Relaxations of parameterized LMIs with control applications

Abstract

SUMMARYA wide variety of problems in control system theory fall within the class of parameterized Linear Matrix Inequalities (LMIs), that is, LMIs whose coefficients are functions of a parameter confined to a compact set. However, in contrast to LMIs, parameterized LMI (PLMIs) feasibility problems involve infinitely many LMIs hence are very hard to solve.In this paper, we propose several effective relaxation techniques to replace PLMIs by a finite set of LMIs. The resulting relaxed feasibility problems thus become convex and hence can be solved by very efficient interior point methods. Applications of these techniques to different problems such as robustness analysis, or Linear Parameter-Varying (LPV) control are then thoroughly discussed and illustrated by examples.

A Survey of Linear Parameter-Varying Control Applications Validated by Experiments or High-Fidelity Simulations

“…In [57], the multiconvexity approach is explored, extended to polynomial parameter-dependent LMIs (PLMIs) and relaxed by incorporating slack variables. Other approaches are convex approximations and so-called difference convex representations [58]. The sum-of-squares (SOSs) approach [59], aims at providing the designer with a degree-offreedom to trade conservatism due to the relaxation of PLMIs versus computational complexity.…”

Parameterized linear matrix inequality techniques in fuzzy control system design

“…For it is obvious that one such convex (LMI) relaxation for (14) is obtained as [15] (16) Unfortunately, conditions (16) are practically very restrictive and some potential improvements have been discussed in [14]- [16]. Other convex relaxations techniques solving a general PLMI including (14) as a particular case have been proposed in [17] and [3].…”

L2-Gain Analysis and Control Design of Linear Systems with Input Saturation using Matrix Sum of Squares Relaxation

“…The optimization scheme is performed for each relaxed problem. (13) and (14) are relaxed to (19) and (20) by using matrix SOS polynomials. If h i (θ) on (14) is θ-affine matrix polynomial, it is possible to consider LMIs on vertices on Ω in stead of matrix SOS relaxation.…”

“…The constraints (15) and (14) are correspond to (17) with m F = 2. In (13), T (θ) with m i=1 α i ≤ 1 is adopted.…”