Achieving Minimum Phase Transfer Function for a Noncollocated Single-link Flexible Manipulator

A noncollocated system has the potential of providing more precise tracking, improved disturbance rejection and increased bandwidth at the sensor location, but is considerably more difficult to stabilize than a collocated system due to its nonminimum phase nature. For a flexible manipulator, the problem becomes even more complicated because the system is inherently infinite dimensional. In this paper, a single-link flexible manipulator having a tip payload and a noncollocated actuator and sensor is investigated using a linear distributed parameter model. With the joint torque as the input and the joint angle plus a weighted value of tip deflection as the measured output, an exact transfer function involving transcendental functions is derived. Using the methods of infinite product expansion, the root locus, and the asymptotic property of the roots of the transcendental equation, a necessary and sufficient condition in terms of the weighting factor of tip deflection is obtained such that the transfer function does not have any open right-half plane zeros. This condition depends neither on the physical properties of the link nor on the mass properties of the tip payload. In order to correct the misinterpretation which has occurred in some closely-related works concerning the stability of the infinite-dimensional zero-dynamics, the equivalence of the zeros of the transfer function and the eigenvalues of zero-dynamics is also verified. Numerical results for the closed-loop performance of a single-link flexible arm using collocated and noncollocated PD controllers are given to show the efficacy of the proposed minimum phase transfer function approach.