts will also be demonstrated [1, 2, 3, 11, 12, 13, 14]. Proceedings of the 13th WSEAS International Conference on SYSTEMS ISSN: 1790-2769 202 ISBN: 978-960-474-097-0