An Algorithm for Synthesizing Interval Plant Models of Uncertain Systems

All systems contain uncertainties. In the case of electromechanical systems these uncertainties include dimensional tolerances and knowledge of material parameters, e.g. density, only within a given range. The extent to which these tolerances affect the accuracy of a model is difficult to quantify. Beginning with the work of Kharitonov, a large body of theory concerning interval polynomials has been developed. With a model expressed in the form of a ratio of interval polynomials, robust controllers can be designed and system performance information such as the bounds of Nyquist plots and bounds of Bode plots can be readily obtained. To address the problem of evaluating uncertain electromechanical systems, we’ve developed an algorithm that synthesizes an interval plant (a ratio of interval polynomials) of the system. The algorithm consists of three steps: first, a constant coefficient model is synthesized that has necessary and sufficient complexity for a given frequency range of interest. Next, transfer function coefficients are replaced with symbolic design parameters. Finally, constrained optimization routines are employed to determine the maximum and minimum values of the transfer function coefficients. The result is a ratio of interval polynomials, which is the necessary form for the body of theory built around Kharitonov polynomials. The paper includes the algorithm and a demonstration. We conclude that the procedure described here will enable modelers to incorporate real uncertainties into their models and, by having realistic predictions of the bounds of performance, actually have more reliable predictions of system performance. Furthermore, the interval plant models will enable engineers to use results based on interval polynomials.