ON THE STRONG RELATIVE DEGREE OF TIME-DELAY SYSTEMS WITH NONCOMMENSURATE DELAYS

The fundamental notion of relative degree is studied for a class of linear time-delay systems of retarded type, where the standard assumption of commensurate delays is dropped. Algebraic conditions are provided that fully exploit the delay dependence structure. Even though the analyzed systems do not have a nontrivial feedthrough from input to output, it is shown that the relative degree may be sensitive to infinitesimal delay perturbations, which is at the basis of a novel notion of relative degree, called strong relative degree. This notion is characterized algebraically and computationally in both a SISO and a MIMO setting. By means of the obtained characterizations and a benchmark problem, which illustrates that invariant zeros may be characterized as zeros of quasi-polynomials of retarded, neutral, or advanced type, new light is shed on existence conditions of a normal form. The novel concepts and theoretical results play a key role in the design and analysis of extended PD controllers, for which an algorithm and numerical experiments are presented.