The ellipsoid algorithm for probabilistic robust controller design

Abstract

If you want to cite this report, please use the following reference instead:S. Kanev, B. De Schutter, and M. Verhaegen, "The ellipsoid algorithm for probabilistic robust controller design," Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, Nevada, pp. 2248-2253, Dec. 2002 This paper presents a new iterative approach to probabilistic robust controller design, which is an alternative to the recently proposed Subgradient Iteration Algorithm (SIA). In its original version [12] the SIA possesses the useful property of guaranteed convergence in a finite number of iterations, but requires that the radius of a non-empty ball contained in the solution set is known a-priori. This rather restrictive assumption was later on released in [3], but only at the expense of an increased number of iterations. The approach in this paper does also not require the knowledge of such a radius, and offers a significant improvement even over the original SIA in terms of the maximum number of possible correction steps that can be executed before a feasible solution is reached. Given an initial ellipsoid that contains the solution set, the approach iteratively generates a sequence of ellipsoids with decreasing volumes, all containing the solution set. A method for finding an initial ellipsoid containing the solution set is also proposed. The approach is illustrated on a real-life diesel actuator benchmark model.

The Scenario Approach to Robust Control Design

“…This is different from the stochastic sequential methods of [19], [24], [31], [37] that have an asymptotic nature. Moreover, a notable improvement upon the stochastic sequential methods is that our result holds for robust optimization problems and not only for feasibility.…”

“…Alternatively, when the original synthesis problem is convex (which includes many, albeit not all, relevant control problems) the sequential approaches based on stochastic gradients [19], 0018-9286/$20.00 © 2006 IEEE [24], [37] or ellipsoid iterations [31], may be applied with success. However, these methods are currently limited to feasibility problems, and have not yet been extended to deal satisfactorily with optimization.…”

“…However, the probabilistic approach has found to date limited application in robust control synthesis. Basically, two different methodologies are currently available for probabilistic robust control synthesis: the approach based on the Vapnik-Chervonenkis theory of learning [34], [49], [50], and the sequential methods based on stochastic gradient iterations [19], [24], [37] or ellipsoid iterations, [31].…”

A probabilistic analytic center cutting plane method for feasibility of uncertain LMIs

“…In this general setting, no algorithm is known that solves the problem in an exact and efficient way. There are, however, efficient probabilistic algorithms that are guaranteed with high probability to return a solution P 0 that may fail to satisfy the matrix inequalities (1) at most on a subset of D having arbitrarily small probability measure, see for instance Calafiore and Campi (2005), Calafiore and Polyak (2001), Kanev, De Schutter, and Verhaegen (2003), Oishi (2003) and Oishi and Kimura (2003).…”

“…These algorithms were based on an iterative scheme in which the current solution is updated towards a descent direction obtained by a random gradient of a suitable feasibility violation function. Later, improved algorithms for probabilistic feasibility, based on the use of the ellipsoid method (Nemirovski & Yudin, 1983;Shor, 1970), have been developed in Kanev et al (2003) and further analyzed in Oishi (2003).…”