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prof. Ing. Didier Henrion, Ph.D.

Všechny publikace

Cone-Copositive Lyapunov Functions for Complementarity Systems: Converse Result and Polynomial Approximation

  • Autoři: Souaiby, M., Tanwani, A., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: IEEE Transactions on Automatic Control. 2022, 67(3), 1253-1268. ISSN 0018-9286.
  • Rok: 2022
  • DOI: 10.1109/TAC.2021.3061557
  • Odkaz: https://doi.org/10.1109/TAC.2021.3061557
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    This article establishes the existence of Lyapunov functions for analyzing the stability of a class of state-constrained systems, and it describes algorithms for their numerical computation. The system model consists of a differential equation coupled with a set-valued relation that introduces discontinuities in the vector field at the boundaries of the constraint set. In particular, the set-valued relation is described by the subdifferential of the indicator function of a closed convex cone, which results in a cone-complementarity system. The question of analyzing the stability of such systems is addressed by constructing cone-copositive Lyapunov functions. As a first analytical result, we show that exponentially stable complementarity systems always admit a continuously differentiable cone-copositive Lyapunov function. Putting some more structure on the system vector field, such as homogeneity, we can show that the aforementioned functions can be approximated by a rational function of cone-copositive homogeneous polynomials. This latter class of functions is seen to be particularly amenable for numerical computation as we provide two types of algorithms for precisely that purpose. These algorithms consist of a hierarchy of either linear or semidefinite optimization problems for computing the desired cone-copositive Lyapunov function. Some examples are given to illustrate our approach.

Exploiting Sparsity for Semi-Algebraic Set Volume Computation

  • Autoři: Tacchi, M., Weisser, T., Lasserre, J., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: Foundations of Computational Mathematics. 2022, 22(1), 161-209. ISSN 1615-3375.
  • Rok: 2022
  • DOI: 10.1007/s10208-021-09508-w
  • Odkaz: https://doi.org/10.1007/s10208-021-09508-w
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We provide a systematic deterministic numerical scheme to approximate the volume (i.e., the Lebesgue measure) of a basic semi-algebraic set whose description follows a correlative sparsity pattern. As in previous works (without sparsity), the underlying strategy is to consider an infinite-dimensional linear program on measures whose optimal value is the volume of the set. This is a particular instance of a generalized moment problem which in turn can be approximated as closely as desired by solving a hierarchy of semidefinite relaxations of increasing size. The novelty with respect to previous work is that by exploiting the sparsity pattern we can provide a sparse formulation for which the associated semidefinite relaxations are of much smaller size. In addition, we can decompose the sparse relaxations into completely decoupled subproblems of smaller size, and in some cases computations can be done in parallel. To the best of our knowledge, it is the first contribution that exploits correlative sparsity for volume computation of semi-algebraic sets which are possibly high-dimensional and/or non-convex and/or non-connected.

Globally Optimal Solution to Inverse Kinematics of 7DOF Serial Manipulator

  • Autoři: Trutman, P., Safey El Din, M., prof. Ing. Didier Henrion, Ph.D., Pajdla, T.
  • Publikace: IEEE Robotics and Automation Letters. 2022, 7(3), 6012-6019. ISSN 2377-3766.
  • Rok: 2022
  • DOI: 10.1109/LRA.2022.3163444
  • Odkaz: https://doi.org/10.1109/LRA.2022.3163444
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    The Inverse Kinematics (IK) problem is concerned with finding robot control parameters to bring the robot into a desired position under the kinematics and joint limit constraints. We present a globally optimal solution to the IK problem for a general serial 7DOF manipulator with revolute joints and a polynomial objective function. We show that the kinematic constraints due to rotations can be all generated by the second-degree polynomials. This is an important result since it significantly simplifies the further step where we find the optimal solution by Lasserre relaxations of nonconvex polynomial systems. We demonstrate that the second relaxation is sufficient to solve a general 7DOF IK problem. Our approach is certifiably globally optimal. We demonstrate the method on the 7DOF KUKA LBR IIWA manipulator and show that we are, in practice, able to compute the optimal IK or certify infeasibility in 99.9% tested poses. We also demonstrate that by the same approach, we are able to solve the IK problem for any generic (random) manipulator with seven revolute joints.

Graph Recovery from Incomplete Moment Information

  • DOI: 10.1007/s00365-022-09563-8
  • Odkaz: https://doi.org/10.1007/s00365-022-09563-8
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We investigate a class of moment problems, namely recovering a measure supported on the graph of a function from partial knowledge of its moments, as, for instance, in some problems of optimal transport or density estimation. We show that the sole knowledge of first degree moments of the function, namely linear measurements, is sufficient to obtain asymptotically all the other moments by solving a hierarchy of semidefinite relaxations (viewed as moment matrix completion problems) with a specific sparsity-inducing criterion related to a weighted l(1)-norm of the moment sequence of the measure. The resulting sequence of optimal solutions converges to the whole moment sequence of the measure which is shown to be the unique optimal solution of a certain infinite-dimensional linear optimization problem (LP). Then one may recover the function by a recent extraction algorithm based on the Christoffel-Darboux kernel associated with the measure. Finally, the support of such a measure supported on a graph is a meager, very thin (hence sparse) set. Therefore, the LP on measures with this sparsity-inducing criterion can be interpreted as an analogue for infinite-dimensional signals of the LP in super-resolution for (sparse) atomic signals.

Convex Computation of Extremal Invariant Measures of Nonlinear Dynamical Systems and Markov Processes

  • DOI: 10.1007/s00332-020-09658-1
  • Odkaz: https://doi.org/10.1007/s00332-020-09658-1
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We propose a convex-optimization-based framework for computation of invariant measures of polynomial dynamical systems and Markov processes, in discrete and continuous time. The set of all invariant measures is characterized as the feasible set of an infinite-dimensional linear program (LP). The objective functional of this LP is then used to single out a specific measure (or a class of measures) extremal with respect to the selected functional such as physical measures, ergodic measures, atomic measures (corresponding to, e.g., periodic orbits) or measures absolutely continuous w.r.t. to a given measure. The infinite-dimensional LP is then approximated using a standard hierarchy of finite-dimensional semidefinite programming problems, the solutions of which are truncated moment sequences, which are then used to reconstruct the measure. In particular, we show how to approximate the support of the measure as well as how to construct a sequence of weakly converging absolutely continuous approximations. As a by-product, we present a simple method to certify the nonexistence of an invariant measure, which is an important question in the theory of Markov processes. The presented framework, where a convex functional is minimized or maximized among all invariant measures, can be seen as a generalization of and a computational method to carry out the so-called ergodic optimization, where linear functionals are optimized over the set of invariant measures. Finally, we also describe how the presented framework can be adapted to compute eigenmeasures of the Perron-Frobenius operator.

Dual optimal design and the Christoffel-Darboux polynomial

  • Autoři: De Castro, Y., Gamboa, F., prof. Ing. Didier Henrion, Ph.D., Lasserre, J.
  • Publikace: Optimization Letters. 2021, 15(1), 3-8. ISSN 1862-4472.
  • Rok: 2021
  • DOI: 10.1007/s11590-020-01680-2
  • Odkaz: https://doi.org/10.1007/s11590-020-01680-2
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    The purpose of this short note is to show that the Christoffel-Darboux polynomial, useful in approximation theory and data science, arises naturally when deriving the dual to the problem of semi-algebraic D-optimal experimental design in statistics. It uses only elementary notions of convex analysis. Geometric interpretations and algorithmic consequences are mentioned.

Exact algorithms for semidefinite programs with degenerate feasible set

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Naldi, S., Safey El Din, M.
  • Publikace: Journal of Symbolic Computation. 2021, 104 942-959. ISSN 0747-7171.
  • Rok: 2021
  • DOI: 10.1016/j.jsc.2020.11.001
  • Odkaz: https://doi.org/10.1016/j.jsc.2020.11.001
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Given symmetric matrices A0,A1,…,An of size m with rational entries, the set of real vectors x=(x1,…,xn) such that the matrix A0+x1A1+⋯+xnAn has non-negative eigenvalues is called a spectrahedron. Minimization of linear functions over spectrahedra is called semidefinite programming. Such problems appear frequently in control theory and real algebra, especially in the context of nonnegativity certificates for multivariate polynomials based on sums of squares. Numerical software for semidefinite programming are mostly based on interior point methods, assuming non-degeneracy properties such as the existence of an interior point in the spectrahedron. In this paper, we design an exact algorithm based on symbolic homotopy for solving semidefinite programs without assumptions on the feasible set, and we analyze its complexity. Because of the exactness of the output, it cannot compete with numerical routines in practice. However, we prove that solving such problems can be done in polynomial time if either n or m is fixed.

Global optimality in minimum compliance topology optimization of frames and shells by moment-sum-of-squares hierarchy

  • Autoři: Tyburec, M., Zeman, J., Kružík, M., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: Structural and Multidisciplinary Optimization. 2021, 64(4), 1963-1981. ISSN 1615-147X.
  • Rok: 2021
  • DOI: 10.1007/s00158-021-02957-5
  • Odkaz: https://doi.org/10.1007/s00158-021-02957-5
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    The design of minimum-compliance bending-resistant structures with continuous cross-section parameters is a challenging task because of its inherent non-convexity. Our contribution develops a strategy that facilitates computing all guaranteed globally optimal solutions for frame and shell structures under multiple-load cases and self-weight. To this purpose, we exploit the fact that the stiffness matrix is usually a polynomial function of design variables, allowing us to build an equivalent non-linear semidefinite programming formulation over a semi-algebraic feasible set. This formulation is subsequently solved using the Lasserre moment-sum-of-squares hierarchy, generating a sequence of outer convex approximations that monotonically converges from below to the optimum of the original problem. Globally optimal solutions can subsequently be extracted using the Curto-Fialkow flat extension theorem. Furthermore, we show that a simple correction to the solutions of the relaxed problems establishes a feasible upper bound, thereby deriving a simple sufficient condition of global ε-optimality. When the original problem possesses a unique minimum, we show that this solution is found with a zero optimality gap in the limit. These theoretical findings are illustrated on several examples of topology optimization of frames and shells, for which we observe that the hierarchy converges in a finite (rather small) number of steps.

Measures and lmis for validation of an aircraft with mrac and uncertain actuator dynamics

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    It is well known that actuator dynamics with degraded performance can lead to poor stability and tracking for adaptive control. In this paper, we use occupation measures and LMI relaxations (called the moment sums of squares or Lasserre hierarchy) for verification and validation of a longitudinal F-16 aircraft model with MRAC and higher order actuator dynamics with uncertain parameters and deflection saturation. These uncertain parameters are represented explicitly in the space of occupation measures. To improve numerical scalability, we exploit sparsity for ordinary differential equations (ODEs) using parsimony to partition the dynamics. We finally compare these main results to those obtained using Monte-Carlo methods.

Parabolic set simulation for reachability analysis of linear time-invariant systems with integral quadratic constraint

  • DOI: 10.1016/j.ejcon.2020.08.002
  • Odkaz: https://doi.org/10.1016/j.ejcon.2020.08.002
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    This paper describes the computation of reachable sets and tubes for linear time-invariant systems with an unknown input bounded by integral quadratic constraints, modeling e.g. delay, rate limiter, or energy bounds. We define a family of paraboloidal overapproximations. These paraboloids are supported by the reachable tube on touching trajectories. Parameters of each paraboloid are expressed as a solution to an initial value problem. Compared to previous methods based on the classical linear quadratic regulator, our approach can be applied to unstable systems as well. We tested our approach on large scale systems.

Peak Estimation for Uncertain and Switched Systems

  • DOI: 10.1109/CDC45484.2021.9683778
  • Odkaz: https://doi.org/10.1109/CDC45484.2021.9683778
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Peak estimation bounds extreme values of a function of state along trajectories of a dynamical system. This paper focuses on extending peak estimation to continuous and discrete settings with time-independent and time-dependent uncertainty. Techniques from optimal control are used to incorporate uncertainty into an existing occupation measure-based peak estimation framework, which includes special consideration for handling switching-type (polytopic) uncertainties. The resulting infinite-dimensional linear programs can be solved approximately with Linear Matrix Inequalities arising from the moment-SOS hierarchy.

Peak Estimation Recovery and Safety Analysis

  • DOI: 10.1109/LCSYS.2020.3047591
  • Odkaz: https://doi.org/10.1109/LCSYS.2020.3047591
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Peak Estimation aims to find the maximum value of a state function achieved by a dynamical system. This problem has been previously cast as a convex infinite-dimensional linear program on occupation measures, which can be approximately solved by a converging hierarchy of moment relaxations. In this letter, we present an algorithm to approximate optimal trajectories if the solutions to these relaxations satisfy rank constraints. We also extend peak estimation to maximin and safety analysis problems, providing a certificate that trajectories are bounded away from an unsafe set.

Semi-algebraic Approximation Using Christoffel-Darboux Kernel

  • Autoři: Marx, S., Pauwels, E., Weisser, T., prof. Ing. Didier Henrion, Ph.D., Lasserre, J.
  • Publikace: Constructive Approximation. 2021, 54(3), 391-429. ISSN 0176-4276.
  • Rok: 2021
  • DOI: 10.1007/s00365-021-09535-4
  • Odkaz: https://doi.org/10.1007/s00365-021-09535-4
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We provide a new method to approximate a (possibly discontinuous) function using Christoffel-Darboux kernels. Our knowledge about the unknown multivariate function is in terms of finitely many moments of the Young measure supported on the graph of the function. Such an input is available when approximating weak (or measure-valued) solution of optimal control problems, entropy solutions to nonlinear hyperbolic PDEs, or using numerical integration from finitely many evaluations of the function. While most of the existing methods construct a piecewise polynomial approximation, we construct a semi-algebraic approximation whose estimation and evaluation can be performed efficiently. An appealing feature of this method is that it deals with nonsmoothness implicitly so that a single scheme can be used to treat smooth or nonsmooth functions without any prior knowledge. On the theoretical side, we prove pointwise convergence almost everywhere as well as convergence in the Lebesgue one norm under broad assumptions. Using more restrictive assumptions, we obtain explicit convergence rates. We illustrate our approach on various examples from control and approximation. In particular, we observe empirically that our method does not suffer from the Gibbs phenomenon when approximating discontinuous functions.

