Partially Decentralized Design Principle in Large-Scale System Control

Abstract

A number of problems that arise in state control can be reduced to a handful of standard convex and quasi-convex problems that involve matrix inequalities. It is known that the optimal solution can be computed by using interior point methods (Nesterov & Nemirovsky (1994)) which converge in polynomial time with respect to the problem size, and efficient interior point algorithms have recently been developed for and further development of algorithms for these standard problems is an area of active research. For this approach, the stability conditions may be expressed in terms of linear matrix inequalities (LMI), which have a notable practical interest due to the existence of powerful numerical solvers. Some progres review in this field can be found e.g. in Boyd et al. (1994), Hermann et al. (2007), Skelton et al. (1998), and the references therein. Over the past decade, H∞ norm theory seems to be one of the most sophisticated frameworks for robust control system design. Based on concept of quadratic stability which attempts to find a quadratic Lyapunov function (LF), H∞ norm computation problem is transferred into a standard LMI optimization task, which includes bounded real lemma (BRL) formulation (Wu et al. (2010)). A number of more or less conservative analysis methods are presented to assess quadratic stability for linear systems using a fixed Lyapunov function. The first version of the BRL presents simple conditions under which a transfer function is contractive on the imaginary axis of the complex variable plain. Using it, it was possible to determine the H∞ norm of a transfer function, and the BRL became a significant element to shown and prove that the existence of feedback controllers (that results in a closed loop transfer matrix having the H∞ norm less than a given upper bound) is equivalent to the existence of solutions of certain LMIs. Linear matrix inequality approach based on convex optimization algorithms is extensively applied to solve the above mentioned problem (Jia (2003), Kozakova & Veselý (2009)), Pipeleers et al. (2009). For time-varying parameters the quadratic stability approach is preferable utilized (see. e.g. Feron et al. (1996)). In this approach a quadratic Lyapunov function is used which is independent of the uncertainty and which guarantees stability for all allowable uncertainty values. Setting Lyapunov function be independent of uncertainties, this approach guarantees uniform asymptotic stability when the parameter is time varying, and, moreover, using a parameter-dependent Lyapunov matrix quadratic stability may be established by LMI tests over the discrete, enumerable and bounded set of the polytope vertices, which define the uncertainty domain. To include these requirements the equivalent LMI representations of 16