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.

JF - Submitted to CDC 2007 CY - New Orleans, USA ER - TY - ABST T1 - Uniform stabilization for linear systems with persistency of excitation. The neutrally stable and the double integrator cases Y1 - 2007 A1 - A. Chaillet A1 - Y. Chitour A1 - A. Lor A1 - M. Sigalotti KW - Robotics AB -Consider 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.

JF - Submitted to Mathematics of Control, Signals and Systems ER - TY - CONF T1 - On the PE stabilization of time-varying systems: open questions and preliminary answers T2 - Proc. IEEE Int. Conf. on Decision and Control Y1 - 2005 A1 - A. Lor A1 - A. Chaillet A1 - G. Besancon A1 - Y. Chitour KW - Robotics AB -We 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.

JF - Proc. IEEE Int. Conf. on Decision and Control CY - Sevilla, Spain ER - TY - JOUR T1 - Reachability and Steering of Rolling Polyhedra: A Case Study in Discrete Nonholonomy JF - IEEE Trans. on Automatic Control Y1 - 2004 A1 - A. Bicchi A1 - Y. Chitour A1 - A. Marigo KW - Embedded Control KW - Robotics AB -Rolling 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.

VL - 49 ER - TY - CONF T1 - Manipulation of polyhedral parts by rolling T2 - Proc. IEEE Int. Conf. on Robotics and Automation Y1 - 1997 A1 - A. Marigo A1 - Y. Chitour A1 - A. Bicchi KW - Hybrid and Embedded Control Systems KW - Nonlinear Control Systems KW - Quantized Control Systems JF - Proc. IEEE Int. Conf. on Robotics and Automation ER - TY - CHAP T1 - Rolling Polyhedra on a Plane: Analysis of the Reachable Set T2 - Algorithms for Robotic Motion and Manipulation Y1 - 1997 A1 - Y. Chitour A1 - A. Marigo A1 - D Prattichizzo A1 - A. Bicchi ED - J.P. Laumond ED - M. Overmars KW - Embedded Control KW - Robotics JF - Algorithms for Robotic Motion and Manipulation PB - A. K. Peters CY - Wellesley, MA, U.S.A. ER - TY - CONF T1 - Dexterity through Rolling: Towards Manipulation of Unknown Objects T2 - Proc. Third Int. Symp. on Methods and Models for Automation and Robotics Y1 - 1996 A1 - A. Bicchi A1 - Y. Chitour A1 - A. Marigo A1 - D Prattichizzo KW - Nonlinear Control Systems KW - Robot Hands JF - Proc. Third Int. Symp. on Methods and Models for Automation and Robotics CY - Miedzyzdroje, Poland ER - TY - CHAP T1 - Reachability of Rolling Parts T2 - Advances in Robotics: The ERNET Perspective Y1 - 1996 A1 - Y. Chitour A1 - A. Marigo A1 - D Prattichizzo A1 - A. Bicchi ED - C. Bonivento ED - C. Melchiorri ED - H. Tolle KW - Haptics KW - Robotics JF - Advances in Robotics: The ERNET Perspective PB - World Scientific Publisher Corp. CY - Singapore ER -