Mixed LMI/randomized methods for static output feedback control design

Abstract

This paper addresses the problem of stabilization of LTI systems via static output feedback (sof). The objective is not only to compute a stabilizing sof but rather to compute a discrete set of stabilizing sof. Two complementary mixed LMI/randomized algorithms are defined for this purpose. The main idea is to combine a particular relaxed LMI parametrization of stabilizing sof with high efficiency of Hit-and-Run method for generating random points in a given domain. Their respective relevance is analysed on several examples of the COMPleib library which is intended to be the reference library for evaluating performance of reduced-order controller synthesis algorithms. Finally, the paper additionally provides an extensive evaluation of the different relevant instances of the COMPleib library.

Nonsmooth optimization via quasi-Newton methods

“…3, we give an analysis of the Broyden class of quasi-Newton methods on the norm function for n = 2 when the line search is exact. We show that they converge to the origin, spiraling in with a Q-linear rate 1 2 with respect to the number of line searches, independent of the initial Hessian approximation. Numerical evidence indicates that this property extends to n > 2, with a rate of convergence of approximately 1 − 1/ √ 2n.…”

“…Furthermore, it is easy to check that the interval lengths w j = |x k − z j | computed inside the while loop are precisely 2 1− j , a sequence converging to zero with Q-linear rate 1 2 . Since x k and z j have opposite sign within the while loop, we have |z j | < w j , and it follows that the sequence of all function trial values |z j | converges to zero with R-linear rate 1 …”

Fixed-order control of LTI systems subject to polytopic uncertainty via the concept of Strictly Positive Realness

“…The minimum and maximum values of a can be determined by a bisection algorithm. The results of Theorem 1 are compared with ones of [18], [19], [30], [31] in Table I. The proposed methods in Theorem 1 and [18] are both initialized with the same parameter-dependent state feedback K sf (λ).…”

“…As it is reported in [19], since matrix C(λ) is not full row rank, the approaches of [30], [31] are not applicable. Results given in Table I indicate that the proposed method in this paper and [18] lead to the best results among the others.…”

On Application of the Ray-Shooting Method for LQR via Static-Output-Feedback

“…Randomized algorithms are thus natural solutions to this problem. The probabilistic and randomized methods for the constrained SOF problem and robust stabilization via SOFs (among other hard problems) are discussed in [12][13][14][15]. The Ray-Shooting Method was recently introduced in [16], where it was utilized to derive the Ray-Shooting (RS) randomized algorithm for the minimal-gain SOF problem with regional pole-assignment, where the region can be non-convex and unconnected.…”