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International audience; We consider a linear-quadratic deterministic optimal control problem where the control takes values in a two-dimensional simplex. The phase portrait of the optimal synthesis contains second-order singular extremals and exhibits modes of infinite accumulations of switchings in finite time, so-called chattering. We prove the presence of an entirely new phenomenon, namely a chaotic behaviour of the set of optimal trajectories. The set of optimal non-wandering trajectories has the structure of a Cantor set, and the dynamics of the system is described by a topological Markov chain. We compute the entropy and the Hausdorff dimension of the non-wandering set. This behaviour is generic for piece-wise smooth Hamiltonian systems in the vicinity of a junction of three discontinuity hypersurface strata.

In this paper the following problem is studied: design an input signal with the property that the estimated model based on this signal satisfies a given performance level with a prescribed probability. This problem is mathematically translated into a chance constrained optimization problem, which is typically non convex. To solve it, several convex approximations are proposed and compared.

International audience; We consider optimal experiment design for parametric prediction error system identification of linear time-invariant systems in closed loop. The optimisation is performed jointly over the controller and the external input. We use a partial correlation approach, i.e. we parameterize the set of "admissible controller"-"external input" pairs by a finite set of matrix-valued trigonometric moments. Our main contribution is twofold. First we derive a description of the set of admissible finite-dimensional moments by a linear matrix inequality. Optimal input design problems with semi-definite constraints and criteria which are linear in these moments can then be cast as semi-definite programs and solved by standard semi-definite programming packages. Secondly, we develop algorithms to recover the controller and the power spectrum of the external input from the optimal moment vector. This furnishes the user a complete and very general procedure to solve the input design problems of the considered class. Our results can be applied to multi-input multi-output systems, but for pedagogical reasons we present here the single-input single-output case. We also assume that the true system is in the model set.

International audience; In this paper we investigate the problem of designing an input signal for a multi-input multi-output plant to minimize a control-oriented criterion. By employing Ljung's asymptotic (in model order and sample size) covariance formulas, we determine closed form expressions for the optimal input, which provide direct insight into the effect of the principal directions and gains of the open- and closed-loop transfer functions on the kind of experiment to be applied.

International audience; In this paper we consider the problem of input design for open loop identification of linear single input single output finite impulse response systems, under the assumption of full-order modeling. We provide analytic formulas for the information matrix of the optimal input when the design is intended to identify a scalar function of the impulse response coefficients. In particular, we characterize the cases when the optimal information matrix is singular and the optimal input is necessarily a multisine signal. Our results have surprising consequences for the robustness properties of optimal input designs. We show that, contrary to intuition, the choice of an optimal regular information matrix might pose robustness problems if there exist also singular solutions.