A MOMENT APPROACH FOR ENTROPY SOLUTIONS TO NONLINEAR HYPERBOLIC PDES

  • Autoři: Marx, S., Tillmann, W., prof. Ing. Didier Henrion, Ph.D., Lasserre, J.B.
  • Publikace: Mathematical Control and Related Fields. 2020, 10(1), 113-140. ISSN 2156-8472.
  • Rok: 2020
  • DOI: 10.3934/mcrf.2019032
  • Odkaz: https://doi.org/10.3934/mcrf.2019032
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We propose to solve hyperbolic partial differential equations (PDEs) with polynomial flux using a convex optimization strategy. This approach is based on a very weak notion of solution of the nonlinear equation, namely the measure-valued (mv) solution, satisfying a linear equation in the space of Borel measures. The aim of this paper is, first, to provide the conditions that ensure the equivalence between the two formulations and, second, to introduce a method which approximates the infinite-dimensional linear problem by a hierarchy of convex, finite-dimensional, semidefinite programming problems. This result is then illustrated on the celebrated Burgers equation. We also compare our results with an existing numerical scheme, namely the Godunov scheme.

Approximating regions of attraction of a sparse polynomial differential system

  • Autoři: Tacchi, M., Cardozo, C., prof. Ing. Didier Henrion, Ph.D., Lasserre, J.B.
  • Publikace: Proceedings of the IFAC World Congress 2020. Laxenburg: IFAC, 2020. p. 3266-3271. IFAC-PapersOnLine. vol. 53. ISSN 2405-8963.
  • Rok: 2020
  • DOI: 10.1016/j.ifacol.2020.12.1488
  • Odkaz: https://doi.org/10.1016/j.ifacol.2020.12.1488
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Motivated by stability analysis of large scale power systems, we describe how the Lasserre (moment - sums of squares, SOS) hierarchy can be used to generate outer approximations of the region of attraction (ROA) of sparse polynomial differential systems, at the price of solving linear matrix inequalities (LMI) of increasing size. We identify specific sparsity structures for which we can provide numerically certified outer approximations of the region of attraction in high dimension. For this purpose, we combine previous results on non-sparse ROA approximations with sparse semi-algebraic set volume computation.

Computation of lyapunov functions under state constraints using semidefinite programming hierarchies

  • Autoři: Souaiby, M., Tanwani, A., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: Proceedings of the IFAC World Congress 2020. Laxenburg: IFAC, 2020. p. 6281-6286. IFAC-PapersOnLine. vol. 53. ISSN 2405-8963.
  • Rok: 2020
  • DOI: 10.1016/j.ifacol.2020.12.1746
  • Odkaz: https://doi.org/10.1016/j.ifacol.2020.12.1746
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We provide algorithms for computing a Lyapunov function for a class of systems where the state trajectories are constrained to evolve within a closed convex set. The dynamical systems that we consider comprise a differential equation which ensures continuous evolution within the domain, and a normal cone inclusion which ensures that the state trajectory remains within a prespecified set at all times. Finding a Lyapunov function for such a system boils down to finding a function which satisfies certain inequalities on the admissible set of state constraints. It is well-known that this problem, despite being convex, is computationally difficult. For conic constraints, we provide a discretization algorithm based on simplicial partitioning of a simplex, so that the search of desired function is addressed by constructing a hierarchy (associated with the diameter of the cells in the partition) of linear programs. Our second algorithm is tailored to semi-algebraic sets, where a hierarchy of semidefinite programs is constructed to compute Lyapunov functions as a sum-of-squares polynomial.

GLOBAL TOPOLOGY STIFFNESS OPTIMIZATION OF FRAME STRUCTURES BY MOMENT-SUM-OF-SQUARES HIERARCHY

  • Autoři: Tyburec, M., Zeman, J., Kružík, M., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: Engineering Mechanics 2020: Book of full texts. Prague: Institute of Thermomechanics, AS CR, v.v.i., 2020. p. 492-495. ISSN 1805-8248. ISBN 978-80-214-5896-3.
  • Rok: 2020
  • DOI: 10.21495/5896-3-492
  • Odkaz: https://doi.org/10.21495/5896-3-492
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    This contribution develops an efficient formulation for the topology optimization of frame structures with fixed-aspect-ratio cross-sections, solvable to global optimality by the moment-sum-of-squares hierarchy. While the hierarchy generates a sequence of non-decreasing lower-bounds, we develop a sequence of feasible upper-bounds, allowing us to assess the optimized design quality in each relaxation. Finally, these bounds provide a means of establishing a new sufficiency condition of global ε-optimality.

Measures and LMIs for Lateral F-16 MRAC Validation

  • DOI: 10.23919/ACC45564.2020.9147855
  • Odkaz: https://doi.org/10.23919/ACC45564.2020.9147855
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Occupation measures and linear matrix inequality (LMI) relaxations (called the moment sums of squares or Lasserre hierarchy) are state-of-the-art methods for verification and validation (VV) in aerospace. In this document, we extend these results to a full F-16 closed-loop nonlinear dutch roll polynomial model complete with model reference adaptive control (MRAC). This is done through a new technique of approximating the reference trajectory by exploiting sparse ordinary differential equations (ODEs) with parsimony. The VV problem is then solved directly using moment LMI relaxations and off-the-shelf-software. The main results are then compared to their numerical counterparts obtained using traditional Monte-Carlo simulations.

Moment-SOS Hierarchy, The: Lectures in Probability, Statistics, Computational Geometry, Control and Nonlinear PDEs

  • DOI: 10.1142/q0252
  • Odkaz: https://doi.org/10.1142/q0252
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    The Moment-SOS hierarchy is a powerful methodology that is used to solve the Generalized Moment Problem (GMP) where the list of applications in various areas of Science and Engineering is almost endless. Initially designed for solving polynomial optimization problems (the simplest example of the GMP), it applies to solving any instance of the GMP whose description only involves semi-algebraic functions and sets. It consists of solving a sequence (a hierarchy) of convex relaxations of the initial problem, and each convex relaxation is a semidefinite program whose size increases in the hierarchy.The goal of this book is to describe in a unified and detailed manner how this methodology applies to solving various problems in different areas ranging from Optimization, Probability, Statistics, Signal Processing, Computational Geometry, Control, Optimal Control and Analysis of a certain class of nonlinear PDEs. For each application, this unconventional methodology differs from traditional approaches and provides an unusual viewpoint. Each chapter is devoted to a particular application, where the methodology is thoroughly described and illustrated on some appropriate examples.The exposition is kept at an appropriate level of detail to aid the different levels of readers not necessarily familiar with these tools, to better know and understand this methodology.

ON OPTIMUM DESIGN OF FRAME STRUCTURES

  • Autoři: Tyburec, M., Zeman, J., Kružík, M., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: NMM 2019 Nano & Macro Mechanics. Praha: Czech Technical University in Prague, 2020. p. 117-125. Acta Polytechnica CTU Proceedings. vol. 26. ISSN 2336-5382. ISBN 978-80-01-06720-8.
  • Rok: 2020
  • DOI: 10.14311/APP.2020.26.0117
  • Odkaz: https://doi.org/10.14311/APP.2020.26.0117
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Optimization of frame structures is formulated as a non-convex optimization problem, which is currently solved to local optimality. In this contribution, we investigate four optimization approaches: (i) general non-linear optimization, (ii) optimality criteria method, (iii) non-linear semidefinite programming, and (iv) polynomial optimization. We show that polynomial optimization solves the frame structure optimization to global optimality by building the (moment-sums-of-squares) hierarchy of convex linear semidefinite programming problems, and it also provides guaranteed lower and upper bounds on optimal design. Finally, we solve three sample optimization problems and conclude that the local optimization approaches may indeed converge to local optima, without any solution quality measure, or even to infeasible points. These issues are readily overcome by using polynomial optimization, which exhibits a finite convergence, at the prize of higher computational demands.

Real root finding for low rank linear matrices

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Naldi, S., Din, M.S.E.
  • Publikace: Applicable Algebra in Engineering, Communication and Computing. 2020, 31(2), 101-133. ISSN 0938-1279.
  • Rok: 2020
  • DOI: 10.1007/s00200-019-00396-w
  • Odkaz: https://doi.org/10.1007/s00200-019-00396-w
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We consider m× s matrices (with m≥ s) in a real affine subspace of dimension n. The problem of finding elements of low rank in such spaces finds many applications in information and systems theory, where low rank is synonymous of structure and parsimony. We design computer algebra algorithms, based on advanced methods for polynomial system solving, to solve this problem efficiently and exactly: the input are the rational coefficients of the matrices spanning the affine subspace as well as the expected maximum rank, and the output is a rational parametrization encoding a finite set of points that intersects each connected component of the low rank real algebraic set. The complexity of our algorithm is studied thoroughly. It is polynomial in (n+m(s-r)n). It improves on the state-of-the-art in computer algebra and effective real algebraic geometry. Moreover, computer experiments show the practical efficiency of our approach.

Inner approximations of the maximal positively invariant set for polynomial dynamical systems

  • DOI: 10.1109/LCSYS.2019.2916256
  • Odkaz: https://doi.org/10.1109/LCSYS.2019.2916256
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    The Lasserre or moment-sum-of-square hierarchy of linear matrix inequality relaxations is used to compute inner approximations of the maximal positively invariant set for continuous-time dynamical systems with polynomial vector fields. Convergence in volume of the hierarchy is proved under a technical growth condition on the average exit time of trajectories. Our contribution is to deal with inner approximations in infinite time, while former work with volume convergence guarantees proposed either outer approximations of the maximal positively invariant set or inner approximations of the region of attraction in finite time.

Maximal Positively Invariant Set Determination for Transient Stability Assessment in Power Systems

  • Autoři: Oustry, A., Cardozo, C., Pantiatici, P., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: Proceedings of the 58th Conference on Decision and Control. Piscataway, NJ: IEEE, 2019. p. 6572-6577. ISSN 0743-1546. ISBN 978-1-7281-1398-2.
  • Rok: 2019
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    This paper assesses the transient stability of a synchronous machine connected to an infinite bus through the notion of invariant sets. The problem of computing a conservative approximation of the maximal positively invariant set is formulated as a semi-definitive program based on occupation measures and Lasserre's relaxation. An extension of the proposed method into a robust formulation allows us to handle Taylor approximation errors for non-polynomial systems. Results show the potential of this approach to limit the use of extensive time domain simulations provided that scalability issues are addressed.

Measures and LMIs for Adaptive Control Validation

  • DOI: 10.1109/CDC40024.2019.9029254
  • Odkaz: https://doi.org/10.1109/CDC40024.2019.9029254
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Occupation measures and linear matrix inequality (LMI) relaxations (called the moment sums of squares or Lasserre hierarchy) have been used previously as a means for solving control law verification and validation (VV) problems. However, these methods have been restricted to relatively simple control laws and a limited number of states. In this document, we extend these methods to model reference adaptive control (MRAC) configurations typical of the aircraft industry. The main contribution is a validation scheme that exploits the specific nonlinearities and structure of MRAC. A nonlinear F-16 plant is used for illustration. LMI relaxations solved by off-the-shelf-software are compared to traditional Monte-Carlo simulations.

Optimal control problems with oscillations, concentrations and discontinuities

  • DOI: 10.1016/j.automatica.2019.01.030
  • Odkaz: https://doi.org/10.1016/j.automatica.2019.01.030
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Optimal control problems with oscillations (chattering controls) and concentrations (impulsive controls) can have integral performance criteria such that concentration of the control signal occurs at a discontinuity of the state signal. Techniques from functional analysis (anisotropic parametrized measures) are applied to give a precise meaning of the integral cost and to allow for the sound application of numerical methods. We show how this can be combined with the Lasserre hierarchy of semidefinite programming relaxations.

Parabolic Set Simulation for Reachability Analysis of Linear Time Invariant Systems with Integral Quadratic Constraint

  • Autoři: Rousse, P., Garoche, P., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: Proceedings of the 18th European Control Conference. Piscataway, NJ: IEEE, 2019. p. 4301-4306. ISBN 978-3-907144-00-8.
  • Rok: 2019
  • DOI: 10.23919/ECC.2019.8795930
  • Odkaz: https://doi.org/10.23919/ECC.2019.8795930
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    This work extends reachability analyses based on ellipsoidal techniques to Linear Time Invariant (LTI) systems subject to an integral quadratic constraint (IQC) between the past state and disturbance signals, interpreted as an input-output energetic constraint. To compute the reachable set, the LTI system is augmented with a state corresponding to the amount of energy still available before the constraint is violated. For a given parabolic set of initial states, the reachable set of the augmented system is overapproximated with a time-varying parabolic set. Parameters of this paraboloid are expressed as the solution of an Initial Value Problem (IVP) and the overapproximation relationship with the reachable set is proved. This paraboloid is actually supported by the reachable set on so-called touching trajectories. Finally, we describe a method to generate all the supporting paraboloids and prove that their intersection is an exact characterization of the reachable set. This work provides new practical means to compute overapproximation of reachable sets for a wide variety of systems such as delayed systems, rate limiters or energy-bounded linear systems.

