Flatness and Monge parameterization of two-input systems, control-affine with 4 states or general with 3 states

This paper studies Monge parameterization, or differential flatness, of control-affine systems with four states and twocontrols. Some of them are known to be flat, and this implies admitting a Monge parameterization. Focusing on systems outside this class, we describe the only possible structure of such a parameterization for these systems, and give a lower bound on the order of this parameterization, if it exists. This lower-bound is good enough to recover the known results about "(x,u)-flatness" of these systems, with much more elementary techniques

Local Controllability of the Two-Link Magneto-Elastic Micro-Swimmer

A recent promising technique for robotic micro-swimmers is to endow them with a magnetization and apply an external magnetic field to provoke their deformation. In this note we consider a simple planar micro-swimmer model made of two magnetized segments connected by an elastic joint, controlled via a magnetic field. After recalling the analytical model, we establish a local controllability result around the straight position of the swimmer

Addendum to "Local Controllability of the Two-Link Magneto-Elastic Micro-Swimmer"

In the above mentioned note (, , published in IEEE Trans. Autom. Cont., 2017), the first and fourth authors proved a local controllability result around the straight configuration for a class of magneto-elastic micro-swimmers.That result is weaker than the usual small-time local controllability (STLC), and the authors left the STLC question open. The present addendum closes it by showing that these systems cannot be STLC

On the Curves that may be approached by Trajectories of a Smooth Control Affine System

In this paper, we give a characterization of the set of curves that may be approached by trajectories of a smooth control-affine nonlinear system, in the topology of uniform convergence. This characterization is in terms of the drift vector field and the distribution spanned by the Lie algebra generated by the control vector fields. The characterization is valid on open sets where this distribut- ion has constant rank. These results also characterize the support of diffusion processes with smooth coefficients

Sufficient Stability Conditions for Time-varying Networks of Telegrapher's Equations or Difference Delay Equations

We give a sufficient condition for exponential stability of a network of lossless telegrapher's equations, coupled by linear time-varying boundary conditions. The sufficient conditions is in terms of dissipativity of the couplings, which is natural for instance in the context of microwave circuits. Exponential stability is with respect to any $L^p$-norm, $1\leq p\leq\infty$. This also yields a sufficient condition for exponential stability to a special class of linear time-varying difference delay systems which is quite explicit and tractable. One ingredient of the proof is that $L^p$ exponential stability for such difference delay systems is independent of $p$, thereby reproving in a simpler way some results from [3].Comment: To be published in SIAM Journal on Mathematical Analysis, most probably 202

Multi-phase averaging of time-optimal low-thrust transfers

International audienceAn increasing interest in optimal low-thrust orbital transfers was triggered in the last decade by technological progress in electric propulsion and by the ambition of efficiently leveraging on orbital perturbations to enhance the maneuverability of small satellites. The assessment of a control sequence that is capable of steering a satellite from a prescribed initial to a desired final state while minimizing a figure of interest is referred to as maneuver planning. From the dynamical point of view, the necessary conditions for optimality outlined by the infamous Pontryagin maximum principle (PMP) reveal the Hamiltonian nature of the system governing the joint motion of state and control variables. Solving the control problem via so-called indirect techniques, e.g., shooting method, requires the integration of several trajectories of the aforementioned Hamiltonian. In addition , PMP conditions exhibit very high sensitivity with respect to boundary values of the satellite longitude owing to the fast-oscillating nature of orbital motion. Hence, using perturbation theory to facilitate the numerical solution of the planning problem is appealing. In particular, averaging techniques were used since the early space age to gain understanding into the long-term evolution of perturbed satellite trajectories. However, it is not generally possible to treat low-thrust as any other perturbation (whose spectral content is well defined and predictable) because the control variables may introduce additional frequencies in the system. The talk focuses on time optimal maneuvers in a perturbed orbital environment, and it addresses two questions: (1) Is it possible to average the vector field of this problem? Optimal control Hamiltonians are not in the classical form of fast-oscillating systems. However, we demonstrate that averaged trajectories well approximate the original system if the ad-joint variables of the PMP (i.e., conjugate momenta associated to the enforcement of the equations of motion) are adequately transformed before integrating the averaged trajec-tory. We discuss this transformation in detail, and we emphasize fundamental differences with respect to well-known mean-to-osculating transformations of uncontrolled motion. (2) What is the impact of orbital perturbations and their frequencies on the controlled tra-jectory? We show that control variables are highly sensitive to small exogenous forces. Hence, even the crossing of a high-order resonance may trigger a dramatic divergence between trajectories of the averaged and original system. We then discuss how averaged resonant forms may be used to avoid this divergence. The methodology is finally applied to a deorbiting maneuver leveraging on solar radiation pressure. The presence of eclipses make the original planning problem highly challenging. Averaging with respect to satellite and Sun longitudes drastically simplifies the extremal flow yielding an averaged counterpart of the PMP conditions, which is reasonably easy to solve

On the curves that may be approached by trajectories of a smooth control affine system

International audienceIn this paper, we give a characterization of the set of curves that may be approached by trajectories of a smooth control-ane nonlinear system, in the topology of uniform convergence. This characterization is in terms of the drift vector field and the distribution spanned by the Lie algebra generated by the control vector fields. The characterization is valid on open sets where this distribution has constant rank. These results also characterize the support of di↵usion processes with smooth coecients.

A necessary condition for dynamic equivalence

International audienceIf two control systems on manifolds of the same dimension are dynamic equivalent, we prove that either they are static equivalent --i.e. equivalent via a classical diffeomorphism-- or they are both ruled; for systems of different dimensions, the one of higher dimension must ruled. A ruled system is one whose equations define at each point in the state manifold, a ruled submanifold of the tangent space. Dynamic equivalence is also known as equivalence by endogenous dynamic feedback, or by a Lie-Bäcklund transformation when control systems are viewed as underdetermined systems of ordinary differential equations; it is very close to absolute equivalence for Pfaffian systems. It was already known that a differentially flat system must be ruled; this is a particular case of the present result, in which one of the systems is assumed to be "trivial" (or linear controllable)