On the Zeros of Flexible Systems

Abstract

The objective of this thesis is to investigate the genesis of zeros in the single-input single-output (SISO) transfer function of flexible linear time invariant (LTI) systems, and provide necessary and sufficient conditions to specifically guarantee the absence of non-minimum phase zeros. Flexible system dynamics plays a vital role in several motion and vibration control applications such as space structures, rotorcraft blades, hard-disk drives, flexure mechanisms, flexible manipulators, and motion systems with transmission compliance. These applications often require feedback and feedforward controls to achieve desirable dynamic performance, which generally includes high speed, low settling time, effective disturbance rejection, low sensitivity to modeling uncertainties, and stability robustness. Zeros in the transfer function, defined by the actuated load input and sensed displacement output of a flexible system, have a significant impact on its dynamic performance. Non-minimum phase zeros (i.e. zeros in the right half s-plane), in particular, lead to significant tradeoffs among the competing dynamic performance requirements. Therefore, there is a need for physical design strategies that are informed by mathematical conditions to guarantee the absence of non-minimum phase zeros. Comprehensive and precise mathematical conditions do not currently exist in the literature. A well-known mathematical condition states that when all the modal residue signs of an undamped flexible LTI system are the same, the zeros of that system are guaranteed to be minimum phase. However, the same sign of all modal residues is a sufficient condition and not a necessary one. Furthermore, it may not always be possible to achieve the same sign of all modal residues given various practical constraints on the distribution of mass and stiffness and location of actuators and sensors. Apart from mass-stiffness distribution and actuator-sensor placement, one can also explore the use of viscous damping to change the position of zeros. Viscous damping is generally found to be beneficial for the poles of the flexible system because it moves them to the left-hand side of the s-plane leading to smaller overshoot and residual vibration. However, the effect of viscous damping on the zeros has not been adequately investigated in the existing literature and therefore, there does not exist any physical design strategy where viscous damping is used in a deterministic manner to guarantee the absence of non-minimum phase zeros in the transfer functions of flexible LTI systems. In order to fill these various technical gaps, this thesis makes three key contributions: (i) create a mathematical and graphical framework to explore the necessary and sufficient conditions for the occurrence of different types of zeros in the transfer function of flexible LTI systems, with and without viscous damping; (ii) derive the necessary and/or sufficient conditions to guarantee the absence of non-minimum phase zeros for various flexible LTI systems with and without viscous damping; and, (iii) implement design strategies informed by the above mathematical conditions to demonstrate the absence of non-minimum phase zeros with and without viscous damping. The practical utility (from a physical system design standpoint) of the sufficient conditions derived for undamped and viscous damped flexible systems with any arbitrary number of DoFs (modes) is demonstrated in multiple case studies by making informed design choices of physical parameters such as actuator-sensor placement, mass-stiffness distribution and viscous damping strategy that satisfy these sufficient conditions for different flexible systems.