Semidefinite approximations of invariant measures for polynomial systems

  • Autoři: Magron, V., Forets, M., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: Discrete and Continuous Dynamical Systems - B. 2019, 24(12), 6745-6770. ISSN 1531-3492.
  • Rok: 2019
  • DOI: 10.3934/dcdsb.2019165
  • Odkaz: https://doi.org/10.3934/dcdsb.2019165
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We consider the problem of approximating numerically the moments and the supports of measures which are invariant with respect to the dynamics of continuous- and discrete-time polynomial systems, under semialgebraic set constraints. First, we address the problem of approximating the density and hence the support of an invariant measure which is absolutely continuous with respect to the Lebesgue measure. Then, we focus on the approximation of the support of an invariant measure which is singular with respect to the Lebesgue measure. Each problem is handled through an appropriate reformulation into a conic optimization problem over measures, solved in practice with two hierarchies of finite-dimensional semidefinite moment-sum-of-square relaxations, also called Lasserre hierarchies.Under specific assumptions, the first Lasserre hierarchy allows to approximate the moments of an absolutely continuous invariant measure as close as desired and to extract a sequence of polynomials converging weakly to the density of this measure.The second Lasserre hierarchy allows to approximate as close as desired in the Hausdorff metric the support of a singular invariant measure with the level sets of the Christoffel polynomials associated to the moment matrices of this measure.We also present some application examples together with numerical results for several dynamical systems admitting either absolutely continuous or singular invariant measures.

SPECTRA - a Maple library for solving linear matrix inequalities in exact arithmetic

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Naldi, S., Safey, E.D.M.
  • Publikace: Optimization Methods and Software. 2019, 34(1), 62-78. ISSN 1055-6788.
  • Rok: 2019
  • DOI: 10.1080/10556788.2017.1341505
  • Odkaz: https://doi.org/10.1080/10556788.2017.1341505
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    This document describes our freely distributed Maple library spectra, for Semidefinite Programming solved Exactly with Computational Tools of Real Algebra. It solves linear matrix inequalities with symbolic computation in exact arithmetic and it is targeted to small-size, possibly degenerate problems for which symbolic infeasibility or feasibility certificates are required.

Convergence rates of moment-sum-of-squares hierarchies for volume approximation of semialgebraic sets

  • DOI: 10.1007/s11590-017-1186-x
  • Odkaz: https://doi.org/10.1007/s11590-017-1186-x
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Moment-sum-of-squares hierarchies of semidefinite programs can be used to approximate the volume of a given compact basic semialgebraic set K. The idea consists of approximating from above the indicator function of K with a sequence of polynomials of increasing degree d, so that the integrals of these polynomials generate a convergence sequence of upper bounds on the volume of K. Under certain assumptions, we show that the asymptotic rate of this convergence is at least O(1/log log d) in general and O(1/log d) provided that the semialgebraic set is defined by a single inequality.

Convergence rates of moment-sum-of-squares hierarchies for optimal control problems

  • DOI: 10.1016/j.sysconle.2016.11.010
  • Odkaz: https://doi.org/10.1016/j.sysconle.2016.11.010
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We study the convergence rate of the moment-sum-of-squares hierarchy of semidefinite programs for optimal control problems with polynomial data. It is known that this hierarchy generates polynomial under-approximations to the value function of the optimal control problem and that these under-approximations converge in the L1 norm to the value function as their degree d tends to infinity. We show that the rate of this convergence is O(1∕loglogd). We treat in detail the continuous-time infinite-horizon discounted problem and describe in brief how the same rate can be obtained for the finite-horizon continuous-time problem and for the discrete-time counterparts of both problems.

Exact Solutions to Super Resolution on Semi-Algebraic Domains in Higher Dimensions

  • Autoři: De Castro, Y., Gamboa, F., prof. Ing. Didier Henrion, Ph.D., Lasserre, J.-B.
  • Publikace: IEEE Transactions on Information Theory. 2017, 63(1), 621-630. ISSN 0018-9448.
  • Rok: 2017
  • DOI: 10.1109/TIT.2016.2619368
  • Odkaz: https://doi.org/10.1109/TIT.2016.2619368
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We investigate the multi-dimensional super resolution problem on closed semi-algebraic domains for various sampling schemes such as Fourier or moments. We present a new semidefinite programming (SDP) formulation of the ℓ1-minimization in the space of Radon measures in the multi-dimensional frame on semi-algebraic sets. While standard approaches have focused on SDP relaxations of the dual program (a popular approach is based on Gram matrix representations), this paper introduces an exact formulation of the primal ℓ1-minimization exact recovery problem of super resolution that unleashes standard techniques (such as moment-sum-of-squares hierarchies) to overcome intrinsic limitations of previous works in the literature. Notably, we show that one can exactly solve the super resolution problem in dimension greater than 2 and for a large family of domains described by semi-algebraic sets.

Linear conic optimization for nonlinear optimal control

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Pauwels, E.
  • Publikace: Advances and Trends in Optimization with Engineering Applications. Philadelphia: SIAM, 2017. p. 121-134. ISBN 978-1-61197-467-6.
  • Rok: 2017
  • DOI: 10.1137/1.9781611974683.ch10
  • Odkaz: https://doi.org/10.1137/1.9781611974683.ch10
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    In this chapter, we discuss infinite-dimensional conic linear optimization (CLO) for nonlinear optimal control problems. The primal linear problem consists of finding occupation measures supported on optimal relaxed controlled trajectories, whereas the dual linear problem consists of finding the largest lower bound on the value function of the optimal control problem.

Positivity certificates in optimal control

  • Autoři: Pauwels, E., prof. Ing. Didier Henrion, Ph.D., Lasserre, J.B.
  • Publikace: Geometric and Numerical Foundations of Movements. Heidelberg: Springer, 2017. p. 113-131. Springer Tracts in Advanced Robotics. vol. 117. ISSN 1610-7438. ISBN 978-3-319-51546-5.
  • Rok: 2017
  • DOI: 10.1007/978-3-319-51547-2_6
  • Odkaz: https://doi.org/10.1007/978-3-319-51547-2_6
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    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 the most accessible for a broad audience, in particular engineering and robotics communities.

SEMI-DEFINITE RELAXATIONS FOR OPTIMAL CONTROL PROBLEMS WITH OSCILLATION AND CONCENTRATION EFFECTS

  • Autoři: Claeys, M., prof. Ing. Didier Henrion, Ph.D., Kružík, M.
  • Publikace: ESAIM: Control, Optimisation and Calculus of Variations. 2017, 23(1), 95-117. ISSN 1292-8119.
  • Rok: 2017
  • DOI: 10.1051/cocv/2015041
  • Odkaz: https://doi.org/10.1051/cocv/2015041
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Converging hierarchies of finite-dimensional semi-definite relaxations have been proposed for state-constrained optimal control problems featuring oscillation phenomena, by relaxing controls as Young measures. These semi-definite relaxations were later on extended to optimal control problems depending linearly on the control input and typically featuring concentration phenomena, interpreting the control as a measure of time with a discrete singular component modeling discontinuities or jumps of the state trajectories. In this contribution, we use measures introduced originally by DiPerna and Majda in the partial differential equations literature to model simultaneously, and in a unified framework, possible oscillation and concentration effects of the optimal control policy. We show that hierarchies of semi-definite relaxations can also be constructed to deal numerically with nonconvex optimal control problems with polynomial vector field and semialgebraic state constraints.

Simple approximations of semialgebraic sets and their applications to control

  • DOI: 10.1016/j.automatica.2016.11.021
  • Odkaz: https://doi.org/10.1016/j.automatica.2016.11.021
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Many uncertainty sets encountered in control systems analysis and design can be expressed in terms of semialgebraic sets, that is as the intersection of sets described by means of polynomial inequalities. Important examples are for instance the Schur and Hurwitz stability domains. These sets often have very complicated shapes (nonconvex, and even non-connected), which render difficult their manipulation. It is therefore of considerable importance to find simple-enough approximations of these sets, able to capture their main characteristics while maintaining a low level of complexity. For these reasons, in the past years several convex approximations, based for instance on hyperrectangles, polytopes, or ellipsoids have been proposed. In this work, we move a step further, and propose possibly nonconvex yet still simple approximations, based on a small volume polynomial superlevel set of a single positive polynomial of given degree. We show how these sets can be easily approximated by minimizing the L1 norm of the polynomial over the semialgebraic set, subject to positivity constraints. Intuitively, this corresponds to the trace minimization heuristic commonly encountered in minimum volume ellipsoid problems. From a computational viewpoint, we design a hierarchy of convex linear matrix inequality problems to generate these approximations, and we provide theoretically rigorous convergence results, in the sense that the hierarchy of outer approximations converges in volume (or, equivalently, almost everywhere and almost uniformly) to the original set. Finally, we show how the concept of polynomial superlevel set can be used to generate samples uniformly distributed on a given semialgebraic set. The efficiency of the proposed approach is demonstrated by different numerical examples.

Controller design and value function approximation for nonlinear dynamical systems

  • DOI: 10.1016/j.automatica.2016.01.022
  • Odkaz: https://doi.org/10.1016/j.automatica.2016.01.022
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    This work considers the infinite-time discounted optimal control problem for continuous time input-affine polynomial dynamical systems subject to polynomial state and box input constraints. We propose a sequence of sum-of-squares (SOS) approximations of this problem obtained by first lifting the original problem into the space of measures with continuous densities and then restricting these densities to polynomials. These approximations are tightenings, rather than relaxations, of the original problem and provide a sequence of rational controllers with value functions associated to these controllers converging (under some technical assumptions) to the value function of the original problem. In addition, we describe a method to obtain polynomial approximations from above and from below to the value function of the extracted rational controllers, and a method to obtain approximations from below to the optimal value function of the original problem, thereby obtaining a sequence of asymptotically optimal rational controllers with explicit estimates of suboptimality. Numerical examples demonstrate the approach.

EXACT ALGORITHMS FOR LINEAR MATRIX INEQUALITIES

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Naldi, S., Mohab Safey, E.D.
  • Publikace: SIAM JOURNAL ON OPTIMIZATION. 2016, 26(4), 2512-2539. ISSN 1052-6234.
  • Rok: 2016
  • DOI: 10.1137/15M1036543
  • Odkaz: https://doi.org/10.1137/15M1036543
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Let A(x) = A0 +x1A1+xnAn be a linear matrix, or pencil, generated by given symmetric matrices A0;A1;An 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. 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 semide finite 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 Bezout 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.

LINEAR CONIC OPTIMIZATION FOR INVERSE OPTIMAL CONTROL

  • Autoři: Pauwels, E., prof. Ing. Didier Henrion, Ph.D., Lasserre, J.B.
  • Publikace: SIAM Journal on Control and Optimization. 2016, 54(3), 1798-1825. ISSN 0363-0129.
  • Rok: 2016
  • DOI: 10.1137/14099454X
  • Odkaz: https://doi.org/10.1137/14099454X
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We address the inverse problem of Lagrangian identification based on trajectories in the context of nonlinear optimal control. We propose a general formulation of the inverse problem based on occupation measures and complementarity in linear programming. The use of occupation measures in this context offers several advantages from the theoretical, numerical, and statistical points of view. We propose an approximation procedure for which strong theoretical guarantees are available. Finally, the relevance of the method is illustrated on academic examples.

Minimizing the sum of many rational functions

  • Autoři: Bugarin, F., prof. Ing. Didier Henrion, Ph.D., Lasserre, J.B.
  • Publikace: Mathematical Programming Computation. 2016, 8(1), 83-111. ISSN 1867-2949.
  • Rok: 2016
  • DOI: 10.1007/s12532-015-0089-z
  • Odkaz: https://doi.org/10.1007/s12532-015-0089-z
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We consider the problem of globally minimizing the sum of many rational functions over a given compact semialgebraic set. The number of terms can be large (10 to 100), the degree of each term should be small (up to 10), and the number of variables can be relatively large (10 to 100) provided some kind of sparsity is present. We describe a formulation of the rational optimization problem as a generalized moment problem and its hierarchy of convex semidefinite relaxations. Under some conditions we prove that the sequence of optimal values converges to the globally optimal value. We show how public-domain software can be used to model and solve such problems. Finally, we also compare with the epigraph approach and the BARON software.

Modal occupation measures and LMI relaxations for nonlinear switched systems control

  • DOI: 10.1016/j.automatica.2015.11.003
  • Odkaz: https://doi.org/10.1016/j.automatica.2015.11.003
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    This paper presents a linear programming approach for the optimal control of nonlinear switched systems where the control is the switching sequence. This is done by introducing modal occupation measures, which allow to relax the problem as a primal linear programming (LP) problem. Its dual linear program of Hamilton–Jacobi–Bellman inequalities is also characterized. The LPs are then solved numerically with a converging hierarchy of primal–dual moment-sum-of-squares (SOS) linear matrix inequalities (LMI). Because of the special structure of switched systems, we obtain a much more efficient method than could be achieved by applying standard moment/SOS LMI hierarchies for general optimal control problems.

Real root finding for determinants of linear matrices

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Naldi, S., Safey El Din, M.
  • Publikace: Journal of Symbolic Computation. 2016, 74 205-238. ISSN 0747-7171.
  • Rok: 2016
  • DOI: 10.1016/j.jsc.2015.06.010
  • Odkaz: https://doi.org/10.1016/j.jsc.2015.06.010
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Let A0, A1, ..., Anbe given square matrices of size mwith rational coefficients. The paper focuses on the exact computation of one point in each connected component of the real determinantal vari-ety {x ∈Rn:det(A0+x1A1+···+xnAn) =0}. Such a problem finds applications in many areas such as control theory, computational geometry, optimization, etc. Under some genericity assumptions on the coefficients of the matrices, we provide an algorithm solving this problem whose runtime is essentially polynomial in the bino-mial coefficient n+mn. We also report on experiments with a com-puter implementation of this algorithm. Its practical performance illustrates the complexity estimates. In particular, we emphasize that for subfamilies of this problem where mis fixed, the com-plexity is polynomial inn.

