Robust Bode Methods for Feedback Controller Design of Uncertain Systems

In this work, we introduce several novel approaches to feedback controller design, known collectively as the “Robust Bode” methods, which adapt classical control principles to a modern robust control (H∞) framework. These methods, based on specially modified Bode diagrams extend familiar frequency-domain controller design techniques to linear and nonlinear, single–input/single– output (SISO) and multi–input/multi–output (MIMO) systems with parametric and/or unstructured uncertainties. In particular, we introduce the Contoured Robust Controller Bode (CRCBode) plots which show contours (level-sets) of a robust metric on the Bode magnitude and phase plots of the controller. An iterative loop shaping design procedure is then employed in an attempt to eliminate all intersections of the controller frequency response with certain forbidden regions indicating that a robust stability and performance criteria is satisfied. For SISO systems a robust stability and performance criterion is derived using Nyquist arguments leading to the robust metric used in the construction of the CRCBode plots. For open-loop unstable systems and for non-minimum phase systems the Youla parametrization of all internally stabilizing controllers is used to develop an alternative Robust Bode method (QBode). The Youla parametrization requires the introduction of state-space methods for coprime factorization, and these methods lead naturally to an elegant connection between linear-quadratic Gaussian (LQG) optimal control theory and Robust Bode loop-shaping controller design. Finally, the Robust Bode approach is extended to MIMO systems. Utilizing a matrix norm based robustness metric on the MIMO CRCBode plots allows cross-coupling between all input/output channels to be immediately assessed and accounted for during the design process, making sequential MIMO loop-shaping controller design feasible.