LMI Techniques for Optimization Over Polynomials in Control: A Survey

By Graziano Chesi. IEEE Transactions on Automatic Control, 2010

410 citations recorded by the source corpus.

Numerous tasks in control systems involve optimization problems over polynomials, and unfortunately these problems are in general nonconvex. In order to cope with this difficulty, linear matrix inequality (LMI) techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems. This survey aims to provide the reader with a significant overview of the LMI techniques that are used in control systems for tackling optimization problems over polynomials, describing approaches such as decomposition in sum of squares, Positivstellensatz, theory of moments, Pólya's theorem, and matrix dilation. Moreover, it aims to provide a collection of the essential problems in control systems where these LMI techniques are used, such as stability and performance investigations in nonlinear systems, uncertain systems, time-delay systems, and genetic regulatory networks. It is expected that this survey may be a concise useful reference for all readers.

Subfields: Control and Systems Engineering, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/81696da5e8c69295cf0acf8c399c6595b377e1d4

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Citations in the corpus

LMI stability conditions for uncertain rational nonlinear systems (2014, methodology): “stability conditions into new conditions that can be numerically tested [6,11,13].”
A Comprehensive Review of Stability Analysis of Continuous-Time Recurrent Neural Networks (2014, methodology): “A good survey of LMI techniques in stability analysis of delayed systems was presented in [283], and LMI methods in control applications were reviewed in [48] and [60].”
An ISM-based CNF tracking controller design for uncertain MIMO linear systems with multiple time-delays and external disturbances (2015, background): “For any control problem to have satisfactory behavior, two objectives must be achieved, namely stability and performance [38].”
Convex Computation of the Region of Attraction of Polynomial Control Systems (2012, background): “recently [8], [40] and [9] and the references therein.”
Economic-emission dispatch problem: A semi-definite programming approach (2014, background): “Although polynomial optimization problems are generally non-convex and difficult to solve, various hierarchy of convex relaxation of the problem have been proposed which monotonically converge to the exact global optimal solution [33, 34, 22].”
Review on computational methods for Lyapunov functions (2015, methodology): “We refer the reader to Chesi 2010 [55] and Anderson and Papachristodoulou 2015 [9] for an overview.”