Semidefinite approximations of the polynomial abscissa

  • Autoři: Hess, R., prof. Ing. Didier Henrion, Ph.D., Lasserre, J.B., Pham, T.S.
  • Publikace: SIAM Journal on Control and Optimization. 2016, 54(3), 1633-1656. ISSN 0363-0129.
  • Rok: 2016
  • DOI: 10.1137/15M1033198
  • Odkaz: https://doi.org/10.1137/15M1033198
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Given a univariate polynomial, its abscissa is the maximum real part of its roots. The abscissa arises naturally when controlling linear differential equations. As a function of the polynomial coefficients, the abscissa is Hölder continuous, and not locally Lipschitz in general, which is a source of numerical difficulties for designing and optimizing control laws. In this paper we propose simple approximations of the abscissa given by polynomials of fixed degree, and hence controlled complexity. Our approximations are computed by a hierarchy of finite-dimensional convex semidefinite programming problems. When their degree tends to infinity, the polynomial approximations converge in $L^1$ norm to the abscissa, either from above or from below.

Strong duality in Lasserre’s hierarchy for polynomial optimization

  • DOI: 10.1007/s11590-015-0868-5
  • Odkaz: https://doi.org/10.1007/s11590-015-0868-5
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    A polynomial optimization problem (POP) consists of minimizing a multivariate real polynomial on a semi-algebraic set $$K$$K described by polynomial inequalities and equations. In its full generality it is a non-convex, multi-extremal, difficult global optimization problem. More than an decade ago, J. B. Lasserre proposed to solve POPs by a hierarchy of convex semidefinite programming (SDP) relaxations of increasing size. Each problem in the hierarchy has a primal SDP formulation (a relaxation of a moment problem) and a dual SDP formulation (a sum-of-squares representation of a polynomial Lagrangian of the POP). In this note, we show that there is no duality gap between each primal and dual SDP problem in Lasserre’s hierarchy, provided one of the constraints in the description of set $$K$$K is a ball constraint. Our proof uses elementary results on SDP duality, and it does not assume that $$K$$K has a strictly feasible point.

Symmetries and analytical solutions of the Hamilton–Jacobi–Bellman equation for a class of optimal control problems

  • Autoři: Rodrigues, L., prof. Ing. Didier Henrion, Ph.D., Cantwell, B.J.
  • Publikace: OPTIMAL CONTROL APPLICATIONS & METHODS. 2016, 37(4), 749-764. ISSN 0143-2087.
  • Rok: 2016
  • DOI: 10.1002/oca.2190
  • Odkaz: https://doi.org/10.1002/oca.2190
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    The main contribution of this paper is to identify explicit classes of locally controllable second-order systems and optimization functionals for which optimal control problems can be solved analytically, assuming that a differentiable optimal cost-to-go function exists for such control problems. An additional contribution of the paper is to obtain a Lyapunov function for the same classes of systems. The paper addresses the Lie point symmetries of the Hamilton-Jacobi-Bellman (HJB) equation for optimal control of second-order nonlinear control systems that are affine in a single input and for which the cost is quadratic in the input. It is shown that if there exists a dilation symmetry of the HJB equation for optimal control problems in this class, this symmetry can be used to obtain a solution. It is concluded that when the cost on the state preserves the dilation symmetry, solving the optimal control problem is reduced to finding the solution to a first-order ordinary differential equation. For some cases where the cost on the state breaks the dilation symmetry, the paper also presents an alternative method to find analytical solutions of the HJB equation corresponding to additive control inputs. The relevance of the proposed methodologies is illustrated in several examples for which analytical solutions are found, including the Van der Pol oscillator and mass-spring systems. Furthermore, it is proved that in the well-known case of a linear quadratic regulator, the quadratic cost is precisely the cost that preserves the dilation symmetry of the equation.

Rank-Constrained Fundamental Matrix Estimation by Polynomial Global Optimization Versus the Eight-Point Algorithm

  • Autoři: Bugarin, F., Bartoli, A., prof. Ing. Didier Henrion, Ph.D., Lasserre, J.B., Orteu, J.J., Sentenac, T.
  • Publikace: Journal of Mathematical Imaging and Vision. 2015, 53(1), 42-60. ISSN 0924-9907.
  • Rok: 2015
  • DOI: 10.1007/s10851-014-0545-9
  • Odkaz: https://doi.org/10.1007/s10851-014-0545-9
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    The fundamental matrix can be estimated from point matches. The current gold standard is to bootstrap the eight-point algorithm and two-viewprojective bundle adjustment. The eight-point algorithm first computes a simple linear least squares solution by minimizing an algebraic cost and then projects the result to the closest rank-deficient matrix. We propose a single-step method that solves both steps of the eight-point algorithm. Using recent results from polynomial global optimization, our method finds the rank-deficient matrix that exactly minimizes the algebraic cost. In this special case, the optimizationmethod is reduced to the resolution of very short sequences of convex linear problems which are computationally efficient and numerically stable. The current gold standard is known to be extremely effective but is nonetheless outperformed by our rank-constrained method for bootstrapping bundle adjustment. This is here demonstrated on simulated and standard real datasets.With our initialization, bundle adjustment consistently finds a better local minimum (achieves a lower reprojection error) and takes less iterations to converge.

SEMIDEFINITE APPROXIMATIONS OF PROJECTIONS AND POLYNOMIAL IMAGES OF SEMIALGEBRAIC SETS

  • Autoři: Magron, V., prof. Ing. Didier Henrion, Ph.D., LASSERRE, J.B.
  • Publikace: SIAM JOURNAL ON OPTIMIZATION. 2015, 25(4), 2143-2164. ISSN 1052-6234.
  • Rok: 2015
  • DOI: 10.1137/140992047
  • Odkaz: https://doi.org/10.1137/140992047
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Given a compact semialgebraic set S subset of R-n and a polynomial map f : R-n -> R-m, we consider the problem of approximating the image set F = f (S) subset of R-m. This includes in particular the projection of S on R-m for n >= m. Assuming that F subset of B, with B subset of R-m being a "simple" set (e.g., a box or a ball), we provide two methods to compute certified outer approximations of F. Method 1 exploits the fact that F can be defined with an existential quantifier, while Method 2 computes approximations of the support of image measures. The two methods output a sequence of superlevel sets defined with a single polynomial that yield explicit outer approximations of F. Finding the coefficients of this polynomial boils down to computing an optimal solution of a convex semidefinite program. We provide guarantees of strong convergence to F in L-1 norm on B, when the degree of the polynomial approximation tends to infinity. Several examples of applications are provided, together with numerical experiments.

Stable Radial Distortion Calibration by Polynomial Matrix Inequalities Programming

  • Autoři: Heller, J., prof. Ing. Didier Henrion, Ph.D., Pajdla, T.
  • Publikace: Lecture Notes in Computer Science - Computer Vision ACCV2014, PTI. Berlin: Springer, 2015, pp. 307-321. ISSN 0302-9743. ISBN 978-3-319-16864-7.
  • Rok: 2015
  • DOI: 10.1007/978-3-319-16865-4_20
  • Odkaz: https://doi.org/10.1007/978-3-319-16865-4_20
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Polynomial and rational functions are the number one choice when it comes to modeling of radial distortion of lenses. However, several extrapolation and numerical issues may arise while using these functions that have not been covered by the literature much so far. In this paper, we identify these problems and show how to deal with them by enforcing nonnegativity of certain polynomials. Further, we show how to model these nonnegativities using polynomial matrix inequalities (PMI) and how to estimate the radial distortion parameters subject to PMI constraints using semidefinite programming (SDP). Finally, we suggest several approaches on how to incorporate the proposed method into the overall camera calibration procedure.

Approximating Pareto curves using semidefinite relaxations

  • Autoři: Magron, V., prof. Ing. Didier Henrion, Ph.D., Lasserre, J.B.
  • Publikace: Operations Research Letters. 2014, 42(6-7), 432-437. ISSN 0167-6377.
  • Rok: 2014
  • DOI: 10.1016/j.orl.2014.07.007
  • Odkaz: https://doi.org/10.1016/j.orl.2014.07.007
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We approximate as closely as desired the Pareto curve associated with bicriteria polynomial optimization problems. We use three formulations (including the weighted sum approach and the Chebyshev approximation) and each of them is viewed as a parametric polynomial optimization problem. For each case is associated a hierarchy of semidefinite relaxations and from an optimal solution of each relaxation one approximates the Pareto curve by solving an inverse problem (first two cases) or by building a polynomial underestimator (third case).

CONVEX COMPUTATION OF THE MAXIMUM CONTROLLED INVARIANT SET FOR POLYNOMIAL CONTROL SYSTEMS

  • DOI: 10.1137/130914565
  • Odkaz: https://doi.org/10.1137/130914565
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We characterize the maximum controlled invariant (MCI) set for discrete- as well as continuous-time nonlinear dynamical systems as the solution of an infinite-dimensional linear programming problem. For systems with polynomial dynamics and compact semialgebraic state and control constraints, we describe a hierarchy of finite-dimensional linear matrix inequality (LMI) relaxations whose optimal values converge to the volume of the MCI set; dual to these LMI relaxations are sum-of-squares (SOS) problems providing a converging sequence of outer approximations to the MCI set. The approach is simple and readily applicable in the sense that the approximations are the outcome of a single semidefinite program with no additional input apart from the problem description. A number of numerical examples illustrate the approach.

Convex Computation of the Region of Attraction of Polynomial Control Systems

  • DOI: 10.1109/TAC.2013.2283095
  • Odkaz: https://doi.org/10.1109/TAC.2013.2283095
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    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. The approach is demonstrated on several numerical examples

Design of Marx generators as a structured eigenvalue assignment

  • Autoři: Galeani, S., prof. Ing. Didier Henrion, Ph.D., Jacquemard, A., Zaccarian, L.
  • Publikace: Automatica. 2014, 50(10), 2709-2717. ISSN 0005-1098.
  • Rok: 2014
  • DOI: 10.1016/j.automatica.2014.09.003
  • Odkaz: https://doi.org/10.1016/j.automatica.2014.09.003
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We consider the design problem for a Marx generator electrical network, a pulsed power generator. We show that the components design can be conveniently cast as a structured real eigenvalue assignment with significantly lower dimension than the state size of the Marx circuit. Then we present two possible approaches to determine its solutions. A first symbolic approach consists in the use of Gröbner basis representations, which allows us to compute all the (finitely many) solutions. A second approach is based on convexification of a nonconvex optimization problem with polynomial constraints. We also comment on the conjecture that for any number of stages the problem has finitely many solutions, which is a necessary assumption for the proposed methods to converge. We regard the proof of this conjecture as an interesting challenge of general interest in the real algebraic geometry field.

Mean Squared Error Minimization for Inverse Moment Problems

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Lasserre, J.B., Mevissen, M.
  • Publikace: Applied Mathematics & Optimization. 2014, 70(1), 83-110. ISSN 0095-4616.
  • Rok: 2014
  • DOI: 10.1007/s00245-013-9235-z
  • Odkaz: https://doi.org/10.1007/s00245-013-9235-z
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We consider the problem of approximating the unknown density of a measure on , absolutely continuous with respect to some given reference measure , only from the knowledge of finitely many moments of . Given and moments of order , we provide a polynomial which minimizes the mean square error over all polynomials of degree at most . If there is no additional requirement, is obtained as solution of a linear system. In addition, if is expressed in the basis of polynomials that are orthonormal with respect to , its vector of coefficients is just the vector of given moments and no computation is needed. Moreover in as . In general nonnegativity of is not guaranteed even though is nonnegative. However, with this additional nonnegativity requirement one obtains analogous results but computing that minimizes now requires solving an appropriate semidefinite program. We have tested the approach on some applications arising from the reconstruction of geometrical objects and the approximation of solutions of nonlinear differential equations. In all cases our results are significantly better than those obtained with the maximum entropy technique for estimating .

Measures and LMIs for Impulsive Nonlinear Optimal Control

  • Autoři: Claeys, M., Arzelier, D., prof. Ing. Didier Henrion, Ph.D., Lasserre, J.B.
  • Publikace: IEEE Transactions on Automatic Control. 2014, 59(5), 1374-1379. ISSN 0018-9286.
  • Rok: 2014
  • DOI: 10.1109/TAC.2013.2292735
  • Odkaz: https://doi.org/10.1109/TAC.2013.2292735
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    This note shows how to use semi-definite programming to find lower bounds on a large class of nonlinear optimal control problems with polynomial dynamics and convex semialgebraic state constraints and an affine dependence on the control. This is done by relaxing an optimal control problem into a linear programming problem on measures, also known as a generalized moment problem. The handling of measures by their moments reduces the problem to a convergent series of standard linear matrix inequality relaxations. When the optimal control consists of a finite number of impulses, we can recover simultaneously the actual impulse times and amplitudes by simple linear algebra. Finally, our approach can be readily implemented with standard software, as illustrated by a numerical example.

Convex computation of the maximum controlled invariant set for discrete-time polynomial control systems

  • Autoři: Korda, M., prof. Ing. Didier Henrion, Ph.D., Jones, C.N.
  • Publikace: Proceedings of the 52nd IEEE Conference on Decision and Control. Piscataway: IEEE, 2013. p. 7107-7112. ISSN 0191-2216. ISBN 978-1-4673-5717-3.
  • Rok: 2013
  • DOI: 10.1109/CDC.2013.6761016
  • Odkaz: https://doi.org/10.1109/CDC.2013.6761016
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We characterize the maximum controlled invariant (MCI) set for discrete-time systems as the solution of an infinite-dimensional linear programming problem. In the case of systems with polynomial dynamics and semialgebraic state and control constraints, we describe a hierarchy of finitedimensional linear matrix inequality relaxations of this problem that provides outer approximations with guaranteed set-wise convergence to the MCI set. The approach is compact and readily applicable in the sense that the approximations are the outcome of a single semidefinite program with no additional input apart from the problem description.

