Parameterization of the full-order compensator in the frequency domain

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Abstract
After a brief survey of the relations between time-and frequency-domain representations of the observer-based linear state feedback loop, the parameterization of the compensator is discussed. It is shown that the free parameters specified by the state feedback matrix K and the output error feedback matrix L in the time domain appear as coefficients in the polynomial matrices [Dtilde](s) and (s), characterizing the closed-loop dynamics in the frequency domain. These polynomial matrices can either be chosen arbitrarily (pole-placement problem) or they may be derived as solutions from the corresponding optimal control or the optimal estimation problems, respectively. Starting from the polynomial matrices [Dtilde](s) and (s), the doubly coprime fractional representations of the system and of the compensator transfer matrices can be computed directly without recurrence to the time domain results. From these representations, the right and left coprime compensator matrix fraction descriptions can easily be evaluated using standard algorithms.

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