Laboratoire d’analyse et d’architecture des systèmes
A.OUSTRY, C.CARDOZO, P.PANCIATICI, D.HENRION
LIX, RTE, MAC
Rapport LAAS N°18377, Novembre 2018, 6p.
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 positive 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.
G.DAVY, E.FERON, P-L.GAROCHE, D.HENRION
ONERA, Georgia Institute, MAC
Manifestation avec acte : International Conference on Logic for Programming Artificial Intelligence and Reasoning ( LPAR ) 2018 du 16 novembre au 21 novembre 2018, Awassa (Ethiopie), Novembre 2018, 17p. , N° 18009
With the increasing power of computers, real-time algorithms tends to become more complex and therefore require better guarantees of safety. Among algorithms sustaining autonomous embedded systems, model predictive control (MPC) is now used to compute online trajec-tories, for example in the SpaceX rocket landing. The core components of these algorithms, such as the convex optimization function, will then have to be certified at some point. This paper focuses specifically on that problem and presents a method to formally prove a primal linear programming implementation. We explain how to write and annotate the code with Hoare triples in a way that eases their automatic proof. The proof process itself is performed with the WP-plugin of Frama-C and only relies on SMT solvers. Combined with a framework producing all together both the embedded code and its annotations, this work would permit to certify advanced autonomous functions relying on online optimization.
Rapport LAAS N°18346, Novembre 2018, 16p.
We interpret some wrong results (due to numerical inaccuracies) already observed when solving SDP-relaxations for polynomial optimization on a double precision floating point SDP solver. It turns out that this behavior can be explained and justified satisfactorily by a relatively simple paradigm. In such a situation, the SDP solver (and not the user) performs some `robust optimization' without being told to do so. Instead of solving the original optimization problem with nominal criterion f, it uses a new criterion f~ which belongs to a ball B∞(f,ε) of small radius ε>0, centered at the nominal criterion f in the parameter space. In other words the resulting procedure can be viewed as a `max−min' robust optimization problem with two players (the solver which maximizes on B∞(f,ε) and the user who minimizes over the original decision variables). A mathematical rationale behind this `autonomous' behavior is described.
M.CAPELLE, M.J.HUGUET, N.JOZEFOWIEZ, X.OLIVE
ROC, LCOMS, Thalès Alenia Space
Revue Scientifique : Networks, Vol.73, N°2, p.234-253, Novembre 2018, DOI : 10.1002/net.21859 , N° 18338
Free space optical communications are becoming a mature technology to cope with the needs of high data rate payloads for future low‐earth orbiting observation satellites. However, they are strongly impacted by clouds. In this paper, we aim to find a network of optical ground stations maximizing the percentage of data acquired by a low‐earth orbiting satellite that can be transferred to the Earth, taking into consideration cloud information. This problem can be separated in two parts and solved hierarchically: the selection of a network of optical ground stations and the assignment of downloads to visibility windows of the stations. We present theoretical and practical results regarding the complexity of the latter subproblem and propose a dynamic programming algorithm to solve it. We combine this algorithm with two methods for the enumeration of the stations, and compare them with a mixed integer linear program (MILP). Results show that even if the MILP can solve scenarios over small horizons, the hierarchical approaches outperform it in term of computation time while still achieving optimality for larger instances.
R.BOURBON, S.U.NGUEVEU , X.ROBOAM, B.SARENI, C.TURPIN, D.HERNANDEZ-TORRES
Revue Scientifique : Mathematics and Computers in Simulation, Vol.158, pp.418-431, Novembre 2018, doi 10.1016/j.matcom.2018.09.022 , N° 18339
This paper aims at optimizing the energy management of a smart power plant composed of wind turbines coupled with a Lithium Ion storage device in order to fulfill a power production commitment to the utility grid. The application of this case study is typically related to islanded electric grids. Our work particularly investigates and compares two classes of energy management strategies for design purpose: a first capable of providing the global optimum of the power flow planning from a Linear Programming (LP) approach thanks to a priori knowledge of future events in the environment; a second, based on a classical control heuristic without any a priori knowledge on the future, applicable in real time. Beyond the future objectives in terms of system design (techno-economical sizing optimization), the comparison of both approaches also aims at improving the predefined heuristic from the analysis of the ideal reference provided by the global LP optimizer. In this scope, a linear power flow model of the power plant is developed in compliance with a LP solver (Cplex). A particular attention is paid to the techno-economic optimization including storage cost evaluation, commitment failure penalties and exploitation gains. Simulations and optimizations are carried out over one year in order to take variability and seasonal features of the wind potential into account.