Estimation of Consistent Parameter Sets for Continuous-time Nonlinear Systems Using Occupation Measures and LMI Relaxations

  • Autoři: Streif, S., Rumschinski, P., prof. Ing. Didier Henrion, Ph.D., Findeisen, R.
  • Publikace: Proceedings of the 52nd IEEE Conference on Decision and Control. Piscataway: IEEE, 2013, pp. 6379-6384. ISSN 0191-2216. ISBN 978-1-4673-5717-3.
  • Rok: 2013
  • DOI: 10.1109/CDC.2013.6760898
  • Odkaz: https://doi.org/10.1109/CDC.2013.6760898
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Obtaining initial conditions and parameterizations leading to a model consistent with available measurements or safety specifications is important for many applications. Examples include model (in-)validation, prediction, fault diagnosis, and controller design. We present an approach to determine inner- and outer-approximations of the set containing all consistent initial conditions/parameterizations for nonlinear (polynomial) continuous-time systems. These approximations are found by occupation measures that encode the system dynamics and measurements, and give rise to an infinitedimensional linear program. We exploit the flexibility and linearity of the decision problem to incorporate unknown-butbounded and pointwise-in-time state and output constraints, a feature which was not addressed in previous works. The infinitedimensional linear program is relaxed by a hierarchy of LMI problems that provide certificates in case no consistent initial condition/parameterization exists. Furthermore, the applied LMI relaxation guarantees that the approximations converge (almost uniformly) to the true consistent set. We illustrate the approach with a biochemical reaction network involving unknown initial conditions and parameters.

Finding largest small polygons with GloptiPoly

  • DOI: 10.1007/s10898-011-9818-7
  • Odkaz: https://doi.org/10.1007/s10898-011-9818-7
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices n. Many instances are already solved in the literature, namely for all odd n, and for n = 4, 6 and 8. Thus, for even n a parts per thousand yen 10, instances of this problem remain open. Finding those largest small polygons can be formulated as nonconvex quadratic programming problems which can challenge state-of-the-art global optimization algorithms. We show that a recently developed technique for global polynomial optimization, based on a semidefinite programming approach to the generalized problem of moments and implemented in the public-domain Matlab package GloptiPoly, can successfully find largest small polygons for n = 10 and n = 12. Therefore this significantly improves existing results in the domain. When coupled with accurate convex conic solvers, GloptiPoly can provide numerical guarantees of global optimality, as well as rigorous guarantees relying on interval arithmetic.

Measures and LMIs for optimal control of piecewise-affine systems

  • Autoři: Abdalmoaty, M.R., prof. Ing. Didier Henrion, Ph.D., Rodrigues, L.F.
  • Publikace: Proceedings of the European Control Conference. Zurich: European Control Association, 2013, pp. 3173-3178. ISBN 978-3-033-03962-9.
  • Rok: 2013
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    This paper considers the class of deterministic continuous-time optimal control problems (OCPs) with piecewise-affine (PWA) vector field, polynomial Lagrangian and semialgebraic input and state constraints. The OCP is first relaxed as an infinite-dimensional linear program (LP) over a space of occupation measures. This LP is then approached by an asymptotically converging hierarchy of linear matrix inequality (LMI) relaxations. The relaxed dual of the original LP returns a polynomial approximation of the value function that solves the Hamilton-Jacobi-Bellman (HJB) equation of the OCP. Based on this polynomial approximation, a suboptimal policy is developed to construct a state feedback in a sample-and-hold manner. The results show that the suboptimal policy succeeds in providing a suboptimal state feedback law that drives the system relatively close to the optimal trajectories and respects the given constraints

Optimal switching control design for polynomial systems: An LMI approach

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Daafouz, J., Claeys, M.
  • Publikace: Proceedings of the 52nd IEEE Conference on Decision and Control. Piscataway: IEEE, 2013. p. 1349-1354. ISSN 0191-2216. ISBN 978-1-4673-5717-3.
  • Rok: 2013
  • DOI: 10.1109/CDC.2013.6760070
  • Odkaz: https://doi.org/10.1109/CDC.2013.6760070
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We propose a new LMI approach to the design of optimal switching sequences for polynomial dynamical systems with state constraints. We formulate the switching design problem as an optimal control problem which is then relaxed to a linear programming (LP) problem in the space of occupation measures. This infinite-dimensional LP can be solved numerically and approximately with a hierarchy of convex finite-dimensional LMIs. In contrast with most of the existing work on LMI methods, we have a guarantee of global optimality, in the sense that we obtain an asympotically converging (i.e. with vanishing conservatism) hierarchy of lower bounds on the achievable performance. We also explain how to construct an almost optimal switching sequence.

Set approximation via minimum-volume polynomial sublevel sets

  • Autoři: Dabbene, F., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: Proceedings of the European Control Conference. Zurich: European Control Association, 2013. pp. 1114-1119. ISBN 978-3-033-03962-9.
  • Rok: 2013
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Motivated by problems of uncertainty propagation and robust estimation we are interested in computing a polynomial sublevel set of fixed degree and minimum volume that contains a given semialgebraic set K. At this level of generality this problem is not tractable, even though it becomes convex e.g. when restricted to nonnegative homogeneous polynomials. Our contribution is to describe and justify a tractable L1-norm or trace heuristic for this problem, relying upon hierarchies of linear matrix inequality (LMI) relaxations when K is semialgebraic, and simplifying to linear constraints when K is a collection of samples, a discrete union of points.

Convex inner approximations of nonconvex semialgebraic sets applied to fixed-order controller design

  • DOI: 10.1080/00207179.2012.675521
  • Odkaz: https://doi.org/10.1080/00207179.2012.675521
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    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 maximisation 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 publicdomain Matlab package solving nonconvex polynomial optimisation problems with the help of convex semidefinite programming (optimisation 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.

Inner Approximations for Polynomial Matrix Inequalities and Robust Stability Regions

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Lasserre, J.-B.
  • Publikace: IEEE Transactions on Automatic Control. 2012, 57(6), 1456-1467. ISSN 0018-9286.
  • Rok: 2012
  • DOI: 10.1109/TAC.2011.2178717
  • Odkaz: https://doi.org/10.1109/TAC.2011.2178717
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Following a polynomial approach, many robust fixedorder controller design problems can be formulated as optimization problems whose set of feasible solutions is modeled by parametrized polynomial matrix inequalities (PMIs). These feasibility sets are typically nonconvex. Given a parametrized PMI set, we provide a hierarchy of linear matrix inequality (LMI) problems whose optimal solutions generate inner approximations modeled by a single polynomial superlevel set. Those inner approximations converge in a well-defined analytic sense to the nonconvex original feasible set, with asymptotically vanishing conservatism. One may also impose the hierarchy of inner approximations to be nested or convex. In the latter case, they do not converge any more to the feasible set, but they can be used in a convex optimization framework at the price of some conservatism. Finally, we show that the specific geometry of nonconvex polynomial stability regions can be exploited to improve convergence of the hierarchy of inner approximations.

Linear control of time-domain constrained systems

  • Autoři: Aangenent, W., Heemels, W., Van de Molengraft, M., prof. Ing. Didier Henrion, Ph.D., Steinbuch, M.
  • Publikace: Automatica. 2012, 48(5), 736-746. ISSN 0005-1098.
  • Rok: 2012
  • DOI: 10.1016/j.automatica.2012.02.017
  • Odkaz: https://doi.org/10.1016/j.automatica.2012.02.017
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    This paper presents a general framework for the design of linear controllers for linear systems subject to time-domain constraints. The design framework exploits sums-of-squares techniques to incorporate the time-domain constraints on closed-loop signals and leads to conditions in terms of linear matrix inequalities (LMIs). This control design framework offers, in addition to constraint satisfaction, also the possibility of including an optimization objective that can be used to minimize steady state (tracking) errors, to decrease the settling time, to reduce overshoot and so on. The effectiveness of the framework is shown via a numerical example.

Positive trigonometric polynomials for strong stability of difference equations

  • DOI: 10.1016/j.automatica.2012.06.021
  • Odkaz: https://doi.org/10.1016/j.automatica.2012.06.021
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We follow a polynomial approach to analyse strong stability of continuous-time linear difference equations with several delays. Upon application of the Hermite stability criterion on the discretetime homogeneous characteristic polynomial, assessing strong stability amounts to deciding positive definiteness of a multivariate trigonometric polynomial matrix. This latter problem is addressed with a converging hierarchy of linear matrix inequalities (LMIs). Numerical experiments indicate that certificates of strong stability can be obtained at a reasonable computational cost for state dimension and number of delays not exceeding 4 or 5.

Projection methods in convex optimization

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Malick, J.
  • Publikace: Handbook on Semidefinite, Conic and Polynomial Optimization. Berlin: Springer-Verlag, 2012. p. 565-600. International Series in Operations Research & Management Science. vol. 166. ISBN 978-1-4614-0768-3.
  • Rok: 2012
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    There exist efficient algorithms to project a point onto the intersection of a convex conic and an affine subspace. Those conic projections are in turn the work-horse of a range of algorithms in conic optimization, having a variety of applications in science, finance and engineering. This chapter reviews some of these algorithms, emphasizing the so-called regularization algorithms for linear conic optimization, and applications in polynomial optimization. This is a presentation of the material of several recent research articles; we aim here at clarifying the ideas, presenting them in a general framework, and pointing out important techniques.

SEMIDEFINITE CHARACTERISATION OF INVARIANT MEASURES FOR ONE-DIMENSIONAL DISCRETE DYNAMICAL SYSTEMS

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Using recent results on measure theory and algebraic geometry, we show how semidefinite programming can be used to construct invariant measures of one-dimensional discrete dynamical systems (iterated maps on a real interval). In particular we show that both discrete measures (corresponding to finite cycles) and continuous measures (corresponding to chaotic behavior) can be recovered using standard software.

Semidefinite programming for optimizing convex bodies under width constraints

  • DOI: 10.1080/10556788.2010.547580
  • Odkaz: https://doi.org/10.1080/10556788.2010.547580
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We consider the problem of minimizing a functional (such as the area, perimeter and surface) within the class of convex bodies whose support functions are trigonometric polynomials. The convexity constraint is transformed via the Fejér-Riesz theorem on positive trigonometric polynomials into a semidefinite programming problem. Several problems such as the minimization of the area in the class of constantwidth planar bodies, rotors and space bodies of revolution are revisited. The approach seems promising to investigate more difficult optimization problems in the class of three-dimensional convex bodies.

A hierarchy of LMI inner approximations of the set of stable polynomials

  • DOI: 10.1016/j.automatica.2011.02.026
  • Odkaz: https://doi.org/10.1016/j.automatica.2011.02.026
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Exploiting spectral properties of symmetric banded Toeplitz matrices, we describe simple sufficient conditions for the positivity of a trigonometric polynomial formulated as linear matrix inequalities (LMIs) 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 sizem) of the nonconvex set of Schur stable polynomials of given degree n < m. It is shown that when m 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 n) already studied in the technical literature. An application to robust controller design is described.

H2 for HIFOO

  • Autoři: Arzelier, D., Deaconu, G., Gumussoy, S., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: Proceedings of the International Conference on Control and Optimization with Industrial Applilcations. Ankara: Bilkent University, 2011, pp. 1-13.
  • Rok: 2011
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    HIFOO is a public-domain Matlab package initially designed for H1 fixed-order controller synthesis, using nonsmooth nonconvex optimization techniques. It was later on extended to multi-objective synthesis, including strong and simultaneous stabilization under H1 constraints. In this paper we describe a further extension of HIFOO to H2 performance criteria, making it possible to address mixed H2/H1 synthesis. We give implementation details and report our extensive benchmark results.

Hermite matrix in Lagrange basis for scaling static output feedback polynomial matrix inequalities

  • Autoři: Delibasi, A., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: Proceedings of the 2nd IMA Conference on Numerical Linear Algebra and Optimization. University of Birmingham, 2011,
  • Rok: 2011
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Using Hermite's formulation of polynomial stability conditions, static output feedback (SOF) controller design can be formulated as a polynomial matrix inequality (PMI), a (generally nonconvex) nonlinear semidefinite programming problem that can be solved (locally) with PENNON, an implementation of a penalty and augmented Lagrangian method. Typically, Hermite SOF PMI problems are badly scaled and experiments reveal that this has a negative impact on the overall performance of the solver. In this note we recall the algebraic interpretation of Hermite's quadratic form as a particular Bezoutian and we use results on polynomial interpolation to express the Hermite PMI in a Lagrange polynomial basis, as an alternative to the conventional power basis. Numerical experi- ments on benchmark problem instances show the substantial improvement brought by the approach, in terms of problem scaling, number of iterations and convergence behavior of PENNON.

Minimizing the sum of many rational functions

Moment and SDP relaxation techniques for smooth approximations of problems involving nonlinear differential equations

  • Autoři: Mevissen, M., Lasserre, J.B., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: Proceedings of the 18th IFAC World Congress, 2011. Bologna: IFAC, 2011, pp. 10887-10892. ISSN 1474-6670. ISBN 978-3-902661-93-7.
  • Rok: 2011
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Combining recent moment and sparse semidefinite programming (SDP) relaxation techniques, we propose an approach to find smooth approximations for solutions of problems involving nonlinear differential equations. Given a system of nonlinear differential equations, we apply a technique based on finite differences and sparse SDP relaxations for polynomial optimization problems (POP) to obtain a discrete approximation of its solution. In a second step we apply maximum entropy estimation (using moments of a Borel measure associated with the discrete solution) to obtain a smooth closed-form approximation. The approach is illustrated on a variety of linear and nonlinear ordinary differential equations (ODE), partial differential equations (PDE) and optimal control problems (OCP), and preliminary numerical results are reported.

