Laboratoire d’analyse et d’architecture des systèmes
M.A.DAVO, F.GOUAISBAUT, A.BANOS, S.TARBOURIECH, A.SEURET
GIPSA-Lab, MAC, Murcia
Revue Scientifique : Nonlinear Analysis: Hybrid Systems , Vol.29, pp.133-146, Août 2018 , N° 18133
The paper deals with the stability analysis of time delay reset control systems, for which the resetting law is assumed to satisfy a time-dependent condition. A stability analysis of the closed-loop system is performed based on an appropriate sampled-data system. New linear matrix inequality (LMI) conditions are proposed to ensure the exponential stability of the closed-loop system resulting from the connection of a plant with a proportional and integral controller together with a reset integrator (PI+RI).
A.TANWANI, S.MARX, C.PRIEUR
Manifestation avec acte : International Symposium on Mathematical Theory of Networks and Systems ( MTNS ) 2018 du 16 juillet au 20 juillet 2018, Hong Kong (Chine), Juillet 2018, 6p. , N° 18143
The problem of robust stabilization with bounded feedback control is considered for a scalar reaction-diffusion system with uncertainties in the dynamics. The maximum value of the control input acting on one of the boundary points has to respect a given bound at each time instant. It is shown that, if the initial condition and the disturbance satisfy the certain bounds (computed as a function of the bound imposed on the control input), then the proposed control respects the desired saturation level and renders the closed-loop system locally input-to-state stable, that is, the trajectories with certain bound on the initial condition converge to a ball parameterized by certain norm of the disturbance.
D.HENRION, S.NALDI, M.SAFEY EL DIN
MAC, XLIM, Limoges, LIP6-CNRS
Manifestation avec acte : International Symposium on Symbolic and Algebraic Computation ( ISSAC ) 2018 du 16 juillet au 19 juillet 2018, New York city (USA), Juillet 2018, 17p. , N° 18038
Let A 0 ,. .. , A n be m × m symmetric matrices with entries in Q, and let A(x) be the linear pencil A 0 + x 1 A 1 + · · · + x n A n , where x = (x 1 ,. .. , x n) are unknowns. The linear matrix inequality (LMI) A(x) 0 defines the subset of R n , called spectrahedron, containing all points x such that A(x) has non-negative eigenvalues. The minimization of linear functions over spectrahedra is called semidefinite programming (SDP). 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 solving SDP are mostly based on the interior point method, assuming some non-degeneracy properties such as the existence of interior points in the admissible set. 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 but we prove that solving such problems can be done in polynomial time if either n or m is fixed.
V.MAGRON, M.FORETS, D.HENRION
LIP6-CNRS, VERIMAG, MAC
Rapport LAAS N°18183, Juillet 2018, 28p.
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 linear 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.
M.PARK, S.H.LEE, O.KWON, A.SEURET
Chungbuk Natinonal University, MAC
Revue Scientifique : IEEE Transactions on Cybernetics, Vol.48, N°7, pp.2192-2205, Juillet 2018, doi 10.1109/TCYB.2017.2729164 , N° 17535
This paper investigates synchronization in complex dynamical networks (CDNs) with interval time-varying delays. The CDNs are representative of systems composed of a large number of interconnected dynamical units, and for the purpose of the mathematical analysis, the leading work is to model them as graphs whose nodes represent the dynamical units. At this time, we take note of the importance of each node in networks. One way, in this paper, is that the closeness-centrality mentioned in the field of social science is grafted onto the CDNs. By constructing a suitable Lyapunov-Krasovskii functional, and utilizing some mathematical techniques, the sufficient and closeness-centrality-based conditions for synchronization stability of the networks are established in terms of linear matrix inequalities. Ultimately, the use of the closeness-centrality can be weighted with regard to not only the interconnection relation among the nodes, which was utilized in the existing works but also more information about nodes. Here, the centrality will be added as the concerned information. Moreover, to avoid the computational burden causing the nonconvex term including the square of the time-varying delay, how to deal with it is applied by estimating it to the convex term including time-varying delay. Finally, two illustrative examples are given to show the advantage of the closeness-centrality in point of the robustness on time-delay.
