Fundamental robustness and performance limits for unstable systems: Applications to magnetic bearings

All physical systems which utilize feedback control are limited in their achievable robustness and performance. The nature and severity of the limitation depends on the properties of the system to be controlled. We define a robustness and performance objective based on the $H\infty$ system norm and investigate the limits of achievable robustness and performance which exist for unstable, linear, time-invariant (LTI) systems internally stabilized by an LTI controller. We review existing fundamental system limits for single input/single output systems as well as multiple input/multiple output systems, modifying the results when necessary to present a broad array of fundamental system limits for unstable systems based on the $H\infty$ norm of the system, sensitivity and complementary sensitivity functions. We then develop bounds for underactuated (single input/multiple output) systems using principles of complex variables and system theory. We demonstrate the effectiveness of the bounds in providing accurate measures of achievable robustness and performance by applying them to several underactuated systems where a known level of robustness and performance is achievable. We then provide an extensive application of these bounds to a single degree-of-freedom magnetic bearing system, illustrating how the bounds may be used in the system design stage to determine an appropriate sensor configuration and to relate physical system parameter values to achievable robustness and performance of the resulting controlled system.; In addition, we develop extensive analytical models for two multiple degree-of-freedom magnetic bearing systems. The models are described symbolically so that the structure which exists in the system will be clearly shown and so that the correlation between physical system parameters and resulting system properties such as transfer function poles and zeros can be seen. We then evaluate the system frequency response for the particular system under consideration and compare it to an estimated frequency response derived from swept-sine data taken from the system in closed-loop. Fitting a multiple input/multiple output transfer function to the swept-sine data allows us to compare system properties such as the locations of unstable poles, non-minimum phase zeros, resonances, etc. of the identified model to those of the analytical model. This comparison provides a means for validating both the analytical model and the system estimated frequency response. Based on this comparison, we make recommendations for improvements to both the analytical modeling and the system identification.