On the stabilization of persistently excited linear systems

We consider control systems of the type $\dot x = A x +\alpha(t)bu$, where $u\in\R$, $(A,b)$ is a controllable pair and $\alpha$ is an unknown time-varying signal with values in $[0,1]$ satisfying a persistent excitation condition i.e., \int_t^{t+T}\al(s)ds\geq \mu for every $t\geq 0$, with $0<\mu\leq T$ independent on $t$. We prove that such a system is stabilizable with a linear feedback depending only on the pair $(T,\mu)$ if the eigenvalues of $A$ have non-positive real part. We also show that stabilizability does not hold for arbitrary matrices $A$. Moreover, the question of whether the system can be stabilized or not with an arbitrarily large rate of convergence gives rise to a bifurcation phenomenon in dependence of the parameter $\mu/T$

Time Optimal Synthesis for Left--Invariant Control Systems on SO(3)

Consider the control system given by $\dot x=x(f+ug)$, where $x\in SO(3)$, $|u|\leq 1$ and $f,g\in so(3)$ define two perpendicular left-invariant vector fields normalized so that \|f\|=\cos(\al) and \|g\|=\sin(\al), \al\in ]0,\pi/4[. In this paper, we provide an upper bound and a lower bound for $N(\alpha)$, the maximum number of switchings for time-optimal trajectories. More precisely, we show that N_S(\al)\leq N(\al)\leq N_S(\al)+4, where N_S(\al) is a suitable integer function of \al which for \al\to 0 is of order $\pi/(4\alpha).$ The result is obtained by studying the time optimal synthesis of a projected control problem on $R P^2$, where the projection is defined by an appropriate Hopf fibration. Finally, we study the projected control problem on the unit sphere $S^2$. It exhibits interesting features which will be partly rigorously derived and partially described by numerical simulations

Rolling Manifolds: Intrinsic Formulation and Controllability

In this paper, we consider two cases of rolling of one smooth connected complete Riemannian manifold $(M,g)$ onto another one (\hM,\hg) of equal dimension $n\geq 2$. The rolling problem $(NS)$ corresponds to the situation where there is no relative spin (or twist) of one manifold with respect to the other one. As for the rolling problem $(R)$, there is no relative spin and also no relative slip. Since the manifolds are not assumed to be embedded into an Euclidean space, we provide an intrinsic description of the two constraints "without spinning" and "without slipping" in terms of the Levi-Civita connections $\nabla^{g}$ and \nabla^{\hg}. For that purpose, we recast the two rolling problems within the framework of geometric control and associate to each of them a distribution and a control system. We then investigate the relationships between the two control systems and we address for both of them the issue of complete controllability. For the rolling $(NS)$, the reachable set (from any point) can be described exactly in terms of the holonomy groups of $(M,g)$ and (\hM,\hg) respectively, and thus we achieve a complete understanding of the controllability properties of the corresponding control system. As for the rolling $(R)$, the problem turns out to be more delicate. We first provide basic global properties for the reachable set and investigate the associated Lie bracket structure. In particular, we point out the role played by a curvature tensor defined on the state space, that we call the \emph{rolling curvature}. In the case where one of the manifolds is a space form (let say (\hM,\hg)), we show that it is enough to roll along loops of $(M,g)$ and the resulting orbits carry a structure of principal bundle which preserves the rolling $(R)$ distribution. In the zero curvature case, we deduce that the rolling $(R)$ is completely controllable if and only if the holonomy group of $(M,g)$ is equal to SO(n). In the nonzero curvature case, we prove that the structure group of the principal bundle can be realized as the holonomy group of a connection on $TM\oplus \R$, that we call the rolling connection. We also show, in the case of positive (constant) curvature, that if the rolling connection is reducible, then $(M,g)$ admits, as Riemannian covering, the unit sphere with the metric induced from the Euclidean metric of $\R^{n+1}$. When the two manifolds are three-dimensional, we provide a complete local characterization of the reachable sets when the two manifolds are three-dimensional and, in particular, we identify necessary and sufficient conditions for the existence of a non open orbit. Besides the trivial case where the manifolds $(M,g)$ and (\hM,\hg) are (locally) isometric, we show that (local) non controllability occurs if and only if $(M,g)$ and (\hM,\hg) are either warped products or contact manifolds with additional restrictions that we precisely describe. Finally, we extend the two types of rolling to the case where the manifolds have different dimensions

Generalized robust shrinkage estimator and its application to STAP detection problem

Recently, in the context of covariance matrix estimation, in order to improve as well as to regularize the performance of the Tyler's estimator [1] also called the Fixed-Point Estimator (FPE) [2], a "shrinkage" fixed-point estimator has been introduced in [3]. First, this work extends the results of [3,4] by giving the general solution of the "shrinkage" fixed-point algorithm. Secondly, by analyzing this solution, called the generalized robust shrinkage estimator, we prove that this solution converges to a unique solution when the shrinkage parameter $\beta$ (losing factor) tends to 0. This solution is exactly the FPE with the trace of its inverse equal to the dimension of the problem. This general result allows one to give another interpretation of the FPE and more generally, on the Maximum Likelihood approach for covariance matrix estimation when constraints are added. Then, some simulations illustrate our theoretical results as well as the way to choose an optimal shrinkage factor. Finally, this work is applied to a Space-Time Adaptive Processing (STAP) detection problem on real STAP data

Rolling Manifolds of Different Dimensions

If $(M,g)$ and (\hM,\hg) are two smooth connected complete oriented Riemannian manifolds of dimensions $n$ and \hn respectively, we model the rolling of $(M,g)$ onto (\hM,\hg) as a driftless control affine systems describing two possible constraints of motion: the first rolling motion $\Sigma_{NS}$ captures the no-spinning condition only and the second rolling motion $\Sigma_{R}$ corresponds to rolling without spinning nor slipping. Two distributions of dimensions (n + \hn) and $n$, respectively, are then associated to the rolling motions $\Sigma_{NS}$ and $\Sigma_{R}$ respectively. This generalizes the rolling problems considered in \cite{ChitourKokkonen1} where both manifolds had the same dimension. The controllability issue is then addressed for both $\Sigma_{NS}$ and $\Sigma_{R}$ and completely solved for $\Sigma_{NS}$. As regards to $\Sigma_{R}$, basic properties for the reachable sets are provided as well as the complete study of the case (n,\hn)=(3,2) and some sufficient conditions for non-controllability

Singular trajectories of control-affine systems

When applying methods of optimal control to motion planning or stabilization problems, some theoretical or numerical difficulties may arise, due to the presence of specific trajectories, namely, singular minimizing trajectories of the underlying optimal control problem. In this article, we provide characterizations for singular trajectories of control-affine systems. We prove that, under generic assumptions, such trajectories share nice properties, related to computational aspects; more precisely, we show that, for a generic system -- with respect to the Whitney topology --, all nontrivial singular trajectories are of minimal order and of corank one. These results, established both for driftless and for control-affine systems, extend previous results. As a consequence, for generic systems having more than two vector fields, and for a fixed cost, there do not exist minimizing singular trajectories. We also prove that, given a control system satisfying the LARC, singular trajectories are strictly abnormal, generically with respect to the cost. We then show how these results can be used to derive regularity results for the value function and in the theory of Hamilton-Jacobi equations, which in turn have applications for stabilization and motion planning, both from the theoretical and implementation issues