Lumpability and observability of linear systems

Abstract

Large scale dynamic system models are plagued by problems of dimensionality and limitations of measurement. It is common practice in applications to lump similar system state components together to get a reduced number of aggregate states. An example of this approach is the PONA scheme for modelling catalytic cracking of oil. The thousands of chemical species involved in this process are lumped into four groups: paraffins. olefins, naphthenes, and aromatics (PONA). For a survey of the use of lumping in the cracking problem, the reader is referred to Weekman [ 14971. Attention has focused on conditions under which such lumping preserves structural properties of the system dynamics. In particular, when do the lumped states of a linear system satisfy linear dynamics? If this occurs, the system is said to be exactly lumpable. For finite dimensional systems, this issue has been resolved [2, 1.51, but in practice the conditions specified are almost never satisfied. In this paper, we show that it is fruitful to consider the lumped variables as observations of the original system. To our knowledge, this viewpoint has not been taken previously. We establish a relationship between lumpability and observability of the lumped observation system for both finite and infinite dimensional linear systems. This relationship allows us to deduce the existence of a minimal size system which gives a true representation of the dynamics of the lumped variables. Definitions of observability and lumpability are given in Sections 2 and 3, including extensions to infinite dimensional systems. The relationship between these two concepts is discussed and we establish the main result concerning lumpability in Section 4 in the case of finite dimensional systems. In Section 5 this discussion is carried over to infinite dimensional systems. A summary and discussion follow in Section 6.