Non-minimum Phase Zeros in Multi-DoF Undamped Flexible Systems

This paper investigates non-minimum phase zeros (NMP) in the transfer function, between actuated load input and measured position output, of multi degree of freedom (DoF) undamped flexible systems. The transfer function of an undamped flexible system is modally decomposed into second order modes where each mode is characterized by its modal residue and modal frequency. It is well known that when all the modal residue signs are the same, all the zeros are minimum phase. However, it is not always possible to guarantee the same sign for all modal residues, given practical constraints on actuator and sensor placement. This paper presents a new sufficient condition for the absence of NMP zeros when all modal residue signs are not the same. First, a sufficient condition for the absence of only complex NMP zeros is derived in terms of the sequence of modal residue signs. Once this sufficient condition is obtained, the sufficient condition for the absence of all NMP zeros, i.e. complex as well as real NMP zeros, is derived. Its efficacy is then demonstrated theoretically and experimentally via two case studies. They show that the sufficient condition allows a large design space for the choice of physical parameters, is mathematically easy to use, is robust to parametric variations and modeling uncertainty, and is applicable even in the presence of small amount of damping. These attributes make the sufficient condition useful in several motion control applications where the dynamic performance is limited by the presence of NMP zeros.