Numerical Design of Feedback Systems with Backlash for Inputs Restricted in Magnitude and Slope

Abstract

This thesis develops a practical method for designing unity feedback systems where a linear time-invariant (either lumped- or distributed- parameter) plant is in cascade connection with a backlash and a controller. The problem considered is to design a controller so as to ensure that the error and the controller output stay within prescribed bounds for all time and for all inputs satisfying given bounding conditions. The thesis comprises two main parts. Part 1 investigates the BIBO stability of the system subject to two different classes of inputs; the first class characterizes a type of transient signals while the other class is appropriate for characterizing persistent signals. In Part 2, since the original design criteria are computationally intractable, we derive a practical condition in the form of inequalities that can be solved by numerical methods so as to determine the controller for two cases in which the plant is fixed and in which the plant parameters have uncertainties, respectively. In essence, the backlash is replaced with a constant gain and a bounded disturbance, thereby resulting in an auxiliary linear system. By applying the multi-valued version of the fixed-point theorem, we have shown that if a controller satisfies such inequalities for the associated linear system, then it is also a solution of the original design problem. Numerical examples are provided in order to illustrate the usefulness of the method developed here.