Workshop
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
Abstract
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.
Authors
The workshop is organized by Fabrizio Dabbene, Didier Henrion, Dimitri Peaucelle and Roberto Tempo.
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.
Register
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.
In this workshop, we describe recent advancements centered on the development of suitable
relaxation techniques, that have the objective to broaden signicantly the classes of uncertain
systems that can be studied. In particular, we focus on two different and complementary methodologies based on polynomial optimization techniques and on probabilistic methods, respectively.
While polynomial techniques are of deterministic type, and are based on the so-called generalized problem of moments (GPM) [3], the probabilistic methods assume that the underlying
uncertainty is random, and the obtained results are necessarily stated in terms of the probability
of satisfaction of given control specification, see e.g. [4]. An example of a satellite attitude stabilization problem [5] will be used during the workshop for testing and comparing these methods.
Outline
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 [3]. 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 [4]. 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 [6], 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.
RoMulOC This toolbox, see [7, 8], implements a collection of theoretical results obtained
in robust control. The aim is to construct some simple functions for manipulating uncertain systems and building LMI optimization problems related to robust multi-objective
control problems. Two types of uncertain models can be easily manipulated: affine polytopic and LFT (Linear Fractional Transformations). LMIs are build by the toolbox in
YALMIP syntax and can be solved by any available SDP solver. Coded LMIs allow analysis of multiple performances (stability, pole location, H1, H2 and impulse-to-peak) as
well as multi-objective state-feedback design. The package can be freely downloaded from
projects.laas.fr/OLOCEP/romuloc.
RACT This toolbox, see [9], provides convenient uncertain object manipulation and
implementation of randomized methods using state-of-the-art theoretical and algorithmic
results studied in [4] Two main features of the package are a functional approach with m-file
templates and a definition of design problems in generic LMI format using the widely used
YALMIP syntax. RACT features currently include easy and fast sampling of uncertain objects of almost any type, randomized algorithms for probabilistic performance verification
and for feasibility of uncertain LMIs using stochastic gradient, ellipsoid or cutting plane
methods. The package can be freely downloaded from ract.sourceforge.net
References
[1] A. Ben-Tal and A. Nemirovski, ``Robust Convex Optimization'', Mathematics of Operations Research, Vol. 23, pages 769-805, 1998.
[2] D. Peaucelle, D. Arzelier, D. Henrion and F. Gouaisbaut, ``Quadratic Separation for Feedback Connection of an Uncertain Matrix and an Implicit Linear Transformation'', Automatica, Vol. 43, pages 795-804, 2007.
[3] J. B. Lasserre, ``A Semidefinite Programming Approach to the Generalized Problem of Moments'', Mathematical Programming, Vol. 112, pp. 65-92, 2008.
[4] R. Tempo, G. Calafiore and F. Dabbene, ``Randomized Algorithms for Analysis and Control of Uncertain Systems, with Applications'', Second Edition, Springer-Verlag, London, 2013, ISBN
978-1-4471-4609-4.
[5] C. Pittet and D. Arzelier, ``DEMETER: A Benchmark for Robust Analysis and Control of the Attitude of Flexible Microsatellites'', Proc. of the IFAC Symposium on Robust Control Design, Toulouse, France, July 2006.
[6] D. Henrion, J. B. Lasserre and J. Loefberg, ``GloptiPoly 3: Moments, Optimization and Semidefinite Programming'', Optimization Methods and Software, Vol. 24, pp. 761-779, 2009.
[7] G. Chevarria, D. Peaucelle, D. Arzelier and G. Puyou, ``Robust Analysis of the Longitudinal Control of a Civil Aircraft using RoMulOC'', Proc. of the IEEE Conference on Computer Aided
Control System Design, Yokohama, Japan, September 2010.
[8] D. Peaucelle and D. Arzelier, ``Robust Multi-Objective Control Toolbox'', Proc. of the IEEE International Symposium on Computer Aided Control Systems Design, Munich, Germany, October 2006.
[9] A. Tremba, G. Calafiore, F. Dabbene, E. Gryazina, B. T. Polyak, P. S. Shcherbakov and R. Tempo, ``RACT: Randomized Algorithms Control Toolbox for MATLAB'', Proc. of the IFAC World Congress, Seoul, Korea, July 2008.