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Observability and related structural results for linear hereditary systems. (English) Zbl 0531.93015
The class of systems considered in this paper is assumed to be modeled by the equations $$\dot x(t)=A(d)x(t)+B(d)u(t),\quad y(t)=C(d)x(t)$$ where A(d), B(d), and C(d) are polynomial matrices in the delay operator d with the property that $$dx(t)=x(t-h),$$ where $$h>0$$ is the delay duration.
For this class of systems a number of different concepts of observability and observers are discussed. The paper successfully unifies results obtained by applying three different approaches towards the observability problem for linear time-invariant systems with delays. Thus the results which arose from a purely algebraic approach, results which were generated by an application of functional analysis techniques, and results which arose from the abstract semigroup approach are nicely bridged together in this paper. A natural consequence of the analysis of the observability problem is the observer design methodology. Design algorithms for different types of observers for time delay systems are developed. These are: finite-time observers, asymptotic observers, and optimal observers. The results have been established only for the case of systems with commensurable delays; however, most of the results of this paper can be generalized for systems with non-commensurable delays and for systems with distributed delays.
Reviewer: S.H.Żak

##### MSC:
 93B07 Observability 34K35 Control problems for functional-differential equations 93B25 Algebraic methods 34K30 Functional-differential equations in abstract spaces 93C05 Linear systems in control theory 93C25 Control/observation systems in abstract spaces
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