Positive trigonometric polynomials for strong stability of difference equations

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Vyhlídal, T.
  • Publikace: Proceedings of the 18th IFAC World Congress, 2011. Bologna: IFAC, 2011, pp. 296-301. ISSN 1474-6670. ISBN 978-3-902661-93-7. Available from: http://www.ifac-papersonline.net/Detailed/47575.html
  • Rok: 2011
  • DOI: 10.3182/20110828-6-IT-1002.01902
  • Odkaz: https://doi.org/10.3182/20110828-6-IT-1002.01902
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We follow a polynomial approach to analyse strong stability of linear difference equations with several independent delays. Upon application of the Hermite stability criterion on the discrete-time homogeneous characteristic polynomial, assessing strong stability amounts to deciding positive definiteness of a multivariate trigonometric polynomial matrix. This latter problem is addressed with a converging hierarchy of linear matrix inequalities (LMIs). Numerical experiments indicate that certificates of strong stability can be obtained at a reasonable computational cost for state dimension and number of delays not exceeding 4 or 5.

Projection methods for conic feasibility problems: applications to polynomial sum-of-squares decompositions

  • DOI: 10.1080/10556780903191165
  • Odkaz: https://doi.org/10.1080/10556780903191165
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    This paper presents a projection-based approach for solving conic feasibility problems. To find a point at the intersection of a cone and an affine subspace, we simply project a point onto this intersection. This projection is computed by dual algorithms operating a sequence of projections onto the cone and generalizing the alternating projection method. We release an easy-to-use Matlab package implementing an elementary dual-projection algorithm. Numerical experiments show that, for solving some semidefinite feasibility problems, the package is competitive with sophisticated conic programming software.We also provide a particular treatment for semidefinite feasibility problems modelling polynomial sum-of-squares decomposition problems.

Semidefinite Representation of Convex Hulls of Rational Varieties

  • DOI: 10.1007/s10440-011-9623-9
  • Odkaz: https://doi.org/10.1007/s10440-011-9623-9
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Using elementary duality properties of positive semidefinite moment matrices and polynomial sum-of-squares decompositions, we prove that the convex hull of rationally parameterized algebraic varieties is semidefinite representable (that is, it can be represented as a projection of an affine section of the cone of positive semidefinite matrices) in the case of (a) curves; (b) hypersurfaces parameterized by quadratics; and (c) hypersurfaces parameterized by bivariate quartics; all in an ambient space of arbitrary dimension.

Advanced LMI based analysis and design for Acrobot walking

  • DOI: 10.1080/00207179.2010.484468
  • Odkaz: https://doi.org/10.1080/00207179.2010.484468
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    This article aims to further improve previously developed design for Acrobot walking based on partial exact feedback linearisation of order 3. Namely, such an exact system transformation leads to an almost linear system where error dynamics along trajectory to be tracked is a 4-dimensional linear time-varying system having three time-varying entries only, the remaining entries being either zero or one. In such a way, exponentially stable tracking can be obtained by quadratically stabilising a linear system with polytopic uncertainty. The current improvement is based on applying linear matrix inequalities (LMI) methods to solve this problem numerically. This careful analysis significantly improves previously known approaches. Numerical simulations of Acrobot walking based on the above-mentioned LMI design are demonstrated as well.

Convex inner approximations of nonconvex semialgebraic sets applied to fixed-order controller design

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Louembet, Ch.
  • Publikace: Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems - MTNS 2010. Budapest: MTA SZTAKI - Hungarian Academy of Sciences, 2010, pp. 1-14. ISBN 978-963-311-370-7.
  • Rok: 2010
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    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 re- gions with concave boundaries. In particular, the algorithm leaves invariant a given convex set. The algorithm is based on Gloptipoly 3, a public-domain Matlab pack- age 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.

Detecting rigid convexity of bivariate polynomials

  • DOI: 10.1016/j.laa.2009.10.033
  • Odkaz: https://doi.org/10.1016/j.laa.2009.10.033
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Given a polynomial x ∈ Rn -> p(x) in n = 2 variables, a symbolicnumerical algorithm is first described for detecting whether the connected component of the plane sublevel set P = {x : p(x) 0} containing the origin is rigidly convex, or equivalently, whether it has a linear matrix inequality (LMI) representation, or equivalently, if polynomial p(x) is hyperbolic with respect to the origin. The problem boils down to checking whether a univariate polynomial matrix is positive semidefinite, an optimization problem that can be solved with eigenvalue decomposition. When the variety C = {x : p(x) = 0} is an algebraic curve of genus zero, a second algorithm based on Be´ zoutians is proposed to detect whether P has an LMI representation and to build such a representation from a rational parametrization of C. Finally, some extensions to positive genus curves and to the case n > 2 are mentioned.

Finding largest small polygons with GloptiPoly

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Messine, F.
  • Publikace: Proceedings of the Toulouse Global Optimization workshop. Toulouse: ENSEEIHT, Ecole Polytechnique, 2010, pp. 1-16.
  • Rok: 2010
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    A small polygon is a convex polygon of unit diameter. We are interested in small polygons which have the largest area for a given number of vertices n. Many instances are already solved in the lit- erature, namely for all odd n, and for n = 4; 6 and 8. Thus, for even n 10, instances of this problem remain open. Finding those largest small polygons can be formulated as nonconvex quadratic pro- gramming problems which can challenge state-of-the-art global opti- mization algorithms. We show that a recently developed technique for global polynomial optimization, based on a semideFinite programming approach to the generalized problem of moments and implemented in the public-domain Matlab package GloptiPoly, can successfully Find largest small polygons for n = 10 and n = 12. Therefore this signif- icantly improves existing results in the domain.

Hermite matrix in Lagrange basis for scaling static output feedback polynomial matrix inequalities

  • DOI: 10.1080/00207179.2010.531397
  • Odkaz: https://doi.org/10.1080/00207179.2010.531397
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Using Hermite's formulation of polynomial stability conditions, static output feedback (SOF) controller design can be formulated as a polynomial matrix inequality (PMI), a (generally nonconvex) nonlinear semidefinite programming problem that can be solved (locally) with PENNON, an implementation of a penalty and augmented Lagrangian method. Typically, Hermite SOF PMI problems are badly scaled and experiments reveal that this has a negative impact on the overall performance of the solver. In this note we recall the algebraic interpretation of Hermite's quadratic form as a particular Be´zoutian and we use results on polynomial interpolation to express the Hermite PMI in a Lagrange polynomial basis, as an alternative to the conventional power basis. Numerical experiments on benchmark problem instances show the improvement brought by the approach, in terms of problem scaling, number of iterations and convergence behaviour of PENNON.

Les coupes des spectraedres

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Les spectraedres sont des objets a la géométrie complexe, une généralisation des polyedres dont les aretes et les faces peuvent etre incurvées. On peut visualiser les spectraedres a condition de les découper en tranches, et c'est l'objet de cet article.

Polynomial LPV synthesis applied to turbofan engines

  • Autoři: Gilbert, W., prof. Ing. Didier Henrion, Ph.D., Bernussou, J., Boyer, D.
  • Publikace: Control Engineering Practice. 2010, 18(9), 1077-1083. ISSN 0967-0661.
  • Rok: 2010
  • DOI: 10.1016/j.conengprac.2008.10.019
  • Odkaz: https://doi.org/10.1016/j.conengprac.2008.10.019
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Results on polynomial fixed-order controller design are extended to SISO gain-scheduling with guaranteed stability and H1 performance over the whole scheduling parameter range. Salient features of the approach are:(a)the use of polynomials as modeling objects;(b)the use of flexible linear matrix inequalities (LMI) conditions allowing polynomial dependence of the open-loop system and controller transfer functions in the scheduling parameters;and(c) the decouplingin the LMI conditions between the Lyapunov variables and the controller variables,allowing both parameter-dependent Lyapunov functions and fixed-order controller design.The synthesis procedure is integrated into the ATOL framework developed by the manufacturer of aircraft and space engines Snecma to systematically design reduced complexity gain-scheduled control laws for aircraft turbofan engines.

Semidefinite Geometry of the Numerical Range

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular, it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI), an affine section of the semidefinite cone, is always dual to the numerical range of a matrix, which is therefore an affine projection of the semidefinite cone. Both primal and dual sets can also be viewed as convex hulls of explicit algebraic plane curve components. Several numerical examples illustrate this interplay between algebra, geometry and semidefinite programming duality. Finally, these techniques are used to revisit a theorem in statistics on the independence of quadratic forms in a normally distributed vector.

An improved Toeplitz algorithm for polynomial matrix null-space computation

  • Autoři: Zuniga Anaya, J.C., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: Applied Mathematics and Computation. 2009, 207(1), 256-272. ISSN 0096-3003.
  • Rok: 2009
  • DOI: 10.1016/j.amc.200810.037
  • Odkaz: https://doi.org/10.1016/j.amc.200810.037
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    In this paper, we present an improved algorithm to compute the minimal null-space basis of polynomial matrices, a problem which has many applications in control and systems theory. This algorithm takes advantage of the block Toeplitz structure of the Sylvester matrix associated with the polynomial matrix. The analysis of algorithmic complexity and numerical stability shows that the algorithm is reliable and can be considered as an efficient alternative to the well-known pencil (state-space) algorithms found in the literature.

Approximate Volume and Integration for Basic Semialgebraic Sets

  • DOI: 10.1137/080730287
  • Odkaz: https://doi.org/10.1137/080730287
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Given a basic compact semialgebraic set K Rn, we introduce a methodology that generates a sequence converging to the volume of K. This sequence is obtained from optimal values of a hierarchy of either semidefinite or linear programs. Not only the volume but also every finite vector of moments of the probability measure that is uniformly distributed on K can be approximated as closely as desired, which permits the approximation of the integral on K of any given polynomial; the extension to integration against some weight functions is also provided. Finally, some numerical issues associated with the algorithms involved are briefly discussed.

Gloptipoly 3: moments, optimization and semidefinite programming

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Lasserre, J.-B., Lofberg, J.
  • Publikace: Optimization Methods and Software. 2009, 2009(4&5), 761-779. ISSN 1055-6788.
  • Rok: 2009

LMI based design for the Acrobot walking

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    This paper aims to further improve previously developed design for Acrobot walking based on partial exact feedback linearization of order 3. Namely, such an exact system transformation leads to an almost linear system where error dynamics along trajectory to be tracked is a 4-dimensional linear time-varying system having 3 time-varying entries only, the remaining entries being either zero or one. In such a way, exponentially stable tracking can be obtained by quadratically stabilizing a linear system with polytopic uncertainty. The current improvement is based on applying LMI methods to solve this problem numerically. This careful analysis significantly improves previously known approaches. Numerical simulations of Acrobot walking based on the above mentioned LMI design are demonstrated as well.

Multiobjective Robust Control with HIFOO 2.0

  • Autoři: Gumussoy, S., prof. Ing. Didier Henrion, Ph.D., Millstone, M., Overton, M.
  • Publikace: Preprints of ROCOND'09. Haifa: Technion-israel Institute of Technology, Faculty of EE, 2009. pp. 1-6. ISBN 978-3-902661-45-6.
  • Rok: 2009
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Multiobjective control design is known to be a difficult problem both in theory and practice. Our approach is to search for locally optimal solutions of a nonsmooth optimization problem that is built to incorporate minimization objectives and constraints for multiple plants. We report on the success of this approach using our public-domain matlab toolbox hifoo 2.0, comparing our results with benchmarks in the literature.

Optimal Low-Frequency Filter Design for Uncertain 2-1 Sigma-Delta Modulators

  • Autoři: McKernan, J., Gani, M., Yang, F., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: IEEE SIGNAL PROCESSING LETTERS. 2009, 16(5), 362-365. ISSN 1070-9908.
  • Rok: 2009
  • DOI: 10.1109/LSP.2009.2016456
  • Odkaz: https://doi.org/10.1109/LSP.2009.2016456
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Variability in the analogue components of integrators in cascaded 2-1 Sigma-Delta modulators causes imperfect cancellation of first stage quantization noise, and reduced signal-to-noise ratio in analogue-to-digital converters. Design of robust matching filters based on low-frequency weighted convex optimization over uncertain linearized representations are mathematically very complex and computationally intensive, and offer little insight into the solution. This letter describes a design method based on formal optimization of a low-frequency uncertain linearized model of the modulator, and leads to a simple intuitive result which can shed light on the more complex models. Simulation results confirm the optimal properties of the filter.

POCP: a package for polynomial optimal control problems

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Lasserre, J., Savorgnan, C.
  • Publikace: European Control Conference 2009 - ECC 09. Budapest: MTA SZTAKI - Hungarian Academy of Sciences, 2009, pp. 1-17. ISBN 978-963-311-369-1.
  • Rok: 2009
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    POCP is a new Matlab package running jointly with GloptiPoly 3 and, optionally, YALMIP. It is aimed at nonlinear optimal control problems for which all the problem data are polynomial, and provides an approximation of the optimal value as well as some control policy. Thanks to a user-friendly interface, POCP reformulates such control problems as generalized problems of moments, in turn converted by GloptiPoly 3 into a hierarchy of semidefinite programming problems whose associated sequence of optimal values converges to the optimal value of the polynomial optimal control problem. In this paper we describe the basic features of POCP and illustrate them with some numerical examples.

STRONG STABILITY OF NEUTRAL EQUATIONS WITH AN ARBITRARY DELAY DEPENDENCY STRUCTURE

  • Autoři: Michiels, W., Vyhlídal, T., Zítek, P., Nijmeijer, H., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: SIAM Journal on Control and Optimization. 2009, 48(2), 763-786. ISSN 0363-0129.
  • Rok: 2009
  • DOI: 10.1137/080724940
  • Odkaz: https://doi.org/10.1137/080724940
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    The stability theory for linear neutral equations subjected to delay perturbations is addressed. It is assumed that the delays cannot necessarily vary independently of each other, but depend on a possibly smaller number of independent parameters. As a main result, necessary and sufficient conditions for strong stability are derived along with bounds on the spectrum, which take into account the precise dependency structure of the delays. In the derivation of the stability theory, results from realization theory and determinantal representations of multivariable polynomials play an important role. The observations and results obtained in the paper are first illustrated and validated with a numerical example. Next, the effects of small feedback delays on the stability of a boundary controlled hyperbolic partial differential equation and of a control system involving state derivative feedback are analyzed.

