Randomized algorithms for probabilistic robustness with real and complex structured uncertainty

Abstract

Abstract-In recent years, there has been a growing interest in developing randomized algorithms for probabilistic robustness of uncertain control systems. Unlike classical worst case methods, these algorithms provide probabilistic estimates assessing, for instance, if a certain design specification is met with a given probability. One of the advantages of this approach is that the robustness margins can be often increased by a considerable amount, at the expense of a small risk. In this sense, randomized algorithms may be used by the control engineer together with standard worst case methods to obtain additional useful information.The applicability of these probabilistic methods to robust control is presently limited by the fact that the sample generation is feasible only in very special cases which include systems affected by real parametric uncertainty bounded in rectangles or spheres. Sampling in more general uncertainty sets is generally performed through overbounding, at the expense of an exponential rejection rate.In this paper, randomized algorithms for stability and performance of linear time invariant uncertain systems described by a general -1 configuration are studied. In particular, efficient polynomial-time algorithms for uncertainty structures 1 consisting of an arbitrary number of full complex blocks and uncertain parameters are developed.

A probabilistic framework for problems with real structured uncertainty in systems and control

“…However, the results in this paper hold not only for the uniform distribution, but for the more general class of unitarily invariant distributions over B , see (Calaÿore et al, 2000a). The independence assumption reduces the problem of generating samples of uncertainty in the set to the problem of generating a single block in the set B .…”

“…The relation among the uniform pdf of and the pdfs of U; ; V was studied in Calaÿore et al (2000a). That key result is reported below for easier reference of the reader.…”

“…Due to dimensionality problems, classical generation methods based on rejection of samples from an outer bounding set (Devroye, 1986) are highly ine cient for the problem at hand, as already remarked in Calaÿore et al (2000a) for the case of complex matrices. Indeed, the rejection rate -deÿned as the expected number of samples that should be generated in the outer bounding set in order to ÿnd one sample in the desired set-increases exponentially with the dimension n. Let for instance ∈ R n; n , and consider the sets A F :…”

“…In Table 1, we report the rejection rates for samples uniformly generated in the above deÿned sets, computed as the ratio between the volumes of the outer bounding and the desired sets. The required formulas for the volumes have been derived in Calaÿore et al (1999aCalaÿore et al ( , 2000a. Table 1 shows the ine ciency of methods based on rejection, due to the "curse of dimensionality", and motivates the need for more sophisticated techniques for direct generation of samples in B 1 .…”

“…The diculty of this sample generation problem may be easily overlooked, therefore a brief discussion on the failing of ÿrst-attempt techniques is reported in Section 2. The uncertainty sample generation problem for the case of complex matrix blocks was studied in depth in Calaÿore, Dabbene, and Tempo (2000a), while the case of vector uncertainty has been treated in Calaÿore, Dabbene, and Tempo (1999a). However, the sample generation problem in the case of real matrix blocks raised further technical di culties that could not be overcome within the technical framework introduced in Calaÿore et al (2000a), and was therefore left as an open problem.…”