Safe approximations of chance constrained sets by probabilistic scaling

Motivated by problems arising in robust control, we develop a sampling-based methodology to obtain an inner approximation of the region of the design space that satisfies a given set of probabilistic constraints (chance constrained set). Given a set of manageable complexity centered at a point satisfying the probabilistic constraints, we show how to scale it around its center to obtain, with a user defined probability, a region that is included in the chance constrained set. The proposed approach does not require any assumption on the Vapnik-Chervonenkis dimension of the robust problem under consideration. Moreover, its sample complexity does not depend on the dimension of the design space. We show the advantages of the proposed approach by means of an illustrative example.

[1]  Eduardo F. Camacho,et al.  Improved sample size bounds for probabilistic robust control design: A pack-based strategy , 2007, 2007 46th IEEE Conference on Decision and Control.

[2]  Giuseppe Carlo Calafiore,et al.  Research on probabilistic methods for control system design , 2011, Autom..

[3]  Frank Allgöwer,et al.  A general sampling-based SMPC approach to spacecraft proximity operations , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[4]  B. Arnold,et al.  A first course in order statistics , 1994 .

[5]  Vladimir Vapnik,et al.  Statistical learning theory , 1998 .

[6]  Marco C. Campi,et al.  A Sampling-and-Discarding Approach to Chance-Constrained Optimization: Feasibility and Optimality , 2011, J. Optim. Theory Appl..

[7]  Yasuaki Oishi,et al.  Polynomial-time algorithms for probabilistic solutions of parameter-dependent linear matrix inequalities , 2007, Autom..

[8]  Eduardo F. Camacho,et al.  Randomized Strategies for Probabilistic Solutions of Uncertain Feasibility and Optimization Problems , 2009, IEEE Transactions on Automatic Control.

[9]  Eduardo F. Camacho,et al.  Robust Design Through Probabilistic Maximization , 2018 .

[10]  Lorenzo Fagiano,et al.  Randomized Solutions to Convex Programs with Multiple Chance Constraints , 2012, SIAM J. Optim..

[11]  Giuseppe Carlo Calafiore,et al.  Random Convex Programs , 2010, SIAM J. Optim..

[12]  Roberto Tempo,et al.  Randomized methods for design of uncertain systems: Sample complexity and sequential algorithms , 2013, Autom..

[13]  John Lygeros,et al.  On the Road Between Robust Optimization and the Scenario Approach for Chance Constrained Optimization Problems , 2014, IEEE Transactions on Automatic Control.

[14]  Mohammad Ahsanullah,et al.  An Introduction to Order Statistics , 2013 .

[15]  R. Tempo,et al.  Randomized Algorithms for Analysis and Control of Uncertain Systems , 2004 .

[16]  B. Arnold,et al.  A first course in order statistics , 2008 .

[17]  Frank Allgöwer,et al.  Stochastic MPC with offline uncertainty sampling , 2016, Autom..

[18]  Mathukumalli Vidyasagar,et al.  A Theory of Learning and Generalization: With Applications to Neural Networks and Control Systems , 1997 .

[19]  Qing-Guo Wang,et al.  Sequential Randomized Algorithms for Convex Optimization in the Presence of Uncertainty , 2013, IEEE Transactions on Automatic Control.

[20]  John Lygeros,et al.  Stochastic Model Predictive Control using a combination of randomized and robust optimization , 2013, 52nd IEEE Conference on Decision and Control.