Controller Synthesis Subject to Structure and Order Constraints

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The principal aim of the research outlined in this proposal is to develop a systematic theory for the design of fixed order controllers. Specifically, we plan to focus on characterizing the set of all controllers of a prescribed order that stabilize a given plant and then optimize various design criteria over that stabilizing set. The above problem which is motivated from our earlier research and recent interactions with industry, is probably one of the most outstanding unsolved problems in the area of automatic control. Control Theory has witnessed tremendous theoretical advances during the last four decades-the largely adhoc classical design procedures of the 1940's and 50's have given way to the development of elegant control synthesis techniques such as H2, Hoo, L1, and u. However, most of these design methods cannot be applied to many industrial situations, such as in process control, where low order and fixed structure controllers such as PID's are the preferred controllers of choice. This, of course, is not surprising given the fact that the modern approaches for designing optimal and robust controllers cannot incorporate the controller structure and/or order constraints into their design methods. To make matters worse, some of the adhoc design methods currently used in industry, in lieu of the modern approaches, can lead to designs that are dangerously close to instability. Thus there is a very critical need for developing a systematic theory for optimal and robust controller design subject to constraints on the controller order and structure. We believe that the research being proposed here will fulfill this need. Preliminary results, indicating a clear research direction, have already been obtained. Specifically, we have obtained a fundamental result generalizing the classical Hermit-Biehler Theorem. Furthermore, using this result, significant strides have been made in solving some specific cases of the fixed order and structure stabilization and design problems. Based on these, we are confident that the proposed research is very promising and will lead to the development of fundamental theory that is immediately applicable in an industrial setting. In addition to the main thrust outlined above, we also plan to focus on extending our earlier unconstrained, single-input single-output results on adaptive internal model control to the domain of multivariable systems, outlinear systems and systems subject to constraints. This topic is of independent interest, especially to the process control industries. Nevertheless, it fits in very naturally within the broad scope of our projects since internal model control is itself a popular strategy for designing PID controllers.