Consider the controlled system $dx/dt = Ax + \alpha(t)Bu$ where the pair $(A,B)$ is stabilizableand $\alpha(t)$ takes values in $[0,1]$ and is persistently exciting. In particular, when $\alpha(t)$ becomes zero the system dynamics switches to an uncontrollable system. In this paper, we address the following question: is it possible to find a linear time-invariant state-feedback, only depending on $(A,B)$ and the parameters of the persistent excitation, which globally exponentially stabilizes the system? We give a positive answer to this question for two cases: when $A$ is neutrally stable and when the system is the double integrator.

%B Submitted to CDC 2007 %C New Orleans, USA %8 December %G eng %0 Generic %D 2007 %T Uniform stabilization for linear systems with persistency of excitation. The neutrally stable and the double integrator cases %A A. Chaillet %A Y. Chitour %A A. Lor %A M. Sigalotti %K Robotics %XConsider the controlled system $dx/dt = Ax + \alpha(t)Bu$ where the pair $(A,B)$ is stabilizable and $\alpha(t)$ takes values in $[0,1]$ and is persistently exciting. In particular, when $\alpha(t)$ becomes zero the system dynamics switches to an uncontrollable system. In this paper, we address the following question: is it possible to find a linear time-invariant state-feedback, only depending on $(A,B)$ and the parameters of the persistent excitation, which globally asymptotically stabilizes the system? We give a positive answer to this question for two cases: when $A$ is neutrally stable and when the system is the double integrator.

%B Submitted to Mathematics of Control, Signals and Systems %G eng %0 Conference Paper %B Proc. IEEE Int. Conf. on Decision and Control %D 2005 %T On the PE stabilization of time-varying systems: open questions and preliminary answers %A A. Lor %A A. Chaillet %A G. Besancon %A Y. Chitour %K Robotics %XWe address the following fundamental question: given a double integrator and a linear control that stabilizes it exponentially, is it possible to use the {\em same} control input in the case that the control input is multiplied by a time-varying term? Such question has many interesting motivations and generalizations: 1) we can pose the same problem for an input gain that depends on the state and time hence, a specific persistency of excitation property for nonlinear systems must be imposed; 2) the stabilization –with the same method– of chains of integrators of higher order than two is fundamentally more complex and has applications in the stabilization of driftless systems; 3) the popular backstepping method stabilization method for systems with non-invertible input terms. The purpose of this note is two-fold: we present some open questions that we believe are significant in time-varying stabilization and present some preliminary answers for simple, yet challenging case-studies.

%B Proc. IEEE Int. Conf. on Decision and Control %C Sevilla, Spain %P 6847–6852 %8 December %G eng %0 Journal Article %J IEEE Trans. on Automatic Control %D 2004 %T Reachability and Steering of Rolling Polyhedra: A Case Study in Discrete Nonholonomy %A A. Bicchi %A Y. Chitour %A A. Marigo %K Embedded Control %K Robotics %XRolling a ball on a plane is a standard example of nonholonomy reported in many textbooks, and the problem is also well understood for any smooth deformation of the surfaces. For non-smoothly deformed surfaces, however, much less is known. Although it may seem intuitive that nonholonomy is conserved (think e.g. to polyhedral approximations of smooth surfaces), current definitions of ``nonholonomy'' are inherently referred to systems described by ordinary differential equations, and are thus inapplicable to such systems. \İn this paper we study the set of positions and orientations that a polyhedral part can reach by rolling on a plane through sequences of adjacent faces. We provide a description of such reachable set, discuss conditions under which the set is dense, or discrete, or has a compound structure, and provide a method for steering the system to a desired reachable configuration, robustly with respect to model uncertainties. \\Based on ideas and concepts encountered in this case study, and in some other examples we provide, we turn back to the most general aspects of the problem and investigate the possible generalization of the notion of (kinematic) nonholonomy to non-smooth, discrete, and hybrid dynamical systems. To capture the essence of phenomena commonly regarded as ``nonholonomic'', at least two irreducible concepts are to be defined, of ``internal'' and ``external'' nonholonomy, which may coexist in the same system. These definitions are instantiated by examples.

%B IEEE Trans. on Automatic Control %V 49 %P 710-726 %8 May %G eng %0 Conference Paper %B Proc. IEEE Int. Conf. on Robotics and Automation %D 1997 %T Manipulation of polyhedral parts by rolling %A A. Marigo %A Y. Chitour %A A. Bicchi %K Hybrid and Embedded Control Systems %K Nonlinear Control Systems %K Quantized Control Systems %B Proc. IEEE Int. Conf. on Robotics and Automation %G eng %0 Book Section %B Algorithms for Robotic Motion and Manipulation %D 1997 %T Rolling Polyhedra on a Plane: Analysis of the Reachable Set %A Y. Chitour %A A. Marigo %A D Prattichizzo %A A. Bicchi %E J.P. Laumond %E M. Overmars %K Embedded Control %K Robotics %B Algorithms for Robotic Motion and Manipulation %I A. K. Peters %C Wellesley, MA, U.S.A. %P 277-286 %G eng %0 Conference Paper %B Proc. Third Int. Symp. on Methods and Models for Automation and Robotics %D 1996 %T Dexterity through Rolling: Towards Manipulation of Unknown Objects %A A. Bicchi %A Y. Chitour %A A. Marigo %A D Prattichizzo %K Nonlinear Control Systems %K Robot Hands %B Proc. Third Int. Symp. on Methods and Models for Automation and Robotics %C Miedzyzdroje, Poland %G eng %0 Book Section %B Advances in Robotics: The ERNET Perspective %D 1996 %T Reachability of Rolling Parts %A Y. Chitour %A A. Marigo %A D Prattichizzo %A A. Bicchi %E C. Bonivento %E C. Melchiorri %E H. Tolle %K Haptics %K Robotics %B Advances in Robotics: The ERNET Perspective %I World Scientific Publisher Corp. %C Singapore %P 51-60 %G eng