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System Theory: From Classical State Space to Variable Selection and Model Identification
Abstract
System Theory is a powerful paradigm to deal with abstract models of real processes in such a way to be accurate enough to capture the salient underlying dynamics while keeping the mathematical tools easy enough to be manageable. Its typical approach is to describe reality via a reduced subset of ordinary differential equations (ODE) linking the variables. A classical application is the circuits theory, linking the intensive (voltage) and extensive (current) variables across and through each simplified element by means of equilibrium laws at nodes and around elementary circuits. When such relationships are linear (like in ideal capacitors, resistances, and inductors, just to stay in the circuit field), a full battery of theorems does help in understanding the general properties of the ODE system. Positive systems, quite often used in compartmental processes like reservoirs in nature and pharmacologic concentration in medical therapy, enjoy most of the properties of the linear systems, with the nonlinear constraint of non negativity. More general nonlinear systems are less easily treatable unless a simple form of nonlinearity is taken into account like the ideal characteristic of a diode in circuit theory. When the physics of the process is quite known, like in the mentioned examples, it is quite easy to identify a small number of variables whose set would fully describe the dynamics of the process, once their interrelations are properly modeled: this is the classical way to approach such a problem.
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