Laboratoire d’analyse et d’architecture des systèmes
A.BENOIT, M.M.JOLDES, M.MEZZAROBBA
EXT, MAC, LIP6-CNRS
Revue Scientifique : Mathematics of Computation , Vol.86, N°305, pp.1303-1341, Mai 2017 , N° 14329
A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of such methods in the context of rigorous computing (where we need guarantees on the accuracy of the result), and from the complexity point of view. It is well-known that the order-n truncation of the Chebyshev expansion of a function over a given interval is a near-best uniform polynomial approximation of the function on that interval. In the case of solutions of linear differential equations with polynomial coefficients, the coefficients of the expansions obey linear recurrence relations with polynomial coefficients. Unfortunately, these recurrences do not lend themselves to a direct recursive computation of the coefficients, owing among other things to a lack of initial conditions. We show how they can nevertheless be used, as part of a validated process, to compute good uniform approximations of D-finite functions together with rigorous error bounds, and we study the complexity of the resulting algorithms. Our approach is based on a new view of a classical numerical method going back to Clenshaw, combined with a functional enclosure method.
L.BAUDOUIN, A.SEURET, F.GOUAISBAUT, M.DATTAS
Rapport LAAS N°17102, Avril 2017, 6p.
This paper addresses the stability analysis of a system of ordinary differential equations coupled with a classic heat equation using a Lyapunov approach. Relying on recent developments in the area of time-delay systems, a new method to study the stability of such a class of coupled finite/infinite dimensional systems is presented here. It consists in a Lyapunov analysis of the infinite dimensional state of the system using an energy functional enriched by the mean value of the heat variable. The main technical step relies on the use an efficient Bessel-like integral inequality on Hilbert space leading to tractable conditions expressed in terms of linear matrix inequalities. The results are then illustrated on academic examples and demonstrate the potential of this new approach.
Y.ARIBA, D.ARZELIER, L.URBINA IGLESIAS
Rapport LAAS N°17095, Avril 2017, 6p.
L.DAL COL, S.TARBOURIECH, L.ZACCARIAN, M.KIEFFER
Rapport LAAS N°17081, DOI: 10.1002/rnc.3739, Avril 2017, 19p.
In this paper, we provide necessary and sufficient conditifor the quality-fair delivery of multimedia contents to mobile users. We control the encoding rates and the transmission rates of the video streams, delivered through a limited capacity channel. This problem is cast into a problem of consensus among identical discrete-time linear systems, connected through a network with fixed and fully connected topology. The information exchanged over the communication network is the measure of the quality of the encoded videos. Based on a consensus result for identical linear systems, we reduce the problem of designing the proportional and integral gains of the encoding rate and transmission rate controllers to a linear static output feedback. We propose an iterative design technique based on linear matrix inequalities to solve the corresponding nonconvex problem, thereby providing a constructive optimality-based approach to the proportional and integral gains tuning problem. We demonstrate the effectiveness of our method in simulations, where we compare it with pre-existing approaches
L.BRINON ARRANZ, A.SEURET, A.PASCOAL
GIPSA-Lab, MAC, ISR, Lisbonne
Rapport LAAS N°17072, Avril 2017, 6p.
This paper deals with the problem of encircling a moving target with a fleet of unicycle-like vehicles. A new control law is developed to steer the vehicles to a circular formation whose center tracks the target. The novelty of this paper lies in the fact that the control law only uses the velocity of the target and the relative positions of the agents with respect to it, expressed in the local frame of each vehicle. Communication between agents is used to maintain the vehicles equally spaced along the circular formation. Simulation results show the effectiveness of the proposed strategy
Rapport LAAS N°17075, Avril 2017, 6p.
This paper addresses the stability analysis of linear systems subject to a time-varying delay. The contribution of this paper is twofolds. First, we aim at presenting a new matrix inequality, which can be seen as an improved version of the reciprocally convex combination, which provides a more accurate delay-dependent lower bound. When gathering this new inequality with the Wirtinger-based integral inequality, efficient stability conditions expressed in terms of LMI are designed and show a clear reduction of the conservatism with a reasonable associated computational cost. The second original contribution of this paper consists in noting that stability conditions issued from the Wirtinger-based integral inequality depends in an affine manner on the bounds of the delay function and also on its derivative. This allows to refine the definition of allowable delay set and to relax usual convex on the delay function. As a result of this new characterization, the LMI conditions allows obtaining stability regions for slow time-varying delay systems which are very closed to the constant delay case.
