A matrix-pencil-based interpretation of inconsistent initial conditions and system properties of generalized state-space systems

A matrix-pencil-based approach is presented to interpret transition matrices, inconsistent initial conditions, and systems properties of regular generalized state-space (GSS) systmes. On the basis of the well known Weierstrass canonical form of a regular pencil, several definitions of transition matrices for GSS systems are given. Convolution forms of the forced state evolution of GSS systems are also established, both for the case of consistent and of inconsistent initial conditions. Moreover, a fundamental interpretation of inconsistent initial conditions of GSS systems is outlined. Finally, the nation of several types of controllability and observability Gramians of GSS systems is introduced. Relations of these Gramians to the respective controllability and observability properties of GSS systems are examined, and simple and easily checked algebraic criteria based on these Gramians, are estabished. It is pointed out that these results appear to be first in the field of GSS systems.