LMI relaxation to Riccati equations in structured control

Abstract

In this paper we discuss structured H2 control methods for large-scale interconnected systems. Based on a relaxation of Riccati equations, we derive some linear matrix inequality (LMI) conditions for suboptimal controllers in which information structure can be imposed. In particular, we derive controllers by solving low-dimensional LMIs, which are decentralized except for the sharing information between neighbors, as determined by the plant interconnection; also we optimize a performance bound for each of the derived controllers.

On common solutions of Riccati inequalities

“…Corollary 2 The Riccati inequality (12) has positive definite solutions, if and only if there exist T -parameters and X-parameters such that the set of all the X-parameters is a positive cube in R n−1 whose lower and upper limits are described by (14) and (15).…”

An investigation of active structural acoustic control in resonant enclosures

“…What would be helpful is if the strict equality of this ARE could be relaxed and molded into an LMI framework. It can be shown and was in [71] that this can be done an that the solution of the general Riccati (A T P + P A T + C T C − P BB T P = 0) may be found by relaxing the problem to the following LMI's given in equation 2.63…”

Distributed controller synthesis and decision making

“…In other related work, (10; 11) distributed spatially invariant systems are studied using a state space approach and a convex method based on solving constraints in the form of LMIs is presented to obtain structured controllers. In (12), a heuristic method for structured H 2 controllers is presented based on low dimensional LMIs. These efforts do not consider the effect of uncertainty affecting the communication between different sub-controllers.…”

“…12) where y i (k) is the node-value of i th for minimum consensus protocol. Each node updates its value to the present minimum value in its neighborhood.…”