|CORARC - Convex Optimization and Randomized Algorithms for Robust Control|
Relaxation Approaches for Control of Uncertain Complex Systems: Methodologies and Tools
Workshop at 52nd IEEE Conference on Decision and Control - Monday December 9, 2013, Florence
The main objective of this workshop is to present recent developments and software tools in the areas of deterministic and probabilistic relaxation techniques for control of uncertain complex systems.
After a tutorial introduction, the first part of the workshop discusses two successful paradigms, which are focused on polynomial optimization techniques and probabilistic randomized methods, respectively. These methodologies have been developed by researchers with diverse expertise. One of the objectives of this workshop is therefore to bring together two research communities, with the common objective to handle very general classes of uncertain systems by means of various software tools currently available.
The second part of the workshop is devoted to the description of the software tools GloptiPoly (Global optimization over polynomials), RoMulOC (Robust Multi-Objective Control) and RACT (Randomized Algorithms Control Toolbox). In particular, it will be demonstrated how classical control problems subject to ``difficult'' uncertainty structures can be effectively resolved with the techniques previously discussed. Finally, the efforts regarding integration of these tools into a unified package for control of uncertain systems subject to general classes of uncertainty will be described.
CNR-CNRS bilateral cooperation
This workshop is organized in the frame of the international project CORARC (Convex Optimization and Randomized Algorithms for Robust Control). The financial support of CNRS of France and CNR of Italy is gratefully acknowledged.
Participating to the workshop
The workshop is devoted in particular to researchers, engineers and students working on control of uncertain complex systems, optimization and probabilistic methods, and related applications.
Participants of the workshop will be invited to experiment these toolboxes on an satellite example. The considered satellite is named DEMETER and is one of the MYRIADE family produced by CNES. Attitude control of the satellite will be considered in terms of robustness to uncertainties, level of performances, stability to saturations on the actuators.
For this experimental part of the workshop, participants are asked to come with a personal laptop computer that should be equipped with MATLAB. They are also asked to install the YALMIP parser and at least one semidefinite programming solver among the following ones CSDP, SDPA, SDPT3, SEDUMI.
Registration to be done on-line using PaperPlaza, see here for more information.
Workshop structure and main topics
Many problems arising in the context of analysis and control of systems subject to uncertainty
can be recast in the form of robust convex optimization, see, for instance, [1, 2]. Although these
classical deterministic reformulations proved to be computationally very successful in tackling
control problems affected by ``simple'' uncertainty structures, such as affine or linear fractional
transformations, they are not very helpful when the uncertainty is of polynomial or nonlinear
type. In fact, it is well-known that these classical optimization techniques fail to successfully
solve these control problems when the uncertainty structures are of ``difficult type'', also because
of computational issues, such as NP-hardness.
1. Uncertain complex systems
In the first part of the workshop, we provide a tutorial overview of complex control systems subject to uncertainty of different form. We also provide a summary of the main limits, in terms of computational complexity and conservatism of the obtained solution, of a typical robustness approach. Finally, we briefly overview some success stories of classical robustness techniques.
2. Polynomial optimization methods
Polynomial optimization methods have the objective to find the global minimum of a real valued polynomial in a compact set defined by polynomial inequalities, see . This problem reduces to solving an infinite sequence of linear matrix inequalities (LMIs). From a theoretical viewpoint, this approach has a significant impact in various areas of mathematics such as algebra, Fourier analysis, functional analysis, operator theory, probability and statistics. Furthermore, in addition to systems and control, these methods lead to effective computational tools in various fields such as optimization, probability, finance, signal processing, chemistry, crystallography and tomography.
3. Probabilistic methods for control design
Probabilistic methods for control design aim at the development of sequential and non-sequential randomized algorithms for convex and nonconvex control problems affected by ``difficult'' uncertainty structures, see . These techniques are based on Monte Carlo and Las Vegas simulation methods, with specific attention to the structure of the uncertainty entering into the control systems. This requires the study of suitable methods for multivariate sample generation techniques and the computation of the so-called sample complexity. Probabilistic methods are are polynomial-time and are particularly effective when a large number of uncertain parameters enter into the control system in a nonlinear fashion.
4. Software tools: GloptiPoly, RoMulOC and RACT
GloptiPoly, see , is intended to solve, or at least approximate, the Generalized Problem of Moments (GPM), an infinite-dimensional optimization problem which can be viewed as
an extension of the classical problem of moments. The current version of GloptiPoly 3 can
handle moment problems with polynomial data. The software allows to build up a hierarchy of semidefinite programming (SDP), or LMI relaxations of the GPM, whose associated
monotone sequence of optimal values converges to the global optimum. The package can be
freely downloaded from homepages.laas.fr/henrion/software/gloptipoly.
 A. Ben-Tal and A. Nemirovski, ``Robust Convex Optimization'', Mathematics of Operations Research, Vol. 23, pages 769-805, 1998.