A Jacobian singularity based robust controller design for structured uncertainty
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Any real system will have differences between the mathematical model response and the response of the true system it represents. These differences can come from external disturbances, incomplete modeling of the dynamics (unstructured uncertainty) or simply incorrect or changing parameter values (structured uncertainty) in the model. Sources of unstructured uncertainty are unavoidable in real systems, so a controller design must always consider robustness to these effects. In many cases, when the sources of structured uncertainty are addressed as another source of unstructured uncertainty, the resulting controller design is conservative. By accurately addressing and designing a controller for the structured uncertainty, large benefits in the controller performance can be generated since the conservative bound is reduced. The classical approach to output shaping of a system involves a feedback loop since this architecture is more robust to differences between the mathematical model and the true system. This dissertation will present an approach to design a feedback controller which is robust to structured uncertainties in a plant, in an accurate and minimal way. The approach begins by identifying a critical set of system parameters which will be proven to represent the full set of system parameters in the Nyquist plane. This critical set is populated by all parameter vectors which satisfy a developed deficiency condition and is the minimal set which will contribute to the Nyquist plane portraits. The invariance of this critical set to control structure is shown explicitly. An improvement of previous work is the addition of a numerical solution technique which guarantees that all critical points are found. The presented approach will allow for the designer set minimum relative stability margins, such as gain and phase margins, which previous work could not compute accurately or with confidence of the results. A robust controller is designed with respect to this critical set. The presented technique will yield a set of controller gains which will meet the desired performance criteria, allowing the designer to select gains from this set to account for qualitative analysis results. The advantages of the presented technique are highlighted by a direct comparison to previous work. These advantages are also shown through a series of examples. The final example, which is a helicopter in hover, shows that a system with nonlinear coefficient dependencies can be accurately addressed by the current technique considering only a minimal set of system parameters, which cannot be done accurately by the previous work.