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.

10aRobotics1 aChaillet, A1 aChitour, Y1 aLor, A1 aSigalotti, M uhttp://www.centropiaggio.unipi.it/publications/towards-uniform-linear-time-invariant-stabilization-systems-persistency-excitation.html01254nas a2200145 4500008004100000245013000041210006900171520065700240653001300897100001600910700001500926700001100941700001700952856013900969 2007 eng d00aUniform stabilization for linear systems with persistency of excitation. The neutrally stable and the double integrator cases0 aUniform stabilization for linear systems with persistency of exc3 aConsider 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.

10aRobotics1 aChaillet, A1 aChitour, Y1 aLor, A1 aSigalotti, M uhttp://www.centropiaggio.unipi.it/publications/uniform-stabilization-linear-systems-persistency-excitation-neutrally-stable-and-double01634nas a2200169 4500008004100000245009200041210006900133260002900202300001600231520101300247653001301260100001101273700001601284700001601300700001501316856013301331 2005 eng d00aOn the PE stabilization of time-varying systems: open questions and preliminary answers0 aPE stabilization of timevarying systems open questions and preli aSevilla, SpaincDecember a6847–68523 aWe 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.

10aRobotics1 aLor, A1 aChaillet, A1 aBesancon, G1 aChitour, Y uhttp://www.centropiaggio.unipi.it/publications/pe-stabilization-time-varying-systems-open-questions-and-preliminary-answers.html02137nas a2200181 4500008004100000245008900041210006900130260000800199300001200207490000700219520151900226653002101745653001301766100001501779700001501794700001401809856013201823 2004 eng d00aReachability and Steering of Rolling Polyhedra: A Case Study in Discrete Nonholonomy0 aReachability and Steering of Rolling Polyhedra A Case Study in D cMay a710-7260 v493 aRolling 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.

10aEmbedded Control10aRobotics1 aBicchi, A.1 aChitour, Y1 aMarigo, A uhttp://www.centropiaggio.unipi.it/publications/reachability-and-steering-rolling-polyhedra-case-study-discrete-nonholonomy.html00521nas a2200145 4500008004100000245004800041210004800089653004000137653003000177653003000207100001400237700001500251700001500266856009400281 1997 eng d00aManipulation of polyhedral parts by rolling0 aManipulation of polyhedral parts by rolling10aHybrid and Embedded Control Systems10aNonlinear Control Systems10aQuantized Control Systems1 aMarigo, A1 aChitour, Y1 aBicchi, A. uhttp://www.centropiaggio.unipi.it/publications/manipulation-polyhedral-parts-rolling.html00648nas a2200193 4500008004100000245006400041210006300105260004000168300001200208653002100220653001300241100001500254700001400269700002000283700001500303700001700318700001600335856010300351 1997 eng d00aRolling Polyhedra on a Plane: Analysis of the Reachable Set0 aRolling Polyhedra on a Plane Analysis of the Reachable Set aWellesley, MA, U.S.A.bA. K. Peters a277-28610aEmbedded Control10aRobotics1 aChitour, Y1 aMarigo, A1 aPrattichizzo, D1 aBicchi, A.1 aLaumond, J P1 aOvermars, M uhttp://www.centropiaggio.unipi.it/publications/rolling-polyhedra-plane-analysis-reachable-set.html00593nas a2200157 4500008004100000245007100041210006900112260002500181653003000206653001600236100001500252700001500267700001400282700002000296856011900316 1996 eng d00aDexterity through Rolling: Towards Manipulation of Unknown Objects0 aDexterity through Rolling Towards Manipulation of Unknown Object aMiedzyzdroje, Poland10aNonlinear Control Systems10aRobot Hands1 aBicchi, A.1 aChitour, Y1 aMarigo, A1 aPrattichizzo, D uhttp://www.centropiaggio.unipi.it/publications/dexterity-through-rolling-towards-manipulation-unknown-objects.html00593nas a2200205 4500008004100000245003400041210003400075260004800109300001000157653001200167653001300179100001500192700001400207700002000221700001500241700001700256700001800273700001300291856008300304 1996 eng d00aReachability of Rolling Parts0 aReachability of Rolling Parts aSingaporebWorld Scientific Publisher Corp. a51-6010aHaptics10aRobotics1 aChitour, Y1 aMarigo, A1 aPrattichizzo, D1 aBicchi, A.1 aBonivento, C1 aMelchiorri, C1 aTolle, H uhttp://www.centropiaggio.unipi.it/publications/reachability-rolling-parts.html