225 research outputs found
A hierarchy of LMI inner approximations of the set of stable polynomials
Exploiting spectral properties of symmetric banded Toeplitz matrices, we
describe simple sufficient conditions for positivity of a trigonometric
polynomial formulated as linear matrix inequalities (LMI) in the coefficients.
As an application of these results, we derive a hierarchy of convex LMI inner
approximations (affine sections of the cone of positive definite matrices of
size ) of the nonconvex set of Schur stable polynomials of given degree . It is shown that when tends to infinity the hierarchy converges to a
lifted LMI approximation (projection of an LMI set defined in a lifted space of
dimension quadratic in ) already studied in the technical literature
Convex inner approximations of nonconvex semialgebraic sets applied to fixed-order controller design
We describe an elementary algorithm to build convex inner approximations of
nonconvex sets. Both input and output sets are basic semialgebraic sets given
as lists of defining multivariate polynomials. Even though no optimality
guarantees can be given (e.g. in terms of volume maximization for bounded
sets), the algorithm is designed to preserve convex boundaries as much as
possible, while removing regions with concave boundaries. In particular, the
algorithm leaves invariant a given convex set. The algorithm is based on
Gloptipoly 3, a public-domain Matlab package solving nonconvex polynomial
optimization problems with the help of convex semidefinite programming
(optimization over linear matrix inequalities, or LMIs). We illustrate how the
algorithm can be used to design fixed-order controllers for linear systems,
following a polynomial approach
Convex computation of the region of attraction of polynomial control systems
We address the long-standing problem of computing the region of attraction
(ROA) of a target set (e.g., a neighborhood of an equilibrium point) of a
controlled nonlinear system with polynomial dynamics and semialgebraic state
and input constraints. We show that the ROA can be computed by solving an
infinite-dimensional convex linear programming (LP) problem over the space of
measures. In turn, this problem can be solved approximately via a classical
converging hierarchy of convex finite-dimensional linear matrix inequalities
(LMIs). Our approach is genuinely primal in the sense that convexity of the
problem of computing the ROA is an outcome of optimizing directly over system
trajectories. The dual infinite-dimensional LP on nonnegative continuous
functions (approximated by polynomial sum-of-squares) allows us to generate a
hierarchy of semialgebraic outer approximations of the ROA at the price of
solving a sequence of LMI problems with asymptotically vanishing conservatism.
This sharply contrasts with the existing literature which follows an
exclusively dual Lyapunov approach yielding either nonconvex bilinear matrix
inequalities or conservative LMI conditions. The approach is simple and readily
applicable as the outer approximations are the outcome of a single semidefinite
program with no additional data required besides the problem description
Decomposed Structured Subsets for Semidefinite and Sum-of-Squares Optimization
Semidefinite programs (SDPs) are standard convex problems that are frequently
found in control and optimization applications. Interior-point methods can
solve SDPs in polynomial time up to arbitrary accuracy, but scale poorly as the
size of matrix variables and the number of constraints increases. To improve
scalability, SDPs can be approximated with lower and upper bounds through the
use of structured subsets (e.g., diagonally-dominant and scaled-diagonally
dominant matrices). Meanwhile, any underlying sparsity or symmetry structure
may be leveraged to form an equivalent SDP with smaller positive semidefinite
constraints. In this paper, we present a notion of decomposed structured
subsets}to approximate an SDP with structured subsets after an equivalent
conversion. The lower/upper bounds found by approximation after conversion
become tighter than the bounds obtained by approximating the original SDP
directly. We apply decomposed structured subsets to semidefinite and
sum-of-squares optimization problems with examples of H-infinity norm
estimation and constrained polynomial optimization. An existing basis pursuit
method is adapted into this framework to iteratively refine bounds.Comment: 23 pages, 10 figures, 9 table
Controller Synthesis for Discrete-Time Polynomial Systems via Occupation Measures
In this paper, we design nonlinear state feedback controllers for
discrete-time polynomial dynamical systems via the occupation measure approach.
