Plenary Lecture

On System Representation Based on a New Concept of Abstract State Space Energy

Associate Professor Milan Stork
Department of Applied Electronics and Telecommunications
Faculty of Electrical Engineering
University of West Bohemia
P.O. Box 314
30614 Plzen, Czech Republic
stork@kae.zcu.cz

Abstract: 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. Two typical situations should be distinguished in dynamical systems theory, if an in-stability problem has to be solved. The first one arises if an energy function E[x(t)] of a given system is known in a mathematical form. In such situations the time evolution of internal energy along any system motion can be described, and an energy monotonicity test can be used:

On the other hand, there are certainly even more real world situations in which causality and energy conservation are known to play a crucial role, but any mathematical expression for the system energy evolution is not available.
In many cases a sort of approximation seems to be adequate way to avoid this difficulty. If effects of parameter changes are neglected and a technique of approximate linearization is used, then a form of algebraic stability test based on the explicit knowledge of the solution of differential equations may help to simplify the solution. A set of necessary and sufficient conditions for roots:

or for coefficients ai of the system characteristic polynomials P(s), such as the well known Hurwitz criterion, for instance, have been frequently used. For the so-called non-critical cases A. M. Lyapunov has legitimated the approximate linearization approach by his first method, also called Indirect, in the year 1892.
Unfortunately, it is more an exception as a rule that a real world system can be a’priory considered as non-critical.
In fact, more appreciated became the Second Lyapunov Method, which instead of the physical energy E works with a set of axiomatically defined scalar functions V of the state x(t), called Lyapunov functions. Fundamental drawback it lack of a reliable technique of Lyapunov function generation.
The main goal of the paper is to present an alternative method for stability analysis. Instead of Lyapunov functions a state space metric has been introduced as an abstract measure of the total energy accumulated in the system state. The essence of the new approach is demonstrated by variety of examples. In next part, the chaotic systems are also presented.
It is well known that in the case of zero-input continuous-time causal systems, the existence of chaos is impossible if the order n of that system is lower than n=3. Another necessary condition of the chaos creation is existence a proper strong non-linearity. Thus in any situation in which the system representation under examination is considered to be linear or its internal structure does not contain sufficiently high number of independent energy accumulators fails.
If we start from the realistic assumption that in the real world globally linear system does not exist, we can say that at the most their linear mathematical models can be interpreted as some specific local approximation of their real behaviour. One of the most important external demonstrations of strongly non-linear internal system interactions can be some irregularities in observed real systems behaviour, which are commonly called chaotic phenomena. It is evident that observed real systems behaviour is determined by the structure of its internal energetic interactions. As a typical example of chaotic behaviour the turbulent fluid flow can be mentioned..
One of the most essential characteristics of the chaotic behaviour is its long-term non-predictability. Problems connected with chaos detection, modelling and analysis are extraordinary difficult in general. The main reason is that adequate research methods must not be based on the standard assumption of superposition principle validity. From that fact some paradoxes appear. Among the most significant chaos paradoxes belongs that the chaotic behaviour cannot be predicted, but it can be successfully controlled. It is also remarkable that a pair of chaotic signals can be synchronised as well.
The problems of chaos control and synchronization attract attention of researchers and engineers since the early 1990’s. It turns out that the methods describing chaotic behaviour occur in many areas of science and technology, and in many situations proved to be more suitable for describing indeterminacy or irregular oscillations than the stochastic and probabilistic methods. There is a paradoxical fact that chaos cannot be forecasted, but it can be controlled. Perhaps this may be one of the main reasons, which gave rise to an explosive interest of many researchers in the topic, as well as for a large number of publications appeared in the last decade. However, despite of numerous publications and many successful applications, only few strict generally valid facts were established there, and many issues remain open. It is another paradox that the basic principles of chaos have been widely applied without their deep understanding.
It is known, that besides standard physical approach to deterministic chaos theory that follows from the classical theory of Navier-Stokes partial differential equations also procedures based on classical method of Poincare sections and/or Lyapunov exponents method can be applied and are broadly used. From the mathematical point of view the main source of chaotic phenomena can be seen in bifurcations of the state space trajectories. Often it seems possible to use modern mathematical tools of bifurcation theory and fractal dimensions theory, when practically important problems need to be solved. However, engineering applications of those methods slightly fall behind their potentiality, presumably because of their theoretical complexity.
In the proposed paper a physically motivated signal-system-theoretic approach to chaotic phenomena, based on a generalisation of the well known Tellegen’s principle of electrical circuits will be presented and used as a fundamental tool to solve problems of chaos detection, analysis, synthesis and control from a unique physically plausible point of view.
Two fundamental concepts are of crucial importance in the proposed approach. The first one is the concept of strongly non-linear power-informational interactions, and the second one is the notion of state space energy vector, inducing the system state-space topology.
All computations, including numerical solutions of differential equations, were done using MATLAB.

Brief Biography of the Speaker:
Milan Stork received the M.Sc. degree in electrical engineering from the Technical University of Plzen, Czech Republic at the department of Applied electronics in 1974 and Ph.D. degree in automatic control systems at the Czech Technical University in Prague in 1985. In 1997, he became as Associate Professor at the Department of Applied Electronics and Telecommunication, faculty of electrical engineering on University of West Bohemia in Plzen, Czech Republic. He has numerous journal and conference publications. He is member of editorial board magazine "Physician and Technology". His research interest includes analog/digital linear and nonlinear systems, control systems, signal processing and biomedical engineering, especially cardiopulmonary stress tests systems.