Biblio

This paper considers a decentralized switched control problem where exact conditions for controller synthesis are obtained in the form of semidefinite programming (SDP). The formulation involves a discrete-time switched linear plant that has a nested structure, and whose system matrices switch between a finite number of values according to finite-state automation. The goal of this paper is to synthesize a commensurately nested switched controller to achieve a desired level of 2-induced norm performance. The nested structures of both plant and controller are characterized by block lower-triangular system matrices. For this setup, exact conditions are provided for the existence of a finite path-dependent synthesis. These include conditions for the completion of scaling matrices obtained through an extended matrix completion lemma.When individual controller dimensions are chosen at least as large as the plant, these conditions reduce to a set of linear matrix inequalities. The completion lemma also provides an algorithm to complete closed-loop scaling matrices, leading to inequalities for  ontroller synthesis that are solvable either algebraically or numerically through SDP.

This paper is concerned with the tradeoffs between low-cost heterogenous designs and optimality. We study a class of constrained myopic strategic games on networks which approximate the solutions to a constrained quadratic optimization problem; the Nash equilibria of these games can be found using best-response dynamical systems, which only use local information. The notion of price of heterogeneity captures the quality of our approximations. This notion relies on the structure and the strength of the interconnections between agents. We study the stability properties of these dynamical systems and demonstrate their complex characteristics, including abundance of equilibria on graphs with high sparsity and heterogeneity. We also introduce the novel notions of social equivalence and social dominance, and show some of their interesting implications, including their correspondence to consensus. Finally, using a classical result of Hirsch [1], we fully characterize the stability of these dynamical systems for the case of star graphs with asymmetric interactions. Various examples illustrate our results.