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.     

Stability robustness to unstructured uncertainties of linear systems controlled on the basis of the multirate sampling of the plant output

The stability robustness of stable feedback loops designed onthe basis of multirate-output controllers (MROCs) is analysedin this paper. For MROC-based feedback loops, designed to achievestabilization through pole placement or deterministic linear-quadratic(LQ) optimal regulation, we characterize additive or multiplicativenorm-bounded perturbations of the loop transfer-function matrixthat do not destabilize the closed-loop system. New sufficientstability conditions in terms of the elementary MROC matricesare presented, for both static and (stable) dynamic MROCs. Moreover,lower bounds for the minimum singular values of the return-differenceand of the inverse return-difference matrices are suggestedfor all cases of the aforementioned MROC-based stable feedbackdesigns. Also suggested are guaranteed stability margins forMROC-based pole placers and LQ optimal regulators. A comparisonbetween the suggested stability margins for static and (stable)dynamic MROCs is presented, while the superiority of these marginsover known stability margins for deterministic LQ optimal regulatorsis identified. Finally, an analysis of the deficiency of theaforementioned stablity margins is presented for cases wherethe MROC feedback gains become very large, and useful guidelinesare suggested for the choice of the sampling period and of theoutput multiplicities of the sampling to avoid this deficiency. E-mail: karvan{at}control.ece.ntua.gr