M.COCETTI, A.SERRANI, L.ZACCARIAN
Trento, OSU, MAC
Revue Scientifique : Automatica, Vol.97, pp.214-225, Novembre 2018 , N° 18584
This paper considers the linear output regulation problem for uncertain over-actuated plants. The general form of input redundancy considered in this work implies the existence of multiple control inputs and state trajectories compatible with a prescribed reference for the output. On-line selection, according to certain performance criteria, of the most suitable of these inputs-state trajectories leads to a linear output regulation problem with dynamic redundancy allocation. We present a solution that augments the well known internal model control scheme with two additional dynamical systems. The first one, named annihilator, parametrizes the inputs and the corresponding state trajectories that are invisible from the output. The second one, named redundancy allocator, dynamically selects the best solution according to a predefined performance criterion. We derive explicit solutions for the performance criterion equal to relaxed 1, 2, and ∞-norms of the plant input. This setup is a particular case of the dynamic redundancy allocation problem named dynamic input allocation. The proposed solutions can be implemented in an error feedback form and are especially suitable for optimizing sparsity, power and amplitude of the control input. Finally, structural stability, robustness and existence of a unique steady-state are proven.
Revue Scientifique : European Journal of Operational Research, Novembre 2018 , N° 18543
Various optimization problems result from the introduction of nonlinear terms into combinatorial optimization problems. In the context of energy optimization for example, energy sources can have very different characteristics in terms of power range and energy demand/cost function, also known as efficiency function or energy conversion function. Introducing these energy sources characteristics in combinatorial optimization problems, such as energy resource allocation problems or energy-consuming activity scheduling problems may result into mixed integer nonlinear problems neither convex nor concave. Approximations via piecewise linear functions have been proposed in the literature. Non-convex optimization models and heuristics exist to compute optimal breakpoint positions under a bounded absolute error-tolerance. We present an alternative solution method based on the upper and lower bounding of nonlinear terms using non necessarily continuous piecewise linear functions with a relative epsilon-tolerance. Conditions under which such approach yields a pair of mixed integer linear programs with a performance guarantee are analyzed. Models and algorithms to compute the non necessarily continuous piecewise linear functions with absolute and relative tolerances are also presented. Computational evaluations performed on energy optimization problems for hybrid electric vehicles show the efficiency of the method with regards to the state of the art.
A.BISOFFI, F.FORNI, M.DA LIO, L.ZACCARIAN
Trento, Univ of Cambridge, MAC
Revue Scientifique : Automatica, Vol.97, pp.104-114, Novembre 2018 , N° 18585
This work explores the potential of relay-based control on a one-degree-of-freedom nonlinear mechanical system, in the contexts of both sustaining and damping oscillations. For both cases we state our main results building upon a simple reset formulation (relay feedback) and providing intuitive basic equations from classical mechanics. With a more rigorous description following a hybrid system formalism, we establish then the global asymptotic stability of the corresponding (compact-set) attractors through hybrid Lyapunov tools. The aspects of sustaining and damping oscillation are seen as complementary, because they reduce to a suitable mirroring of the reset surface. Finally, we discuss two applications of our results to the case of a hopping mass and an automotive suspension.
Texas A&M University, MAC
Revue Scientifique : Journal of Approximation Theory, Vol.235, pp.74-91, Novembre 2018 , N° 18013
The long-standing problem of minimal projections is addressed from a computational point of view. Techniques to determine bounds on the projection constants of univariate polynomial spaces are presented. The upper bound, produced by a linear program, and the lower bound, produced by a semidefinite program exploiting the method of moments, are often close enough to deduce the projection constant with reasonable accuracy. The implementation of these programs makes it possible to find the projection constant of several three-dimensional spaces with five digits of accuracy, as well as the projection constants of the spaces of cubic, quartic, and quintic polynomials with four digits of accuracy. Beliefs about uniqueness and shape-preservation of minimal projections are contested along the way.
F.FERRANTE, F.GOUAISBAUT, R.G.SANFELICE, S.TARBOURIECH
GIPSA-Lab, MAC, Arizona
Revue Scientifique : IEEE Transactions on Automatic Control, 17p., Novembre 2018 , N° 18343
This paper deals with the problem of estimating the state of a nonlinear time-invariant system in the presence of sporadically available measurements and external perturbations. An observer with a continuous intersample injection term is proposed. Such an intersample injection is provided by a linear dynamical system, whose state is reset to the measured output estimation error whenever a new measurement is available. The resulting system is augmented with a timer triggering the arrival of a new measurement and analyzed in a hybrid system framework. The design of the observer is performed to achieve exponential convergence with a given decay rate of the estima- tion error. Robustness with respect to external perturbations and L2-external stability from plant perturbations to a given performance output are considered. Computationally efficient algorithms based on the solution to linear matrix inequalities are proposed to design the observer. Finally, the effectiveness of the proposed methodology is shown in an example.