A rational function whose denominator polynomial has no zeros inside the unit circle, is called a generalized polynomial, and has properties similar to a polynomial. The ring of generalized polynomials is denoted by
R∑(σ). A generalized polynomial matrix has the
∑-Smith form similar to the Smith form of a polynomial matrix. Linear neutral time-delay systems can be represented as systems over the field of rational functions whose argument is σ, where σ is a fixed delay operator.
R∑(σ)-controllability which is similar to
R[σ]-controllability for a time-delay system of the retarded type, is defined. Theorem 1 gives a necessary and sufficient condition for
R∑(σ)-controllability. This condition is represented by the
∑-Smith form for the controllability matrix.
∑-pole assignability which is the extension of pole assignability for a retarded system and guarantees asymptotic stability of the closed loop system by phiscally realizable state feedback, is defined. Theorem 2 states that
R∑(σ)-controllability is equivalent to
∑-pole assignability. Theorem 3 gives a sufficient condition for decoupling by physically realizable state feedback. Finally, it is shown that under the condition of Theorem 3 and
R∑(σ)-controllability, there exists a differential-difference compensator of the neutral type to achieve decoupling and arbitrary
∑-pole assignment.
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