Abstract

This dissertation deals with performance analysis and robust stabilizability verification of large-scale interconnected systems with respect to the class of linear time-invariant (LTI) decentralized controllers. These problems are formulated and tackled in four phases. First, an interconnected system with some unstable decentralized fixed modes (DFM) is considered. It is well-known that there is no stabilizing LTI decentralized controller for such a system; hence, a method is proposed to change the structure of the controller from decentralized to a proper overlapping form, with respect to which the system is stabilizable. This change in the control configuration is carried out by introducing some interactions among the isolated controllers, which leads to the elimination of the undesirable DFMs. The approach utilized in this thesis is based on the graph theory and, in fact, transforms the knowledge of the system into a number of bipartite graphs. A simple combinatorial algorithm is subsequently proposed to address the problem under consideration. The second problem investigated here is the characterization of all classes of LTI structurally constrained controllers with respect to which a given interconnected system has no fixed modes. Similar to the ideas and notions proposed to handle the preceding problem, an efficient method is presented to tackle the problem in this case. Since establishing a transmission link between a pair of local controllers would certainly incur cost, an implementation expenditure is attributed to each possible link. The proposed approach can also be used to attain the implementation cost associated with any suitable class of controllers obtained. As a by-product of this result, all classes of LTI stabilizing structurally constrained controllers with the minimum implementation cost can be characterized accordingly. A LTI structurally constrained control system is considered next, which is subject to parametric uncertainties. Moreover, a region of uncertainty in the form of a semi-algebraic set is envisioned to parametrize the range of variations for uncertain parameters. It is asserted that if the system is stabilizable via a given constrained controller at the nominal point, then it is almost always stabilizable at any operating point in the region of uncertainty. In other words, the points for which the system has some persistent fixed modes lie on an algebraic variety. A method is subsequently proposed to derive this variety. In the end, it is assumed that a stabilizing decentralized controller is designed for a pseudo-hierarchical large-scale system based on its hierarchical approximation. A LQ cost function is defined to evaluate the effectiveness of this indirect controller design for the system. It is shown that a reasonably tight upper bound on this performance index can be straightforwardly obtained by solving a constrained optimization problem with only three variables.