@article {CHLOscl05, title = {Uniform Semiglobal Practical Asymptotic Stability for Non-autonomous Cascaded Systems and Applications}, journal = {Automatica}, volume = {44}, year = {2008}, pages = {337-347 }, abstract = {

It is due to the modularity they provide that results for cascaded systems have proved their utility in numerous control applications as well as in the development of general control techniques based on {\textquoteleft}{\textquoteleft}adding integrators{\textquoteright}{\textquoteright}. Nevertheless, the standing assumptions in most of the present literature on cascaded systems is that, when \emph{decoupled}, the subsystems constituting the cascade are uniformly globally asymptotically stable (UGAS). Hence existing results fail in the more general case when the subsystems are uniformly semiglobally practically asymptotically stable (USPAS). This situation is often encountered in control practice, {\i}t e.g.}, in control of physical systems with external perturbations, measurement noise, unmodelled dynamics, {\i}t etc}. After giving a rigorous framework for the analysis of such stability properties, this paper generalizes previous results for cascades by establishing that, under a uniform boundedness condition on its solutions, the cascade of two USPAS systems remains USPAS. An analogous result is derived for uniformly semiglobally asymptotically stable (USAS) systems in cascade. Furthermore, we show the utility of our results in the PID control of mechanical systems affected by unknown non-dissipative forces and considering the dynamics of the DC motors.

}, keywords = {Robotics}, doi = {10.1016/j.automatica.2007.05.019}, author = {A. Chaillet and A. Lor} } @booklet {CHCHLOSI07, title = {Uniform stabilization for linear systems with persistency of excitation. The neutrally stable and the double integrator cases}, howpublished = {Submitted to Mathematics of Control, Signals and Systems}, year = {2007}, abstract = {

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.

}, keywords = {Robotics}, author = {A. Chaillet and Y. Chitour and A. Lor and M. Sigalotti} } @article {CHLOEJC06, title = {Uniform global practical asymptotic stability for non-autonomous cascaded systems}, journal = {European Journal of Control}, volume = {12}, number = {6}, year = {2006}, abstract = {

This paper aims to give sufficient conditions for a cascade composed of nonlinear time-varying systems that are uniformly globally practically asymptotically stable (UGPAS) to be UGPAS. These conditions are expressed as relations between the Lyapunov function of the driven subsystem and the interconnection term. Our results generalise previous theorems that establish the uniform global asymptotic stability of cascades.

}, keywords = {Robotics}, author = {A. Chaillet and A. Lor} } @conference {CHLO06, title = {Uniform Global Practical Stability for non-autonomous cascaded systems}, booktitle = {MTNS 2006}, year = {2006}, month = {July}, address = {Kyoto, Japan}, abstract = {

This paper aims to give sufficient conditions for a cascade composed of nonlinear time-varying systems that are uniformly globally practically asymptotically stable (UGPAS) to be UGPAS. These conditions are expressed as relations between the Lyapunov function of the driven subsystem and the interconnection term. Our results generalize previous theorems that establish the uniform global asymptotic stability of cascades.

}, keywords = {Robotics}, author = {A. Chaillet and A. Lor} } @conference {CHLOifac05, title = {Uniform Semiglobal Practical Asymptotic Stability for nonlinear time-varying systems in cascade}, booktitle = {Proc. 16th. IFAC World Congress}, year = {2005}, month = {July}, address = {Praha, Tcheck Republic}, keywords = {Robotics}, author = {A. Chaillet and A. Lor} }