Nonlinear Optimal Control via Occupation Measures and LMI-Relaxations

  • Autoři: Lasserre, J.B., prof. Ing. Didier Henrion, Ph.D., Prieur, Ch., Trelat, E.
  • Publikace: SIAM Journal on Control and Optimization. 2008, 47(4), 1643-1666. ISSN 0363-0129.
  • Rok: 2008

On convexity of the frequency response of a stable polynomial

Plane geometry and convexity of polynomial stability regions

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    The set of controllers stabilizing a linear system is generally non-convex in the parameter space. In the case of two-parameter controller design (e.g. PI control or static output feedback with one input and two outputs), we observe however that quite often for benchmark problem instances, the set of stabilizing controllers seems to be convex. In this note we use elementary techniques from real algebraic geometry (resultants and B´ezoutian matrices) to explain this phenomenon. As a byproduct, we derive a convex linear matrix inequality (LMI) formulation of two-parameter fixed-order controller design problem, when possible.

Robust Filter Design for Uncertain 2-1 Sigma-Delta Modulators via the Central Polynomial Method

  • Autoři: McKernan, J., Gani, M., prof. Ing. Didier Henrion, Ph.D., Yang, F.
  • Publikace: IEEE SIGNAL PROCESSING LETTERS. 2008, 15(12), 737-740. ISSN 1070-9908.
  • Rok: 2008

Fixed-Order and Structure H-Infinity Control with Model Based Feedforward for Elastic Web Winding Systems

  • Autoři: Knittel, D., prof. Ing. Didier Henrion, Ph.D., Millstone, M., Vedrines, M.
  • Publikace: In Preprints of the 11th IFAC/IFORS/IMACS/IFIP Symposium on Large Scale Systems (Theory and Applications. Gdansk: Gdansk University of Technology, Faculty of Electrical and Control Engineering, 2007.
  • Rok: 2007
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    In web transport systems, the main concern is to control independently speed and tension in spite of perturbations such as radius variations and changes of setting point. This paper presents the application of nonsmooth nonconvex optimization techniques to design centralized fixed order and decentralized fixed order and fixed structure H-infinity controllers with model based feedforward for web winding systems. The approach provides improved web tension and velocity regulation. First, mathematical models of fundamental elements in a web process line are presented. A state space model is developed which enables calculation of the phenomenological model feedforward signals and helps in the synthesis of H-infinity controllers around the set points given by the reference signals. The H-infinity control strategies with additive feedforward have been validated on a nonlinear simulator identified on a 3-motor winding test bench.

Fixed-Order Robust H Controller Design With Regional Pole Assignment

  • Autoři: Yang, F., Gani, M., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: IEEE Transactions on Automatic Control. 2007, 52(10), 1959-1963. ISSN 0018-9286.
  • Rok: 2007
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    In this technical note, the problem of designing fixed-order robust H controllers is considered for linear systems affected by polytopic uncertainty. A polynomial method is employed to design a fixed-order controller that guarantees that all the closed-loop poles reside within a given region of the complex plane. In order to utilize the freedom of the controller design, an H performance specification is also enforced by using the equivalence between robust stability and H norm constraint. The design problem is formulated as a linear matrix inequality (LMI) constraint whose decision variables are controller parameters. An illustrative example demonstrates the feasibility of the proposed design methods.

Globally Optimal Estimates for Geometric Reconstruction Problems

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We introduce a framework for computing statistically optimal estimates of geometric reconstruction problems. While traditional algorithms often suffer from either local minima or non-optimality-or a combination of both-we pursue the goal of achieving global solutions of the statistically optimal cost-function. Our approach is based on a hierarchy of convex relaxations to solve non-convex optimization problems with polynomials. These convex relaxations generate a monotone sequence of lower bounds and we show how one can detect whether the global optimum is attained at a given relaxation. The technique is applied to a number of classical vision problems: triangulation, camera pose, homography estimation and last, but not least, epipolar geometry estimation. Experimental validation on both synthetic and real data is provided. In practice, only a few relaxations are needed for attaining the global optimum.

Polynomial LPV Synthesis Applied to Turbofan Engines

  • Autoři: Gilbert, W., prof. Ing. Didier Henrion, Ph.D., Bernussou, J., Boyer, D.
  • Publikace: Proceedings of the 17th IFAC Symposium on Automatic Control in Aerospace. Laxenburg: IFAC, 2007. pp. 1-2.
  • Rok: 2007
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Recent results on fixed order controller synthesis are extended to SISO LPV syntehsis with guarantee of H-infinity performance on the whole scheduling parameter variation range. The salient features of the approach are: (a) the use of polynomials for system modeling; (b) the use of flexible LMI conditions allowing a polynomial dependence on the scheduling parameter of the open-loop system and controller transfer functions; (c) a decoupling between the Lyapunov variables and the controller variables in the LMI conditions, allowing the simultaneous use of parameter-dependent Lyapunov functions and fixed-order controller synthesis. The synthesis procedure is integrated into the simultation tool ATOL developed by Snecma, with the aim of implementing reduced-complexity systematic LPV synthesis techniques for aircraft turbofan engine control.

Quadratic Separation for Feedback Connection of an Uncertain Matrix and an Implicit Linear Transformation

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Topological separation is investigated in the case of an uncertain time-invariant matrix interconnected with an implicit linear transformation. A quadratic separator independent of the uncertainty is shown to prove losslessly the closed-loop well-posedness. Several applications for LTI descriptor system analysis are then given. First, some known results for stability and pole location of descriptor systems are demonstrated in a new way. Second, contributions to robust stability analysis and time-delay systems stability analysis are exposed. These prove to be new even when compared to results for usual LTI systems (not in descriptor form). All results are formulated as linear matrix inequalities (LMIs).

Synthese polynomiale LPV appliquée a la commande d'un moteur d'avion

  • Autoři: Gilbert, W., prof. Ing. Didier Henrion, Ph.D., Bernussou, J., Boyer, D.
  • Publikace: Proceedings of the Journées Doctorales - Journées Nationales - Modélisation, Analyse et Conduite des Systemes dynamiques - 2007. Reims: Université de Reims, 2007,
  • Rok: 2007
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Des resultats existant sur la synthese de correcteurs d'ordre fixe sont etendus au cas de la synthese LPV SISO avec garantie de performance H-infini sur l'ensemble de variation du parametre de sequencement. Les caracteristiques principales de l'approche sont: (a) l'utilisation de polynomes pour la modelisation des systemes; (b) l'utilisation de conditions LMI flexibles autorisant une dependance polynomiale en la variable de sequencement des fonctions de transfert du systeme en boucle ouverte et du correcteur; (c) l'existence d'un decouplage au niveau des conditions LMI entre les variables de Lyapunov et les variables du correcteur, qui permet a la fois l'utilisation de fonctions de Lyapunov dependant de parametre et la synthese d'un correcteur d'ordre fixe.

A Toeplitz Algorithm for Polynomial J-Spectral Factorization

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    A block Toeplitz algorithm is proposed to perform the J-spectral factorization of a para-Hermitian polynomial matrix. The input matrix can be singular or indefinite, and can have zeros along the imaginary axis. The key assumption is that the finite zeros of the input polynomial matrix are given as input data. The algorithm is based on numerically reliable operations only, namely computation of the null-spaces of related block Toeplitz matrices, polynomial matrix factor extraction and linear polynomial matrix equations solving.

Convergent Relaxations of Polynomial Matrix Inequalities and Static Output Feedback

  • DOI: 10.1109/TAC.2005.863494
  • Odkaz: https://doi.org/10.1109/TAC.2005.863494
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Using a moment interpretation of recent results on sum-of-squares decompositions of nonnegative polynomial matrices, we propose a hierarchy of convex linear matrix inequality (LMI) relaxations to solve nonconvex polynomial matrix inequality (PMI) optimization problems, including bilinear matrix inequality (BMI) problems. This hierarchy of LMI relaxations generates a monotone sequence of lower bounds that converges to the global optimum. Results from the theory of moments are used to detect whether the global optimum is reached at a given LMI relaxation, and if so, to extract global minimizers that satisfy the PMI. The approach is successfully applied to PMIs arising from static output feedback design problems.

Fixed-order H-infinity decentralized control with model based feedforward for elastic web winding systems

  • Autoři: Knittel, D., Vedrines, M., prof. Ing. Didier Henrion, Ph.D., Pagilla, P.
  • Publikace: Proceedings of the IEEE Industry Applications Society Conference. New Jersey: IEEE, 2006.
  • Rok: 2006
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Fixed-order H-infinity decentralized control with model based feedforward for elastic web winding systems

Global Optimization Toolbox for Maple

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    As surveyed in [2], many mathematical and engineering problems require a complete search. An example is the 300- year-old Kepler problem of finding the densest packing of equal spheres in three-dimensional Euclidean space, for which a computer-assisted proof is discussed in [3]. The proof involves reducing the problem to several thousand linear programs and using interval calculations to ensure rigorous handling of rounding errors for establishing the correctness of inequalities. Many other difficult problems, such as the traveling salesman and protein folding problems, are global optimization problems.

H-infinity controller design on the COMPLIB problems with the Robust Control Toolbox for Matlab

H-infinity controller design on the COMPLIB problems with theRobust Control Toolbox for Matlab

  • Autoři: prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: Proceedings of the 13th IFAC Workshop on Control Applications of Optimization. Heidelberg: IFAC, 2006,
  • Rok: 2006
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    H-infinity controller design on the COMPLIB problems with theRobust Control Toolbox for Matlab

HIFOO - A Matlab package for fixed-order controller design and H-infinity optimization

  • Autoři: Burke, J.V., prof. Ing. Didier Henrion, Ph.D., Lewis, A.S., Overton, M.L.
  • Publikace: Proceedings of the 5th IFAC Symposium on Robust Control Design ROCOND 2006. Toulouse: LAAS-CNRS, 2006.
  • Rok: 2006
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Description of a Matlab HIFOOpackage for fixed-order controller design and H-infinity optimization

Inégalités matricielles quadratiques et stabilité des polynômes

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    De nouvelles relations sont établies entre diverses inégalités matricielles pour tester la stabilité d'un polynôme scalaire. Ces inégalités sont des fonctions quadratiques des coefficients du polynôme, et linéaires (LMI) en une matrice additionnelle. Des liens sont établis entre les criteres de stabilité d'Hermite et de Lyapunov, ainsi qu'entre les techniques espace d'état et polynomiale pour l'analyse de stabilité. Les conditions pourraient servir a la synthese de lois de commande de complexité réduite.

LMIs for constrained polynomial interpolation with application in trajectory planning

  • DOI: 10.1016/j.sysconle.2005.09.011
  • Odkaz: https://doi.org/10.1016/j.sysconle.2005.09.011
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We consider an open-loop trajectory planning problem for linear systems with bound constraints originating from saturations or physical limitations. Using an algebraic approach and results on positive polynomials, we show that this control problem can be cast into a constrained polynomial interpolation problem admitting a convex linear matrix inequality (LMI) formulation.

Robust H-infinity fixed order control strategies for large scale web winding systems

  • Autoři: Knittel, D., Vedrines, M., prof. Ing. Didier Henrion, Ph.D., Pagilla, P.
  • Publikace: Proceedings of the joint IEEE CCA/CACSD/ISIC Conference. Piscataway: IEEE, 2006,
  • Rok: 2006
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Robust H-infinity fixed order control strategies for large scale web winding systems

Solving static output feedback problems by direct search optimization

Stabilization via Nonsmooth, Nonconvex Optimization

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Burke, J.V., Lewis, A.S., Overton, M.L.
  • Publikace: IEEE Transactions on Automatic Control. 2006, 51(11), 1760-1769. ISSN 0018-9286.
  • Rok: 2006
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Nonsmooth variational analysis and related computational methods are powerful tools that can be effectively applied to identify local minimizers of nonconvex optimization problems arising in fixed-order controller design. We support this claim by applying nonsmooth analysis and methods to a challenging "Belgian chocolate" stabilization problem posed in 1994: find a stable, minimum phase, rational controller that stabilizes a specified second-order plant. Although easily stated, this particular problem remained unsolved until 2002, when a solution was found using an eleventh-order controller. Our computational methods find a stabilizing third-order controller without diffi- culty, suggesting explicit formulas for the controller and for the closed loop system, which has only one pole with multiplicity 5.

Switching and periodic control of the Belgian chocolate problem

  • Autoři: Colaneri, P., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: Proceedings of the 5th IFAC Symposium on Robust Control Design ROCOND 2006. Toulouse: LAAS-CNRS, 2006.
  • Rok: 2006
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    The paper aims at providing a switching solution for a parametrized Belgian chocolate stabilization problem posed by Blondel. We show that for any given value of the parameter describing the given transfer function, there exist simple switching and periodic control laws that stabilize the plant. The simplicity in designing these control laws sharply contrasts with the difficulty of designing a non-switching non-periodic control law for this problem.

Application of generalised polynomials to the decoupling of linear multivariable systems

  • Autoři: Ruiz-León, J., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: IEE Procedings-Control Theory and Applications. 2005, 152(4), 435-442. ISSN 1350-2379.
  • Rok: 2005

Contracting Optimally an Interval Matrix without Loosing Any Positive Semi-Definite Matrix Is a Tractable Problem

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    In this paper, we show that the problem of computing the smallest interval submatrix of a given interval matrix [A] which contains all symmetric positive semi-definite (PSD) matrices of [A], is a linear matrix inequality (LMI) problem, a convex optimization problem over the cone of positive semidefinite matrices, that can be solved in polynomial time. From a constraint viewpoint, this problem corresponds to projecting the global constraint PSD (A) over its domain [A]. Projecting such a global constraint, in a constraint propagation process, makes it possible to avoid the decomposition of the PSD constraint into primitive constraints and thus increases the efficiency and the accuracy of the resolution.

