In many real-world control problems, the plant dynamics are dictated not only by the system being controlled, but are also shaped by the properties (parameters and dynamics) of the surrounding environment. While the properties of the controlled system, are normally well characterized at the time of controller design, the environments properties, which can have a significant impact on the overall plant dynamics, may only be known a posteriori. This scenario is seen in many active vibration control problems where the dynamics of the support structure, on which the control system sensors and actuators are mounted, influence the effectiveness of the control system. We have developed a novel approach to this problem, which is called a posteriori gain-scheduling. Under this approach, a controller is designed which is gain-scheduled on the unknown environmental properties, and at installation time, these properties are identified, and entered into the gain-scheduled controller. We have shown that these controllers are able to meet demanding performance specifications for systems in mass production, whose environment properties vary from one system to the next, without the need to re-tune or re-design controllers for every system.
Recently developed implicit gain-scheduling techniques have been extended for the design of a posteriori gain-scheduled controllers. Unlike traditional gain-scheduling, these controllers are continuously gain-scheduled, deliver guaranteed performance for the entire range of the operating parameters, have compact functional descriptions, and do not suffer from interpolation hazards. The popular D-K (D,G-K) iteration methods have also been modified for designing implicit gain-scheduled controllers, and the resulting controllers have a linear fractional dependence (LFT) on the gain-scheduling parameters. In addition, the recently developed linear parameter-varying (LPV) design techniques based on linear matrix inequalities (LMIs) have been modified for time-invariant parameters and have exhibited nearly optimal performance on a benchmark problem. The need for robustness with respect to parameter estimation errors in LPV controllers has also been examined and a second scaling matrix, which reduces conservatism for parametric uncertainties, has been incorporated to yield a new robust LPV synthesis algorithm.