This thesis considers the analysis of systems with uncertainties and the design of controllers to such systems. Uncertainties are treated in a relatively broad sense covering gain-bounded elements that are not known a priori but could be available to the controller in real time.
The uncertainties are in the most general case norm-bounded operators with a given block-diagonal structure. The structure includes parameters, linear time-invariant and time-varying systems as well as nonlinearities. In some applications the controller may have access to the uncertainty, e.g. a parameter that depends on some known condition.
There exist well-known methods for determining stability of systems subject to uncertainties. This thesis is within the framework for structured singular values also denoted by μ. Given a certain class of uncertainties, μ is the inverse of the size of the smallest uncertainty that causes the system to become unstable. Thus, μ is a measure of the system's "structured gain". In general it is not possible to compute μ exactly, but an upper bound can be determined using efficient numerical methods based on linear matrix inequalities.
An essential contribution in this thesis is a new synthesis algorithm for finding controllers when parametric (real) uncertainties are present. This extends previous results on μ synthesis involving dynamic (complex) uncertainties. Specifically, we can design gain scheduling controllers using the new μ synthesis theorem, with less conservativeness than previous methods. Also, algorithms for model reduction of uncertainty systems are given.
A gain scheduling controller is a linear regulator whose parameters are changed as a function of the varying operating conditions. By treating nonlinearities as uncertainties, μ methods can be used in gain scheduling design. In the discussion, emphasis is put on how to take into consideration different characteristics of the time-varying properties of the system to be controlled. Also robustness and its relation with gain scheduling are treated.
In order to handle systems with time-invariant uncertainties, both linear systems and constant parameters, a set of scalings and multipliers are introduced. These are matched to the properties of the uncertainties. Also, multipliers for treating uncertainties that are slowly varying, such that the rate of change is bounded, are introduced. Using these multipliers the applicability of the analysis and synthesis results are greatly extended.