M.SAFI, L.BAUDOUIN, A.SEURET
Rapport LAAS N°17073, Avril 2017, 6p.
The objective of this contribution is to improve recent stability results for a system coupling ordinary differential equations to a vectorial transport partial differential equation by proposing a new structure of Lyapunov functional. Following the same process of most of the investigations in literature, that are based on an a priori selection of Lyapunov functionals and use the usual integral inequalities (Jensen, Wirtinger, Bessel...), we will present an efficient method to estimate the exponential decay rate of this coupled system leading to a tractable test expressed in terms of linear matrix inequalities. These LMI conditions stem from the new design of a candidate Lyapunov functional, but also the inherent properties of the Legendre polynomials, that are used to build a projection of the infinite dimensional part of the state of the system. Based on these polynomials and using the appropriate Bessel-Legendre inequality, we will prove an exponential stability result and in the end, we will show the efficiency of our approach on academic example.
C.NAINER, M.FURCI, A.SEURET, L.ZACCARIAN, A.FRANCHI
Trento, RIS, MAC
Rapport LAAS N°17074, Avril 2017, 6p.
This paper addresses the problem of control allocation applied to an over-actuated hovercraft-type vehicle. A hierarchical control architecture, consisting of a high level controller for trajectory tracking, and a control allocation algorithm, is developed and proved to be effective in tracking a desired trajectory while optimizing some cost related to actuator constraints. The control allocation algorithm exploits the redundancy of the system in order to keep the actuator states inside their saturation limits and tries to minimize the power consumption of the propellers. Unlike other papers on control allocation, actuator dynamics is taken into account. The control architecture is tested through simulations that well illustrate the capabilities of the proposed control design.
S.FORMENTIN, F.DABBENE, R.TEMPO, L.ZACCARIAN, S.M.SAVARESI
Politecnico, CNR-IEIIT, Torino, MAC
Revue Scientifique : IEEE Transactions on Automatic Control, Vol.62, N°4, pp.1575-1589, Avril 2017, DOI 10.1109/TAC.2016.2586606 , N° 16150
In this paper, we address robust static anti-windup compensator design and performance analysis for saturated linear closed loops in the presence of nonlinear probabilistic parameter uncertainties via randomized techniques. The proposed static anti-windup analysis and robust performance synthesis correspond to several optimization goals, ranging from minimization of the nonlinear input/output gain to maximization of the stability region or maximization of the domain of attraction. We also introduce a novel paradigm accounting for uncertainties in the energy of the disturbance inputs. Due to the special structure of linear static anti-windup design, wherein the design variables are decoupled from the Lyapunov certificates, we introduce a significant extension, called scenario with certificates (SwC), of the so-called scenario approach for uncertain optimization problems. This extension is of independent interest for similar robust synthesis problems involving parameter-dependent Lyapunov functions. We demonstrate that the scenario with certificates robust design formulation is appealing because it provides a way to implicitly design the parameter-dependent Lyapunov functions and to remove restrictive assumptions about convexity with respect to the uncertain parameters. Subsequently, to reduce the computational cost, we present a sequential randomized algorithm for iteratively solving this problem. The obtained results are illustrated by numerical examples.
F.DABBENE, D.HENRION, C.LAGOA
CNR-IEIIT, Torino, MAC, Pennsylvania
Revue Scientifique : Automatica, Vol.78, pp.110-118, Avril 2017 , N° 15322
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 solution set of linear matrix inequalities or the Schur/Hurwitz stability domains. These sets often have very complicated shapes (non-convex, and even non-connected), which renders very 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 hyperrect-angles, polytopes, or ellipsoids have been proposed. In this work, we move a step further, and propose possibly non-convex 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 encounter in minimum volume ellipsoid problems. From a computational viewpoint, we design a hierarchy of 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. Two main applications of the proposed approach are considered. The first one aims at reconstruction/approximation of sets from a finite number of samples. In the second one, 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.