Optimization on linear matrix inequalities for polynomial systems control

Henrion, Didier

doi:10.5802/ccirm.17

Cited by 18 publications

(20 citation statements)

References 62 publications

(54 reference statements)

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“…This section presents some basic definitions used throughout the paper, mostly taken from the works of Henrion 31 and Lasserre. 27 Let X ≠ ∅ be a subset of ℝ n and (X) denotes the Borel -algebra.…”

Section: Preliminariesmentioning

confidence: 99%

A semidefinite programming approach for polynomial switched optimal control problems

Davoudi

Hosseini

2019

Optim Control Appl Methods

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This paper considers solving optimal control of switched systems with polynomial data globally, where the number of switches and the switching signal are not preassigned. The problem is transformed into an embedded polynomial form in which a continuous variable controls the switching policy. Then, using occupation measures, the embedded optimal control problem is formulated as an infinite-dimensional linear programming (LP) over the space of measures. A polynomial sum-of-squares strengthening corresponding to conic dual of the measure LP problem provides an approximating optimal feedback control for both classical control and the switching signal. Furthermore, the optimal switching signal is calculated without mode projection. Finally, three simulation experiments are included to confirm the theoretical results in the paper.

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Section: Preliminariesmentioning

confidence: 99%

A semidefinite programming approach for polynomial switched optimal control problems

Davoudi

Hosseini

2019

Optim Control Appl Methods

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“…Applications of sufficient conditions to represent positive polynomials date back to [17] in static optimization and to [19] in optimal control; see also [18,6] for a more recent overview. This methodology can be used for all the control problems described in Section 4 to provide converging hierarchies of semidefinite approximations [19,14,26], see also [12] for a detailed overview.…”

Section: Convergencementioning

confidence: 99%

Positivity Certificates in Optimal Control

Pauwels

Henrion

Lasserre

2017

Springer Tracts in Advanced Robotics

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We propose a tutorial on relaxations and weak formulations of optimal control with their semidefinite approximations. We present this approach solely through the prism of positivity certificates which we consider to be the most accessible for a broad audience, in particular in the engineering and robotics communities. This simple concept allows to express very concisely powerful approximation certificates in control. The relevance of this technique is illustrated on three applications: region of attraction approximation, direct optimal control and inverse optimal control, for which it constitutes a common denominator. In a first step, we highlight the core mechanisms underpinning the application of positivity in control and how they appear in the different control applications. This relies on simple mathematical concepts and gives a unified treatment of the applications considered. This presentation is based on the combination and simplification of published materials. In a second step, we describe briefly relations with broader literature, in particular, occupation measures and Hamilton-Jacobi-Bellman equation which are important elements of the global picture. We describe the Sum-Of-Squares (SOS) semidefinite hierarchy in the semialgebraic case and briefly mention its convergence properties. Numerical experiments on a classical example in robotics, namely the nonholonomic vehicle, illustrate the concepts presented in the text for the three applications considered.

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“…We finally recall that solving LMIs is a basic subroutine of computer algorithms in systems control and optimization, especially in linear systems robust control [9,33], but also for the analysis or synthesis of nonlinear dynamical systems [67], or in nonlinear optimal control with polynomial data [37,11].…”

Section: Motivationsmentioning

confidence: 99%

Exact Algorithms for Linear Matrix Inequalities

Henrion¹,

Naldi²,

Din³

2016

SIAM J. Optim.

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Let A(x) = A 0 + x 1 A 1 + · · · + x n A n be a linear matrix, or pencil, generated by given symmetric matrices A 0 , A 1 , . . . , A n of size m with rational entries. The set of real vectors x such that the pencil is positive semidefinite is a convex semialgebraic set called spectrahedron, described by a linear matrix inequality (LMI). We design an exact algorithm that, up to genericity assumptions on the input matrices, computes an exact algebraic representation of at least one point in the spectrahedron, or decides that it is empty. The algorithm does not assume the existence of an interior point, and the computed point minimizes the rank of the pencil on the spectrahedron. The degree d of the algebraic representation of the point coincides experimentally with the algebraic degree of a generic semidefinite program associated to the pencil. We provide explicit bounds for the complexity of our algorithm, proving that the maximum number of arithmetic operations that are performed is essentially quadratic in a multilinear Bézout bound of d. When m (resp. n) is fixed, such a bound, and hence the complexity, is polynomial in n (resp. m). We conclude by providing results of experiments showing practical improvements with respect to state-of-the-art computer algebra algorithms.The central object of this paper is the subset of x ∈ R n such that the eigenvalues of A(x) are all nonnegative, that is the spectrahedronHere 0 means "positive semidefinite" and A(x) 0 is called a linear matrix inequality (LMI). The set S is convex closed basic semi-algebraic. This paper addresses the following decision problem for the spectrahedron S :Problem 1 (Feasibility of semidefinite programming) Compute an exact algebraic representation of at least one point in S , or decide that S is empty.We present a probabilistic exact algorithm for solving Problem (1). The algorithm depends on some assumptions on input data that are specified later. If S is not empty, the expected output is a rational parametrization (see e.g. [55]) of a finite set Z ⊂ C n meeting S in at least one point x * . This is given by a vector (q 0 , . . . , q n+1 ) ⊂ Z[t] n+2 and a linear form λ = λ 1 x 1 + · · · + λ n x n ∈ Q[t] such that deg(q n+1 ) = ♯Z, deg(q i ) < deg(q n+1 ) for 0 ≤ i ≤ n, gcd(q n+1 , q 0 ) = 1 and Z coincides with the set

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Optimization on linear matrix inequalities for polynomial systems control

Abstract: Many problems of systems control theory boil down to solving polynomial equations, polynomial inequalities or polyomial differential equations. Recent advances in convex optimization and real algebraic geometry can be combined to generate approximate solutions in floating point arithmetic.

Cited by 18 publications

References 62 publications

A semidefinite programming approach for polynomial switched optimal control problems

A semidefinite programming approach for polynomial switched optimal control problems

Positivity Certificates in Optimal Control

Exact Algorithms for Linear Matrix Inequalities

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