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 transitionmatrices, inconsistent initial conditions, and systems propertiesof regular generalized state-space (GSS) systmes. On the basisof the well known Weierstrass canonical form of a regular pencil,several definitions of transition matrices for GSS systems aregiven. Convolution forms of the forced state evolution of GSSsystems are also established, both for the case of consistentand of inconsistent initial conditions. Moreover, a fundamentalinterpretation of inconsistent initial conditions of GSS systemsis outlined. Finally, the nation of several types of controllabilityand observability Gramians of GSS systems is introduced. Relationsof these Gramians to the respective controllability and observabilityproperties of GSS systems are examined, and simple and easilychecked algebraic criteria based on these Gramians, are estabished.It is pointed out that these results appear to be first in thefield of GSS systems.