Set approximation via minimum-volume polynomial sublevel sets

Motivated by problems of uncertainty propagation and robust estimation we are interested in computing a polynomial sublevel set of fixed degree and minimum volume that contains a given semialgebraic set K. At this level of generality this problem is not tractable, even though it becomes convex e.g. when restricted to nonnegative homogeneous polynomials. Our contribution is to describe and justify a tractable L1-norm or trace heuristic for this problem, relying upon hierarchies of linear matrix inequality (LMI) relaxations when K is semialgebraic, and simplifying to linear constraints when K is a collection of samples, a discrete union of points.

[1]  M. B. Zarrop.  Robustness in Identification and Control , 1991 .

[2]  Aharon Ben-Tal,et al.  Lectures on modern convex optimization , 1987 .

[3]  Didier Henrion,et al.  Convex inner approximations of nonconvex semialgebraic sets applied to fixed-order controller design , 2012, Int. J. Control.

[4]  J. Shamma,et al.  Approximate set-valued observers for nonlinear systems , 1997, IEEE Trans. Autom. Control..

[5]  Jean B. Lasserre,et al.  Level Sets and NonGaussian Integrals of Positively Homogeneous Functions , 2011, IGTR.

[6]  R. Kannan,et al.  Convex Sets and their Applications , 2006 .

[7]  Giuseppe Carlo Calafiore,et al.  Robust filtering for discrete-time systems with bounded noise and parametric uncertainty , 2001, IEEE Trans. Autom. Control..

[8]  Didier Henrion,et al.  Approximate Volume and Integration for Basic Semialgebraic Sets , 2009, SIAM Rev..

[9]  Dario Piga,et al.  Polytopic outer approximations of semialgebraic sets , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[10]  Jirí Vlach,et al.  Ellipsoidal method for design centering and yield estimation , 1993, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[11]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

[12]  Leonidas J. Guibas,et al.  Zonotopes as bounding volumes , 2003, SODA '03.

[13]  Vicenç Puig,et al.  Robust fault detection using zonotope‐based set‐membership consistency test , 2009 .

[14]  Inseok Hwang,et al.  Applications of polytopic approximations of reachable sets to linear dynamic games and a class of nonlinear systems , 2003, Proceedings of the 2003 American Control Conference, 2003..

[15]  Eduardo F. Camacho,et al.  Guaranteed state estimation by zonotopes , 2005, Autom..

[16]  Pravin Varaiya,et al.  Ellipsoidal Techniques for Reachability Analysis , 2000, HSCC.

[17]  A. Magnani,et al.  Tractable fitting with convex polynomials via sum-of-squares , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[18]  Constantino M. Lagoa,et al.  On the complexity of randomized approximations of nonconvex sets , 2010, 2010 IEEE International Symposium on Computer-Aided Control System Design.

[19]  Eric Walter,et al.  Trace Versus Determinant in Ellipsoidal Outer-Bounding, with Application to State Estimation , 1996 .

[20]  A. Morozov,et al.  New and old results in resultant theory , 2009, 0911.5278.

[21]  Boris Polyak,et al.  Multi-Input Multi-Output Ellipsoidal State Bounding , 2001 .

[22]  P. McMullen.  Convex Sets and Their Applications , 1982 .

[23]  R. Ash,et al.  Probability and measure theory , 1999 .