دوگانگی در تجزیه و تحلیل سیستم برای مدل های گراف باند

Duality as a general notion has been discussed previously in multiple domains. The field of system modeling, analysis and control has used it for quite some time. The duality between the controllability and the observability is well known, especially in the case of the linear time-invariant systems. But when it comes to the linear systems in general, time-invariant or otherwise, the definitions become ambiguous. Even though there have been papers which use the state space representation or the module theoretical approach, a unified description has not been found yet. This paper is meant to fill in this gap, by using the bond graph representation. The bond graph perspective offers a global overview because the bond graph is a graphical tool which can be seen both as a state space representation and as a module.

Usually a large range of procedures and algorithms for calculating best suited control laws are proposed to system control engineers. The techniques can be very different in their substance and the concepts can be understood only by skilled engineers or researchers. One goal is to link different kinds of model properties or control law properties. This link can be pointed out with the concept of duality. The concept of duality in the linear systems has been discussed from the state space representation point of view in [1] and [2] in the 1980s. In the late 1990s, the module theoretical approach offered a new perspective on the duality in linear systems [3]. The present paper brings a new perspective by using the bond graph, a graphical tool which can be seen at the same time as a state space representation and as a module. Therefore the procedures developed in both approaches can be applied on the bond graph models. The interest of the article is in the duality in system analysis. The first part offers a recall of the duality from the two approaches, state space and module theoretical representations. The second part is focused on the definition of the dual bond graph model. In the third part the procedures used to prove the duality in the system analysis from the bond graph perspective are presented. In the final section, we have gathered the conclusions and some perspectives.

In this section, we have introduced a methodology for studying the duality in system analysis for LTV models. Our study of duality begun by defining the dual graphical representations for bond graphs. The most important point of this procedure is the time-dependent state variable change View the MathML sourcex¯=F(t)x. Contrary to the mathematical approach, either state space representation or module theoretical approach, on the dual bond graph model the state variables have a physical meaning, they are the efforts of the C elements in integral causality and the flows of the I elements in integral causality. With the state variable change on the dual bond graph model, we had to develop some new rules for calculating the gains of the causal loops and causal paths. In the last part of the article, we tackled the system analysis problem. In order to study the duality between controllability and observability, we have introduced some graphical methods for studying these properties. Firstly, we provide some graphical computational methods for calculating the controllability and observability matrices. These procedures are applied on the controllability/observability bond graph, respectively. Secondly, we present some graphical direct methods for determining these structural properties. The controllability procedure applied in an LTV bond graph model introduced in [9] is reused here. For the observability procedure, we use the dual bond graph model on which we apply the controllability procedure. This paper is a gateway to a new perspective of dealing with the linear systems and opens new perspectives on other duality. Future work involves (A,B)(A,B)-invariance and (C,A)(C,A)-invariance, system decoupling and disturbance rejection by state feedback and output injection, as well as possible generalization for the nonlinear case using the variational bond graph model. The duality for the nonlinear bond graph models is the subject of further work. On the long range the aim is to provide a comprehensive tool for the engineers which are proficient in either command or observers and who, based on their strong point, can easily build a good regulation system using the duality of the system and the procedures which are most familiar to them.