Linear matrix inequalities for robust control: Theory, algorithms, and applications

Robust control methods, such as {dollar}{lcub}cal H{rcub}sb{lcub}infty{rcub}{dollar}, are interested in the stability and performance of systems with respect to unmodeled dynamics and uncertain parameters. A nominal model and uncertainty descriptions are used to describe a set of plant models to be controlled. Analysis techniques, such as the structured singular value ({dollar}mu{dollar}), provide bounds on the amount of uncertainty allowed in a system to guarantee that desired stability and performance characteristics are achieved.; This thesis formulates {dollar}{lcub}cal H{rcub}sb{lcub}infty{rcub}{dollar} and {dollar}mu{dollar} analysis and synthesis problems in the linear matrix inequality (LMI) framework. LMIs are convex in the free parameters and may be solved using finite dimensional convex optimization algorithms. Two of these algorithms, the Ellipsoid Method and the Method of Centers, are utilized in this thesis. Convergence properties of each algorithm are analyzed.; The structured singular value is difficult to compute, therefore upper and lower bounds are formulated. An upper bound for {dollar}mu{dollar} is formulated as an LMI. Robustness with respect to structured uncertainty is computed as an optimization over structured scaling matrices. Structured uncertainty can be linear time-invariant (LTI) and linear time-varying (LTV) parameters. The {dollar}mu{dollar} upper bound accounts for LTI and LTV uncertainty elements.; Optimal {dollar}{lcub}cal H{rcub}sb{lcub}infty{rcub}{dollar} controller synthesis is formulated as an LMI for full information (FI) feedback. Full information feedback provides direct measurement of all states and disturbances. Synthesis algorithms are developed with respect to complex uncertainty and extended to directly account for real uncertainty.; Examples are presented to demonstrate the synthesis and analysis algorithms. Optimal missile autopilots are designed for full information feedback and analyzed with respect to complex and real uncertainty. Accounting for real uncertainty reduces conservatism in the controllers. Full information and output feedback controllers are designed for vibration attenuation of a flexible structure to demonstrate a large order control synthesis problem.