We investigate the exact and approximate spectrum assignment properties associated with realizable output-feedback pole-placement-type controllers for single-input single-output linear time-invariant time-delay systems with commensurate point delays. The controller synthesis problem is discussed through the solvability of a set of coupled Dio-phantine equations of polynomials. An extra complexity is incorporated in the above design to cancel extra unsuitable dynamics being generated when solving the above Dio-phantine equations. Thus, the complete controller tracks any arbitrary prefixed (either finite or delay-dependent) closed-loop spectrum. However, if the controller is simplified by deleting the above-mentioned extra complexity, then robust stability and approximated spectrum assignment are still achievable for a certain sufficiently small amount of delayed dynamics. Finally, the approximate spectrum assignment and robust stability problems are revisited under plant disturbances if the nominal controller is maintained. In the current approach, the finite spectrum assignment is only considered as a particular case of the designer's choice of a (delay-dependent) arbitrary spectrum assignment objective.
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