Inherent design limitations of ill-conditioned plants.

In this thesis, we study inherent design limitations imposed by ill-conditioned plants. It has been conjectured that when the gains of a multivariable plant show strong directional dependence (i.e. when the plant is ill-conditioned), satisfaction of the robust performance specifications is difficult and the design difficulty increases with the size of the plant condition number. We prove this conjecture for special classes of systems. First, we assume that the plant is diagonalizable through orthogonal transformations and that the compensator is chosen suitably so that the input and output transfer functions can be reduced to the same diagonal transfer function via orthogonal transformations. The assumptions of diagonalizability are important since classical integral relations such as Bode's sensitivity integral, the complementary sensitivity integral, and Poisson's integral, can be applied to show that the satisfaction of robust performance goals implies certain bounds upon the plant condition number at all frequencies. The upper bounds are functions of important design specifications such as the descriptions of the uncertainties and the crossover gap, and reveal valuable design insights. Through the use of subharmonic function theory and the multivariable versions of the integral relations, the results are then extended by relaxing all of the assumptions on the compensator structure and by assuming only a pair of the plant singular subspaces is constant with frequency. We also show that the satisfaction of the robust performance specs at the low and high frequency intervals implies a lower bound that the structured singular value must satisfy at intermediate frequencies. The value of this lower bound increases as the plant condition number increases and as the uncertainty descriptions become more severe. Finally, we show that, under the assumption of unstructured uncertainty, scaling the plant to reduce its condition number scales up the size of the uncertainty and the overall design difficulty is invariant to the scaling effects.Ph.D.Electrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/103486/1/9319625.pdfDescription of 9319625.pdf : Restricted to UM users only