Positive Real Systems

Abstract

In the study and control of general dynamical systems the energy stored in the system is an important information. Some autonomous dynamical systems (i.e. with input equal to zero) have the important property that the amount of energy stored in the system remains unchanged or even decreases. This gives rise to the definition of Dissipative systems which is very useful in the synthesis of control systems. Different definitions are introduced to distinguish different classes of systems. Linear systems and nonlinear systems are two important classes of dynamical systems which for simplicity reasons have been usually studied separately in what concerns their dissipative properties. Linear systems whose stored energy remains constant or decreases are called Positive Real (PR) systems or Strictly Positive Real (SPR) respectively while we usually reserve the term dissipative for nonlinear systems in general. We can certainly use the term dissipative for a linear system however the term positive real is only reserved for transfer functions. Furthermore, there exists several important subclasses of linear and nonlinear systems with slightly different properties which are important in the analysis. This will lead us to introduce a series of different definitions of PR and dissipative systems. This chapter deals with Positive Real systems. The next chapter will be devoted to dissipative nonlinear systems.