M.SAFI, L.BAUDOUIN, A.SEURET
Manifestation avec acte : IFAC Workshop on Time Delay Systems ( TDS ) 2018 du 28 juin au 30 juin 2018, Budapest (Hongrie), Juin 2018, 6p. , N° 18137
This paper deals with the stability analysis of time delay systems based on continuous-time approach. The originality of the present paper relies on the construction of several models for a same time-delay systems using the interconnection of an ordinary differential equation and a transport partial differential equation. The stability analysis is then performed using a Lyapunov functional. These models are constructed in order to first reduce potentially the complexity of the resulting stability conditions. Second several models are build in order to be interpreted as a discretization scheme as the one usually used in the Lyapunov functional. The proposed result can be seen as a generalized (N − M) discretization which consists in both a time-discretization of the delay interval into M sub-intervals, and the projection of the state function within each sub-interval on the Legendre polynomials of degree less than N. The efficiency of this novel approach is illustrated on an academic example.
S.TARBOURIECH, A.GIRARD, L.HETEL
MAC, L2S, CRIStAL
Ouvrage (éditeur) : Control Subject to Computational and Communication Constraints: Current Challenges, Springer, N°ISBN 978-3-319-78448-9, Juin 2018, 375p. , N° 18170
This book provides a broad overview of the current problems, challenges and solutions in the field of control theory, communication theory and computational resources management. Recent results on dynamical systems, which open new opportunities for research and challenges to be addressed in the future, are proposed in the context of computational and communication constraints. In order to take into the account complex phenomena, such as nonlinearities, time-varying parameters and limited availability of information, the book proposes new approaches for open problems with both theoretical and practical significance. The contributors’ research is centred on robust stability and performance of control loops that are subject to computational and communication constraints. A particular focus is placed on the presence of constraints in communication and computation, which is a critical issue in networked control systems and cyber-physical systems. The contributions, which rely on the development of novel paradigms are provided are by leading experts in the field from all over the world, thus providing readers with the most accurate solutions for the constraints. Control subject to Computational and Communication Constraints highlights many problems encountered by control researchers, while also informing graduate students of the many interesting ideas at the frontier between control theory, information theory and computational theory. The book is also a useful point of reference for engineers and practitioners, and the survey chapters will assist instructors in lecture preparation.
M.BARREAU, F.GOUAISBAUT, A.SEURET
Manifestation avec acte : European Control Conference ( ECC ) 2018 du 12 juin au 15 juin 2018, Limassol (Chypre), Juin 2018, 7p. , N° 17428
In this paper, the design of a static feedback gain for a linear system subject to an input delay is studied. This synthesis is based on a stability analysis conducted using Lyapunov-Krasovskii theorem and Bessel-Legendre inequalities expressed in terms of LMIs. Some bilinear non-convex matrix inequalities are obtained to go from analysis to synthesis. They are then difficult to solve and an iterative LMI procedure is given which takes advantage of the elimination lemma. Naturally, slack variables are introduced and then, following an optimization process, values for them are proposed to reduce the conservatism. The two main corollaries discuss the static state and output feedback synthesis. Finally, a comparison is proposed and shows that this formulation introduces small conservatism.
Rapport LAAS N°18146, Juin 2018, 14p.
Not every positive functional defined on bi-variate polynomials of a prescribed degree bound is represented by the integration against a positive measure. We isolate a couple of conditions filling this gap, either by restricting the class of polynomials to harmonic ones, or imposing the vanishing of a defect indicator. Both criteria offer constructive cubature formulas and they are obtained via well known matrix analysis techniques involving either the dilation of a contractive matrix to a unitary one or the specific structure of the Hessenberg matrix associated to the multiplier by the underlying complex variable.
S.MARX, Y.CHITOUR, C.PRIEUR
MAC, L2S, GIPSA-Lab
Manifestation avec acte : European Control Conference ( ECC ) 2018 du 12 juin au 15 juin 2018, Limassol (Chypre), Juin 2018, 8p. , N° 17412
This article deals with the derivation of ISS-Lyapunov functions for infinite-dimensional linear systems subject to saturations. Two cases are considered: 1) the saturation acts in the same space as the control space; 2) the saturation acts in another space, especially a Banach space. For the first case, an explicit ISS-Lyapunov function can be derived. For the second case, we can only ensure the existence of an ISS-Lyapunov function.