Section: New Results

Research axis 4: Applications (tools: ALG-HOM-SET)

• Robust set-point tracking control and optimal control algorithms for turbulent flows have been developed in [26] and tested in Wind Tunnel L1 of ONERA, Lille. (https://www.youtube.com/watch?v=b5NnAV2qeno) The set-point tacking control have been patented, FR 1755440, “Dispositif de contrôle actif du recollement d'un écoulement sur un profil”.

• In [75], [77], the development of a robust ($H_{\infty }$) control for parametric systems has been initiated. A general framework based on symbolic computation techniques was proposed. In these two papers, the general approach has been applied to the case of linear systems of order up to four and illustrated with the two mass-spring system with damping. In particular, closed forms for the robust controllers and for the robustness radius were obtained. Finally, the robust stabilization of the line of sight of a stabilized mirror system, modeled by a time-delay fourth order system, was studied in [76].

• Within a collaboration with Safran Tech Laboratory and Safran Electronics & Defense, in [98], we propose a symbolic method for the explicit computation of certain invariant observers studied in navigation theory.

• An experimental synchronization of a family of a recently proposed oscillator model (i.e. the Brockett oscillator) was studied and implemented in [12].

• In [13], high frequency measurements of various water characteristics and nutrients information of the Marel-Carnot sea monitoring station (Boulogne-sur-Mer, France) have been used to identify a physiological model for phytoplankton bloom through the fluorescence signal. An auto-regressive-moving-average with exogenous inputs (ARMAX) model is designed and tested based on the dataset. It was demonstrated that the developed dynamical model can be used for estimating the fluorescence level and for predicting the various states of phytoplankton bloom. Thus, the developed model can be used for monitoring phytoplankton biomass in the water which in turn might give information about unbalanced ecosystem or change in water quality.

• The problem of latency reduction in direct human-computer interaction was considered in [50] and formulated as a trajectory prediction problem. The predictor was constructed as a frequency-domain approximation of the non-casual ideal predictor. This approximation can be computed analytically, or obtained as an optimization task. An adaptive modification of the forecasting algorithm was proposed taking into account possible variations in user behavior.

• In [24], a necessary and sufficient criterion to establish input-to-state stability (ISS) of nonlinear dynamical systems, the dynamics of which are periodic with respect to certain state variables and which possess multiple invariant solutions (equilibria, limit cycles, etc.), is provided. Unlike standard Lyapunov approaches, the condition is relaxed and formulated via a sign-indefinite function with sign-definite derivative, and by taking the system’s periodicity explicitly into account. The new result is established by using the framework of cell structure and it complements the ISS theory of multistable dynamics for periodic systems. The efficiency of the proposed approach is illustrated via the global analysis of a nonlinear pendulum with constant persistent input.

• Conditions for almost global stability of an operating point of a realistic model of a synchronous generator with constant field current connected to an infinite bus are derived in [38]. The analysis is conducted by employing the recently proposed concept of input-to-state stability (ISS)–Leonov functions, which is an extension of the powerful cell structure principle developed by Leonov and Noldus to the ISS framework. Compared with the original ideas of Leonov and Noldus, the ISS–Leonov approach has the advantage of providing additional robustness guarantees. The efficiency of the derived sufficient conditions is illustrated via numerical experiments. This article is part of the themed issue ‘Energy management: flexibility, risk and optimization’.

• Conditions for existence and global attractivity of the equilibria of a realistic model of a synchronous generator with constant field current connected to an infinite bus are derived in [14]. First, necessary and sufficient conditions for existence and uniqueness of equilibrium points are provided. Then, sufficient conditions for local asymptotic stability and almost global attractivity of one of these equilibria are given. The analysis is carried out by employing a new Lyapunov–like function to establish convergence of bounded trajectories, while the latter is proven using the powerful theoretical framework of cell structures pioneered by Leonov and Noldus. The efficiency of the derived sufficient conditions is illustrated via extensive numerical experiments based on two benchmark examples taken from the literature.

• In [96], we propose a new approach for testing the stability of $n$D systems. The standard stability conditions are transformed into algebraic conditions and then checked by means of computer algebra techniques for solving algebraic systems such as Gröbner bases, univariate representations and discriminant varieties. The corresponding results were implemented in Maple .

• In [17], we address the problem of computing stabilizing controllers for a specific class of multidimensional SISO systems. This problem, which was an open problem (i.e., no effective methods were existing for the computation of stabilizing controllers), has been solved using techniques from computer algebra. As a result, an effective test of stabilizability as well as an algorithm for computing stabilizing controllers were developed.

• We have recently proposed a new method for the anchor position self-calibration problem, a rather well-known problem in the signal processing community. In essence, given two sets of wireless communicating devices, i.e. sources and sensors lying in the three dimensional space, the self-calibration algorithm estimates the position of the devices by only using the source–sensor distance measurements. We have first reformulated the problem in terms of certain matrix equalities. They can then be studied in detail by means of computer algebra methods such as Gröbner basis techniques and the package OreModules . Coupling symbolic methods with standard linear algebra techniques, we obtain a general solution in all dimensions. In particular, for a space of dimension three, very compact closed-form solutions are obtained in a particular reference frame. Thanks to these closed-form solutions, the noise effect can then be characterized yielding the synthesis of realtime filtering to mitigate the effect of the measurement noise. Finally, the resulting implementation is rather straightforward and based on real-time operations. Additionally, the underlying numerical tools are standard (least-squares, low-rank factorization, matrix calculus) and well-known. The result of this work is being transferred to a patent. A software prototype AutoCal (https://bil.inria.fr/fr/search/query?terms=AutoCal in the BIL) is available on the server Autocalibrationserver (https://allgo.inria.fr/webapps/166) under the Inria platform AllGO , which allows the user to test the implemented algorithm on his own dataset.