Linear systems approach basic concepts

The concepts of structural observability are the basic tools for developing variable classification strategies. Some approaches presented in Chapter 3 are based on the fact that the classification of process variables results from the topology of the system and the placement of instruments and has nothing to do with the functional form of the balance equations. Thus, the linearity restriction will be removed and efficient reduction of the large-scale problem will be accomplished. 

There has evolved over the past three decades a set of general concepts that have revolutionized the way we regard and study systems in nature. Their basic premises run counter to the Newtonian reductionistic approaches and might thus be labelled post-Newtonian concepts. The central theme of this new philosophy is the recognition that the behaviour and properties of a system are non-linear combinations of the subsystems. Such a system is endowed with complexity and displays specific properties that emerge from dynamic interactions between the subsystems. We discuss briefly complexity and emergence as the two pillars of post-Newtonian thought. 

Manning proposed a linear counterion condensation theory to account for the low activity of counterions in polyelectrolyte solutions 14). The basic idea of the theo is that there is a critical charge density on a polymer chain beyond which some counterions will condense to the polymer chain to lower the charge density, otherwise the ener of the system would approach infinite. The concept of this theory has been widely accepted. The shortcoming of the linear countmon condensation is that it predicts that counterion condensation is independent of ionic strength in the solution, which is not in agreement with experimental observations. Counterion condensation can be obtained duectly by solving the nonlinear Poisson-Boltzmann equation. 

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