Insights on reduced order, H2 optimal controller design methods

There are three paths for designing reduced order controllers: 1) model reduction followed by full-order design, 2) full-order design followed by controller reduction, and 3) direct reduced-order controller design. In general, path 1 does not yield the quality of controllers obtainable using paths 2 and 3; and path 2 does not necessarily yield optimal controllers. Path 3 usually uses descent methods to compute the controllers; the optimality of the resulting controllers depends on the initialization of these decent methods. Usually, controllers resulting from path 2 is used as initializations for path 3. Criteria for choosing path 2 methods for initialization descent methods of path 3 to yield low cost (preferably optimal) controller are not known. Since it is very difficult to establish an analytical solution to this problem, this paper seeks to supply reasonable answers via numerical experimentation. Three reduced order control design problems that appear in the existing literature are investigated. It is seen that because of the prevalence of numerous local minima the best results are not always obtained by initializing a quasi-Newton algorithm of path 3 with a low cost controller. In addition, one of the simplest indirect methods, the modified balanced truncation, often results in the best "optimal" controllers.