We propose the discrete-time controlled Liouville equation, and use it to
formulate the controller synthesis problem as an infinite-dimensional linear
programming problem on measures, which is then relaxed as finite-dimensional
semidefinite programming problems on moments of measures and their duals on
sums-of-squares polynomials. Nonlinear controllers can be extracted from the
solutions to the relaxed problems. The advantage of the occupation measure
approach is that we solve convex problems instead of generally non-convex
problems, and the computational complexity is polynomial in the state and input
dimensions, and hence the approach is more scalable. In addition, we show that
the approach can be applied to over-approximating the backward reachable set of
discrete-time autonomous polynomial systems and the controllable set of
discrete-time polynomial systems under known state feedback control laws. We
illustrate our approach on several dynamical systems
Stability analysis of linear ODE-PDE interconnected systems
Les systĂšmes de dimension infinie permettent de modĂ©liser un large spectre de phĂ©nomĂšnes physiques pour lesquels les variables d'Ă©tats Ă©voluent temporellement et spatialement. Ce manuscrit s'intĂ©resse Ă l'Ă©valuation de la stabilitĂ© de leur point d'Ă©quilibre. Deux Ă©tudes de cas seront en particulier traitĂ©es : l'analyse de stabilitĂ© des systĂšmes interconnectĂ©s Ă une Ă©quation de transport, et Ă une Ă©quation de rĂ©action-diffusion. Des outils thĂ©oriques existent pour l'analyse de stabilitĂ© de ces systĂšmes linĂ©aires de dimension infinie et s'appuient sur une algĂšbre d'opĂ©rateurs plutĂŽt que matricielle. Cependant, ces rĂ©sultats d'existence soulĂšvent un problĂšme de constructibilitĂ© numĂ©rique. Lors de l'implĂ©mentation, une approximation est rĂ©alisĂ©e et les rĂ©sultats sont conservatifs. La conception d'outils numĂ©riques menant Ă des garanties de stabilitĂ© pour lesquelles le degrĂ© de conservatisme est Ă©valuĂ© et maĂźtrisĂ© est alors un enjeu majeur. Comment dĂ©velopper des critĂšres numĂ©riques fiables permettant de statuer sur la stabilitĂ© ou l'instabilitĂ© des systĂšmes linĂ©aires de dimension infinie ? Afin de rĂ©pondre Ă cette question, nous proposons ici une nouvelle mĂ©thode gĂ©nĂ©rique qui se dĂ©compose en deux temps. D'abord, sous l'angle de l'approximation sur les polynĂŽmes de Legendre, des modĂšles augmentĂ©s sont construits et dĂ©coupent le systĂšme original en deux blocs : d'une part, un systĂšme de dimension finie approximant est isolĂ©, d'autre part, l'erreur de troncature de dimension infinie est conservĂ©e et modĂ©lisĂ©e. Ensuite, des outils frĂ©quentiels et temporels de dimension finie sont dĂ©ployĂ©s afin de proposer des critĂšres de stabilitĂ© plus ou moins coĂ»teux numĂ©riquement en fonction de l'ordre d'approximation choisi. En frĂ©quentiel, Ă l'aide du thĂ©orĂšme du petit gain, des conditions suffisantes de stabilitĂ© sont obtenues. En temporel, Ă l'aide du thĂ©orĂšme de Lyapunov, une sous-estimation des rĂ©gions de stabilitĂ© est proposĂ©e sous forme d'inĂ©galitĂ© matricielle linĂ©aire et une sur-estimation sous forme de test de positivitĂ©. Nos deux Ă©tudes de cas ont ainsi Ă©tĂ© traitĂ©es Ă l'aide de cette mĂ©thodologie gĂ©nĂ©rale. Le principal rĂ©sultat obtenu concerne le cas des systĂšmes EDO-transport interconnectĂ©s, pour lequel l'approximation et l'analyse de stabilitĂ© Ă l'aide des polynĂŽmes de Legendre mĂšne Ă des estimations des rĂ©gions de stabilitĂ© qui convergent exponentiellement vite. La mĂ©thode dĂ©veloppĂ©e dans ce manuscrit peut ĂȘtre adaptĂ©e Ă d'autres types d'approximations et exportĂ©e Ă d'autres systĂšmes linĂ©aires de dimension infinie. Ce travail ouvre ainsi la voie Ă l'obtention de conditions nĂ©cessaires et suffisantes de stabilitĂ© de dimension finie pour les systĂšmes de dimension infinie.Infinite dimensional systems allow to model a large panel of physical phenomena for which the state variables evolve both temporally and spatially. This manuscript deals with the evaluation of the stability of their equilibrium point. Two case studies are treated in particular: the stability analysis of ODE-transport, and ODE-reaction-diffusion interconnected