Balanced representations for a class of L2 systems

The author discusses the concept of balanced state-space representations for the class of systems whose behavior can be described as the solution set of a finite number of ordinary linear differential equations. It is shown that the set of square integrable solutions of differential equations of this type always admits a finite-dimensional balanced state-space representation. The notion of balancing is defined without reference to specific state-space representations, and only invokes the introduction of arbitrary norms on the past and the future behavior of the system. It is more general than the prevailing notion in that it is well defined for nonstable systems, independent of a particular representation, and yields a new set of invariants associated with the system. It is shown how this type of balanced representation may be directly obtained by computing the extremal solutions of algebraic Riccati equations