Probabilistic Robustness Analysis—Risks, Complexity, and Algorithms

Abstract

It is becoming increasingly apparent that probabilistic approaches can overcome conservatism and computational complexity of the classical worstcase deterministic framework and may lead to designs that are actually safer. In this paper we argue that a comprehensive probabilistic robustness analysis requires a detailed evaluation of the robustness function and we show that such evaluation can be performed with essentially any desired accuracy and confidence using algorithms with complexity linear in the dimension of the uncertainty space. Moreover, we show that the average memory requirements of such algorithms are absolutely bounded and well within the capabilities of today's computers.In addition to efficiency, our approach permits control over statistical sampling error and the error due to discretization of the uncertainty radius. For a specific level of tolerance of the discretization error, our techniques provide an efficiency improvement upon conventional methods which is inversely proportional to the accuracy level; i.e., our algorithms get better as the demands for accuracy increase.

A new statistical approach for the analysis of uncertain systems

“…By letting n k be the total number of simulations on the direction associated with directional sample U k , k = 1, 2, · · · , N and applying Theorem 1 of [7] and Theorem 5 in this paper to a sample reuse process conditioned upon a direction with grid points r 1 , r 2 , · · · , r m and sample size N = 1, we have E …”

“…where we have used the technique of integration by part in (6) and the fact that P(ρ) is continuous for any ρ > 0 in (7). ∈ (a, b).…”

“…Aimed at overcoming the computational complexity and conservatism of the classical deterministic worst-cast approach, there are growing interests in developing probabilistic methods and randomized algorithms (see, [1][2][3][4][5][6][7][8][9][10][11] and the references therein). Specially, a probabilistic robustness measure, referred to as the confidence degradation function or robustness function is proposed in [3].…”

A statistical approach for performance analysis of uncertain systems

“…For completeness of argument, we need to quote a general complexity result established in [7] as Theorem 5 at below. This theorem concerns the sampling complexity of the Sample Reuse Algorithm proposed in page 1963 of [5].…”

Robustness of feedback stabilization of quasi non-integrable Hamiltonian systems with parametric uncertainty

“…The classic robust control theory was developed in the deterministic worst-case framework and many useful techniques have been proposed to deal with the robustness analysis, such as the Kharitonov theorem (Kharitonov, 1978), the edge theorem (Barlett et al, 1988), structured singular value m analysis (Doyle, 1982;Iorga et al, 2009), the gap metric (Foias et al, 1993;Georgiou and Smith, 1997) and so on. Recently the probabilistic robustness analysis framework has received growing attention (Chen et al, 2008;Field et al, 1996;Ray and Stengel, 1993;Stengel and Ray, 1991;Tempo et al, 1997). In these researches, the guaranteed probability of the robustness requirement is the central problem and generally evaluated by Monte Carlo simulation or other randomized algorithms.…”