Realization theory for differential algebraic input-output systems

The problem considered is that of realization for a given prime differential input-output system γ (which generates a prime differential ideal in the differential algebra K(y,u)). It is shown that states can always be chosen from within the extension field K(y,u: γ) rather than from without. The new concept of a faithful realization is defined, which is thought to be an important addition to realization theory of non-linear differential input-output (IO) control systems. A new definition of observability is presented which differs from previous algebraic definitions, but which is akin to those given in geometric control theory. Existence and uniqueness of a minimal differential algebraic realization for a given differential IO system are proved. These results are also in line with geometric control theory. The paper uses the tools of differential algebra.