Control of Linear Systems Subject to Time-Domain Constraints With Polynomial Pole Placement and LMIs

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Tarbouriech, S., Kučera, V.
  • Publikace: IEEE Transactions on Automatic Control. 2005, 50(9), 1360-1364. ISSN 0018-9286.
  • Rok: 2005
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    This note focuses on the control of continuous-time linear systems subject to time-domain constraints (input amplitude limitation, output overshoot) on closed-loop signals. Using recent results on positive polynomials, it is shown that finding a Youla-Kučera polynomial parameter of fixed degree (hence, a controller of fixed order) such that time-domain constraints are satisfied amounts to solving a convex linear matrix inequality (LMI) optimization problem as soon as distinct strictly negative closed-loop poles are assigned by pole placement. Proceeding this way, time-domain constraints are handled by an appropriate choice of the closed-loop zeros.

Detecting Infinite Zeros in Polynomial Matrices

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Zúńiga, J.-C.
  • Publikace: IEEE Transactions on Circuits and Systems II-Express Briefs. 2005, 52(12), 744-745. ISSN 1057-7130.
  • Rok: 2005
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    A simple necessary and sufficient algebraic condition is given to detect zeros at infinity in a polynomial matrix.

Numerical Stability of Block Toeplitz Algorithms in Polynomial Matrix Computations

  • Autoři: Zúňiga, J.C., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: Proceedings of the 16th World Congress of the International Federation of Automatic Control. Praha: IFAC, 2005. ISSN 1474-6670. ISBN 978-0-08-045108-4.
  • Rok: 2005
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    We study the problem of computing the eigenstructure of a polynomial matrix. Via a backward error analysis we analyze the stability of some block Toeplitz algorithms to obtain this eigenstructure. We also elaborate on the nature of the problem, i.e. conditioning and posedness.

Optimizing Simultaneously Over the Numerator and Denominator Polynomials in the Youla-Kučera Parametrization

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Kučera, V., Molina-Cristóbal, A.
  • Publikace: IEEE Transactions on Automatic Control. 2005, 50(9), 1369-1374. ISSN 0018-9286.
  • Rok: 2005
  • DOI: 10.1109/TAC.2005.854618
  • Odkaz: https://doi.org/10.1109/TAC.2005.854618
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Traditionally, when approaching controller design with the Youla-Kučera parametrization ofall stabilizing controllers, the denominator ofthe rational parameter is fixed to a given stable polynomial, and optimization is carried out over the numerator polynomial. In this note, we revisit this design technique, allowing to optimize simultaneously over the numerator and denominator polynomials. Stability ofthe denominator polynomial, as well as fixed-order controller design with Hinf performance are ensured via the notion ofa central polynomial and linear matrix inequality (LMI) conditions for polynomial positivity.

Quadratic Separation for Feedback Connection of an Uncertain Matrix and an Implicit Linear Transformation

  • Autoři: Peaucelle, D., prof. Ing. Didier Henrion, Ph.D., Arzelier, D.
  • Publikace: Proceedings of the 16th World Congress of the International Federation of Automatic Control. Praha: IFAC, 2005. ISSN 1474-6670. ISBN 978-0-08-045108-4.
  • Rok: 2005
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Topological separation is investigated in the case of an uncertain time-invariant matrix interconnected with an implicit linear transformation. A quadratic separator independent of the uncertainty is shown to prove losslessly the closed-loop well-posedness. Several applications for descriptor systems are then given. First, some known results for stability and pole location are demonstrated in a new way. Second, contributions to robust stability analysis are exposed. All results are formulated as linear matrix inequalities (LMIs).

Real-Time H2 and H_inf Control of a Gyroscope Using a Polynomial Approach

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    In this paper H2 and H_inf control techniques ara applied to the real-time control of a gyroscope with two degrees of freedom. The controllers are designed based on a polynomial approach and using routines from the Polynomial Toolbox for MATLAB. Real-time results are presented, showing a good performance of the controllers.

Robust Pole Placement for Second-Order Systems: an LMI Approach

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Based on recently developed sufficient conditions for stability of polynomial matrices, an LMI technique is described to perform robust pole placement by proportional-derivative feedback on second-order linear systems affected by polytopic or norm-bounded uncertainty. As illustrated by several numerical examples, at the core of the approach is the choice of a nominal, or central quadratic polynomial matrix.

Control of linear systems subject to time-domain constraints with polynomial pole placement and LMIs

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Tarbouriech, S., Kučera, V.
  • Publikace: Proceedings Sixteenth International Symposium on: Mathematical Theory of Networks and Systems. Leuven: Katholieke Universiteit, 2004. ISBN 90-5682-517-8.
  • Rok: 2004

Fixed-order robust controller design with the Polynomial Toolbox 3.0

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    With the help of numerical examples, this paper describes new fixed-order robust controller design functions implemented in version 3.0 of the Polynomial Toolbox for Matlab. The functions use convex optimization over linear matrix inequalities (LMIs) solved with the SeDuMi solver.

On parameter-dependent Lyapunov functions for robust stability of linear systems

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Arzelier, D., Paucelle, D., Lasserre, J.-B.
  • Publikace: Proceedings 43rd IEEE Conference on Decision and Control. New York: IEEE Control System Society, 2004. p. 887-892. ISBN 0-7803-8683-3.
  • Rok: 2004
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    For a linear system affected by real parametric uncertainty, this paper focuses on robust stability analysis via quadratic-in-the-state Lyapunov functions polynomially dependent on the parameters. The contribution is twofold. First, if n denotes the system order and m the number of parameters, it is shown that it is enough to seek a parameterdependent Lyapunov function of given degree 2nm in the parameters. Second, it is shown that robust stability can be assessed by globally minimizing a multivariate scalar polynomial related with this Lyapunov matrix. A hierarchy of LMI relaxations is proposed to solve this problem numerically, yielding simultaneously upper and lower bounds on the global minimum with guarantee of asymptotic convergence.

On the application of displacement structure methods to obtain null-spaces of polynomial matrices

  • Autoři: Zúniga, J.C., prof. Ing. Didier Henrion, Ph.D.,
  • Publikace: Proceedings 43rd IEEE Conference on Decision and Control. New York: IEEE Control System Society, 2004. p. 3406-3411. ISBN 0-7803-8683-3.
  • Rok: 2004
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    In this paper we present different algorithms to obtain the null-space of a polynomial matrix. These algorithms are based on the computation of the constant null-spaces of some associated block Toeplitz matrices. For the case of large block Toeplitz matrices we introduce two fast numerical methods to compute their constant null-spaces and we compare the performance of these fast methods with the classical orthogonal methods. We also discuss briefly the numerical stability of the developed algorithms.

Optimizing simultaneously over the numerator and denominator polynomials in the Youla-Kučera parametrization

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Kučera, V., Molina-Cristóbal, A.
  • Publikace: Proceedings 43rd IEEE Conference on Decision and Control. New York: IEEE Control System Society, 2004. pp. 2177-2181. ISBN 0-7803-8683-3.
  • Rok: 2004
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Traditionally, when approaching controller design with the Youla-Kučera parametrization of all stabilizing controllers, the denominator of the rational parameter is fixed to a given stable polynomial, and optimization is carried out over the numerator polynomial. In this work, we revisit this design technique, allowing to optimize simultaneously over the numerator and denominator polynomials. Stability of the denominator polynomial, as well as fixed-order controller design with Hinf performance are ensured via the notion of a central polynomial and LMI conditions for polynomial positivity.

Overcoming non-convexity in polynomial robust control design

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    When developing efficient and reliable computer-aided control system design (CACSD) tools for low-order robust control systems analysis and synthesis, the main issue faced by theoreticians and practitioners is the non-convexity of the stability domain in the space of polynomial coefficients, or equivalently, in the space of design parameters. In this paper, we survey some of the recently developed techniques to overcome this non-convexity, underlining their respective pros and cons. We also enumerate some related open research problems which, in our opinion, deserve particular attention.

Robust Root-Clustering of a Matrix in Intersections or Unions of Regions

  • Autoři: Bachelier, O., prof. Ing. Didier Henrion, Ph.D., Pradin, B., Mehdi, D.
  • Publikace: SIAM Journal on Control and Optimization. 2004, 43(3), 1078-1093. ISSN 0363-0129.
  • Rok: 2004
  • DOI: 10.1137/S0363012903432365
  • Odkaz: https://doi.org/10.1137/S0363012903432365
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    This paper considers robust stability analysis for a matrix affected by unstructured complex uncertainty. A method is proposed to compute a bound on the amount of uncertainty ensuring robust root-clustering in a combination (intersection and/or union) of several possibly nonsymmetric half planes, discs, and outsides of discs. In some cases to be detailed, this bound is not conservative. The conditions are expressed in terms of linear matrix inequalities (LMIs) and derived through Lyapunov's second method. As a distinctive feature of the approach, the Lyapunov matrices proving robust root-clustering (one per subregion) are not necessarily positive definite but have prescribed inertias depending on the number of roots in the corresponding subregions.

Solving nonconvex optimization problems

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    The objective of this article is to show how GloptiPoly can solve challenging nonconvex optimization problems in robust and nonlinear control. First, we describe several non-convex optimization problems arising in control system analysis and design. These problems involve multivariable polynomial objective functions and constraints. We then review the theoretical background behind GloptiPoly. Next, an example is presented to illustrate the successive LMI relaxations and general features of the GloptiPoly software. Finally, we apply GloptiPoly to several control-related problems.

Controller Design Using Polynomial Matrix Description

Ellipsoidal Approximation of the Stability Domain of a Polynomial

Polynomial and matrix Fraction Dessription

Polynomial Methods and LMI Optimization: New Robust Control Functions for the Polynomial Toolbox 3.0

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Polynomial Methods and LMI Optimization: New Robust Control Functions for the Polynomial Toolbox 3.0

Positive Polynomials and Robust Stabilization with Fixed-Order Controllers

Robust Control Design for Parametric Uncertainties via Polynomial Toolbox for Matlab

Robust Pole Placement for Second-Order Systems: An LMI Approach

Discrete-Time Symmetric Polynomial Equations with Complex coefficients

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Discrete-time symmetric polynomial equations with complex coefficients are studied in the scalar and matrix case. New theoretical results axe derived and several algorithms are proposed and evaluated. Polynomial reduction algorithms are first described to study theoretical properties of the equations. Sylvester matrix algorithms are then developed to solve numerically the equations. The algorithms are implemented in the Polynomial Toolbox for MATLAB.

GloptiPoly: Global Optimization over Polynomials with Matlab and SeDuMi

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Lasserre, J.B.
  • Publikace: Proceedings of the 41st IEEE Conference on decision and Control. New York: IEEE Control System Society, 2002. p. 747-774. ISBN 0-7803-7516-5.
  • Rok: 2002
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    GloptiPoly is a Matlab/SeDuMi add-on to build and solve convex linear matrix inequality relaxations of the (generally non-convex) global optimization problem of minimizing a multivariable polynomial function subject to polynomial inequality, equality or integer constraints. It generates a series of lower bounds monotonically converging to the global optimum. Global optimality is detected and isolated optimal solutions are extracted automatically. Numerical experiments show that for most of the small- and medium-scale problems described in the literature, the global optimum is reached at low computational cost.

Rank-one LMI Approach to Robust Stbility of Polynomial Matrices

An LMI Condition for Robust Stability of Polynomial Matrix Polytopes

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    A new condition for robust stability of polynomial matrix polytopes is presented which is based on LMI

Control of Linear Systems Subject to Input Constraints: a Polynomial Approach

D-stability of polynomial matrices

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Necessary and sufficient conditions are formulated for the zeros of an arbitrary polynomial matrix to belong to a given region D of the complex plane. The conditions stem from a general optimization methodology mixing quadratic and semidefinite programming, LFRs and rank-one LMIs. They are expressed as an LMI feasibility problem that can be tackled with widespread powerful interior-point methods. Most importantly, the D-stability conditions can be combined with other LMI conditions arising in robust stability analysis.

Rank-One LMI Approach to Robust Stability of Polynomial Matrices

Rank-one LMI Approach to Stability of 2-D Polynomial Matrices

An algorithm for polynomial matrix factor extraction

Control of Linear Systems Subject to Input Constraints: a Polynomial Approach. MIMO Case

  • Autoři: prof. Ing. Didier Henrion, Ph.D., Tarbouriech, S., Kučera, V.
  • Publikace: 2000 American Control Conference. Chicago: Illinois Institute of Technology, 2000, pp. 1774-1778. ISBN 0-7803-5522-9.
  • Rok: 2000
  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Control of Linear Systems Subject to Input Constraints: a Polynomial Approach. MIMO case

Stabilization of Affine Polynomial Families: an LMI Approach

Rank-one LMI Approach to Simultaneous Stabilization of Linear Systems

Reliable numerical methods for polynomial matrix triangularization

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    Numerical procedures are proposed for triangularizing polynomial matrices over the field of polynomial fractions and over the ring of polynomials. They are based on two standard polynomial techniques: Sylvester matrices and interpolation, In contrast to other triangularization methods, the algorithms described in this paper only rely on well-worked numerically reliable tools. They can also be used for greatest common divisor extraction, polynomial rank evaluation, or polynomial null-space computation.

Reliable Numerical Methods for Polynomial Matrix Triangularization

Algebraic approach to robust controller design: A geometric interpretation

Efficient numerical method for the discrete-time symmetric matrix polynomial equation

  • Pracoviště: Katedra řídicí techniky
  • Anotace:
    New efficient numerical method for solution of the discrete-time symmetric matrix polynomial equation

Numerical methods for polynomial matrix rank evaluation

Symmetric matrix polynomial equation: Interpolation results

Za stránku zodpovídá: Ing. Mgr. Radovan Suk