Realization Theory Fundamentals

The basic problem in Realization theory is to determine matrices A,B,C,D such that the state space system [A,B,C,D] represents a given input-output map, specified by its impulse response or transfer function.

[A,B,C,D]and [A,B,C,D] are algebraically equivalent if they are of the same dimension and they are related by a similarity(coordinate) transformation, i.e. there exists an invertible matrix T such that

[A,B,C,D] is reducible (non-minimal) if there exists a zero-state equivalent representation of smaller state-space dimension; otherwise[A,B,C,D] is irreducible or mminimal. Notice that algebraic equivalence implies zero-state equivalence. The converse may not be true even if both systems have the same state-space dimension.

In other words, algebraically equivalent realizations have identical controllability, observability and stability properties. An implication of that is that the following two systems cannot be algebraically equivalent even though they are zero-state equivalent and they have the same state-space dimension:

Notice that the first system is c.o. but not .c.c. while the second is c.c. but notc c.o. Some properties of minimal realizations are described by the following theorem. They make use of the so-called Hankel matrix

In dealing with nonmminimal realizations the following result is important, providing both the theoretical framework and a computational approach to construct I/O equivalent minimal realizations.

In dealing with nonmminimal realizations the following result is important, providing both the theoretical framework and a computational approach to construct I/O equivalent minimal realizations.

Where the state partition is compatible with the A-matrix partition. Furthermore, the transformation T can be computed systematically using SVD or QR decompositions of the controllability and observability matrices: R(Qc) is the controllable subspace and N(Qo) is the unobservable subspace.

Using the Kalman Canonical Decomposition a computational procedure to obtain a minimal realization is described below. This approach is not necessarily numerically efficient or reliable; nevertheless it establishes a systematic method to obtain minimal realizations.