systems. Theoretical tools exist for the stability analysis of these infinite-dimensional linear systems and are based on an operator algebra rather than a matrix algebra. However, these existence results raise a problem of numerical constructibility. During implementation, an approximation is performed and the results are conservative. The design of numerical tools leading to stability guarantees for which the degree of conservatism is evaluated and controlled is then a major issue. How can we develop reliable numerical criteria to rule on the stability or instability of infinite-dimensional linear systems? In order to answer this question, one proposes here a new generic method, which is decomposed in two steps. First, from the perspective of Legendre polynomials approximation, augmented models are built and split the original system into two blocks: on the one hand, a finite-dimensional approximated system is isolated, on the other hand, the infinite-dimensional truncation error is preserved and modeled. Then, frequency and time tools of finite dimension are deployed in order to propose stability criteria that have high or low numerical load depending on the approximated order. In frequencies, with the aid of the small gain theorem, sufficient stability conditions are obtained. In temporal, with the aid of the Lyapunov theorem, an under estimate of the stability regions is proposed as a linear matrix inequality and an over estimate as a positivity test. Our two case studies have been treated with this general methodology. The main result concerns the case of ODE-transport interconnected systems, for which the approximation and stability analysis using Legendre polynomials leads to exponentially fast converging estimates of stability regions. The method developed in this manuscript can be adapted to other types of approximations and exported to other infinite-dimensional linear systems. Thus, this work opens the way to obtain necessary and sufficient finite-dimensional conditions of stability for infinite-dimensional systems
SOStab: a Matlab Toolbox for Approximating Regions of Attraction of Nonlinear Systems
This paper presents a novel Matlab toolbox, aimed at facilitating the use of
polynomial optimization for stability analysis of nonlinear systems. Indeed, in
the past decade several decisive contributions made it possible to recast the
difficult problem of computing stability regions of nonlinear systems, under
the form of convex optimization problems that are tractable in modest
dimensions. However, these techniques combine sophisticated frameworks such as
algebraic geometry, measure theory and mathematical programming, and existing
software still requires their user to be fluent in Sum-of-Squares and Moment
programming, preventing these techniques from being used more widely in the
control community. To address this issue, SOStab entirely automates the writing
and solving of optimization problems, and directly outputs relevant data for
the user, while requiring minimal input. In particular, no specific knowledge
of optimization is needed to use it.Comment: 14 pages, 2 figures
Conic Optimization Theory: Convexification Techniques and Numerical Algorithms
Optimization is at the core of control theory and appears in several areas of
this field, such as optimal control, distributed control, system
identification, robust control, state estimation, model predictive control and
dynamic programming. The recent advances in various topics of modern
optimization have also been revamping the area of machine learning. Motivated
by the crucial role of optimization theory in the design, analysis, control and
operation of real-world systems, this tutorial paper offers a detailed overview
of some major advances in this area, namely conic optimization and its emerging
applications. First, we discuss the importance of conic optimization in
different areas. Then, we explain seminal results on the design of hierarchies
of convex relaxations for a wide range of nonconvex problems. Finally, we study
different numerical algorithms for large-scale conic optimization problems.Comment: 18 page
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