Ball J.A., Fang Q., Groenewald G.J., ter Horst S.
Department of Mathematics, Virginia Tech., Blacksburg, VA 24061-0123, United States; Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260-2900, United States; Department of Mathematics, North West University, Potchefstroom 2520, South Africa
Ball, J.A., Department of Mathematics, Virginia Tech., Blacksburg, VA 24061-0123, United States; Fang, Q., Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260-2900, United States; Groenewald, G.J., Department of Mathematics, North West University, Potchefstroom 2520, South Africa; ter Horst, S., Department of Mathematics, Virginia Tech., Blacksburg, VA 24061-0123, United States
One approach to robust control for linear plants with structured uncertainty as well as for linear parameter-varying plants (where the controller has on-line access to the varying plant parameters) is through linear-fractional-transformation models. Control issues to be addressed by controller design in this formalism include robust stability and robust performance. Here robust performance is defined as the achievement of a uniform specified L 2-gain tolerance for a disturbance-to-error map combined with robust stability. By setting the disturbance and error channels equal to zero, it is clear that any criterion for robust performance also produces a criterion for robust stability. Counter-intuitively, as a consequence of the so-called Main Loop Theorem, application of a result on robust stability to a feedback configuration with an artificial full-block uncertainty operator added in feedback connection between the error and disturbance signals produces a result on robust performance. The main result here is that this performance-to-stabilization reduction principle must be handled with care for the case of dynamic feedback compensation: casual application of this principle leads to the solution of a physically uninteresting problem, where the controller is assumed to have access to the states in the artificially-added feedback loop. Application of the principle using a known more refined dynamic-control robust stability criterion, where the user is allowed to specify controller partial-state dimensions, leads to correct robust-performance results. These latter results involve rank conditions in addition to linear matrix inequality conditions. © Springer-Verlag London Limited 2009.
Linear fractional transformations; Multidimensional linear systems; Output feedback; Robust performance; Robust stabilization; Access control; Applications; Block codes; Controllers; Delay control systems; Feedback; Fuzzy control; Linear control systems; Mathematical operators; Mathematical transformations; Robust control; Robustness (control systems); Speed control; Stability criteria; Stabilization; Switching systems; System stability; Uncertain systems; Linear matrix inequalities