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 ``adding integrators''. 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, ıt e.g.}, in control of physical systems with external perturbations, measurement noise, unmodelled dynamics, ıt 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.

%B Automatica %V 44 %P 337-347 %G eng %N 2 %R 10.1016/j.automatica.2007.05.019 %0 Generic %D 2007 %T Towards uniform linear time-invariant stabilization of systems with persistency of excitation %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 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 2006 %T Adaptive output feedback control of spacecraft relative translation %A R. Kristiansen %A A. Lor %A A. Chaillet %A P. J. Nicklasson %K Robotics %XWe address the problem of tracking relative translation in a leader-follower spacecraft formation using feedback from relative position only and under parameter uncertainty (spacecraft mass) and uncertainty in the leader variables (true anomaly rate and rate of change). We only assume boundedness of orbital perturbations and the leader control force but with unknown bounds. Under these conditions we propose a controller that renders the closed-loop system, uniformly semiglobally practically asymptotically stable. In particular, the domain of attraction can be made arbitrarily large by picking convenient gains, and the state errors in the closed-loop system are proved to converge from any initial condition within the domain of attraction to a ball in close vicinity of the origin in a stable way; moreover, this ball can be diminished arbitrarily by increasing the gains in the control law. Simulation results of a leader-follower spacecraft formation using the proposed controller are presented.

%B Proc. IEEE Int. Conf. on Decision and Control %C San Diego, USA %8 December %G eng %0 Conference Paper %B Proc. IEEE Int. Conf. on Decision and Control %D 2006 %T A converse Lyapunov theorem for semiglobal practical asymptotic stability and application to cascades-based control %A A. Chaillet %A A. Lor %K Robotics %XWe present a converse Lyapunov result for nonlinear time-varying systems that are uniformly semiglobally asymptotically stable. This stability property pertains to the case when the size of initial conditions may be arbitrarily enlarged and the solutions of the system converge, in a stable way, to a closed ball that may be arbitrarily diminished by tuning a design parameter of the system (typically but not exclusively, a control gain). This result is notably useful in cascaded-based control when uniform practical asymptotic stability is established without a Lyapunov function, , ıt e.g.} via averaging. We provide a concrete example by solving the stabilization problem of a hovercraft.

%B Proc. IEEE Int. Conf. on Decision and Control %C San Diego, USA %8 December %G eng %0 Journal Article %J Automatica %D 2006 %T Necessary and sufficient conditions for uniform semiglobal practical asymptotic stability: Application to cascaded systems %A A. Chaillet %A A. Lor %K Robotics %XIt is well established that for a cascade of two uniformly globally asymptotically stable (UGAS) systems, the origin remains UGAS provided that the solutions of the cascade are uniformly globally bounded. While this result has met considerable popularity in specific applications it remains restrictive since, in practice, it is often the case that the decoupled subsystems are only uniformly \emph{semiglobally} \emph{practically} asymptotically stable (USPAS). Recently, we established that the cascade of USPAS systems is USPAS under a local boundedness assumption and the hypothesis that one knows a Lyapunov function for the driven subsystem. The contribution of this paper is twofold: 1) we establish USPAS of cascaded systems without the requirement of a Lyapunov function and 2) we present a converse theorem for USPAS. While other converse theorems in the literature cover the case of USPAS ours has the advantage of providing a bound on the gradient of the Lyapunov function, which is fundamental to establish theorems for cascades.

%B Automatica %V 42 %P 1899-1906 %8 November %G eng %0 Book Section %D 2006 %T Output feedback control of relative translation in a leader-follower spacecraft formation %A R. Kristiansen %A A. Lor %A A. Chaillet %A P. J. Nicklasson %K Robotics %XWe present a solution to the problem of tracking relative translation in a leader-follower spacecraft formation using feedback from relative position only. Three controller configurations are presented which enables the follower spacecraft to track a desired reference trajectory relative to the leader. The controller design is performed for different levels of knowledge about the leader spacecraft and its orbit. The first controller assumes perfect knowledge of the leader and its orbital parameters, and renders the equilibrium points of the closed-loop system uniformly globally asymptotically stable (UGAS). The second controller uses the framework of the first to render the closed-loop system uniformly globally practically asymptotically stable (UGPAS), with knowledge of bounds on some orbital parameters, only. That is, the state errors in the closed-loop system are proved to converge from any initial conditions to a ball in close vicinity of the origin in a stable way, and this ball can be diminished arbitrarily by increasing the gains in the control law. The third controller, based on the design of the second, utilizes adaptation to estimate the bounds that were previously assumed to be known. The resulting closed-loop system is proved to be uniformly semiglobally practically asymptotically stable (USPAS). The last two controllers assume boundedness only of orbital perturbations and the leader control force. Simulation results of a leader-follower spacecraft formation using the proposed controllers are presented.

%S Lecture Notes in Control and Information Sciences %I Springer Verlag %C Tromsoe, Norway %P 131–151 %G eng %& in Workshop on Group Coordination and Cooperative Control %0 Conference Paper %B Proc. IEEE Int. Conf. on Decision and Control %D 2006 %T Robustness of PID controlled manipulators with respect to external disturbances %A A. Chaillet %A A. Lor %A R. Kelly %K Robotics %XWe present a robustness analysis for PID-controlled robot manipulators. For robot manipulators under the influence of external disturbances we provide a proof, and a tuning procedure, to establish uniform semiglobal practical asymptotic stability. In particular, in contrasts to other works on robust stability of PIDs, we do not use La Salle's principle but provide a strict Lyapunov function. The perturbations that we consider include discontinuous functions of the state, such as Coulomb friction. As corollaries of our main results, one may conclude the same stability property for the case of motion control using linear PID.

%B Proc. IEEE Int. Conf. on Decision and Control %C San Diego, USA %8 December %G eng %0 Journal Article %J European Journal of Control %D 2006 %T Uniform global practical asymptotic stability for non-autonomous cascaded systems %A A. Chaillet %A A. Lor %K Robotics %XThis 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.

%B European Journal of Control %V 12 %G eng %0 Conference Paper %B MTNS 2006 %D 2006 %T Uniform Global Practical Stability for non-autonomous cascaded systems %A A. Chaillet %A A. Lor %K Robotics %XThis 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.

%B MTNS 2006 %C Kyoto, Japan %8 July %G eng %0 Conference Paper %B Proc. 16th. IFAC World Congress %D 2005 %T Output feedback control via adaptive observers with persistency of excitation %A De Leon Morales, J. %A A. Chaillet %A A. Lor %A G. Besancon %K Robotics %XWe address the problem of adaptive observer design for nonlinear time-varying systems which can be transformed in the so-called output feedback form (linear in the unmeasured variables). The observer design follows up previous work on adaptive observers for linear systems and has the form of the classical Luenberger observers for linear systems except that the observer gain is time-varying. A specific form of persistency of excitation is imposed to guarantee the convergence of the (state and parameter) estimation errors. As for the output feedback loop, we proceed using a cascade approach, i.e., we impose the appropriate conditions so that the closed loop system has a cascaded structure. Uniform global asymptotic stability may then be concluded based on cascaded systems theory.

%B Proc. 16th. IFAC World Congress %C Praha, Tcheck Republic %8 July %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 Conference Paper %B Proc. 16th. IFAC World Congress %D 2005 %T Uniform Semiglobal Practical Asymptotic Stability for nonlinear time-varying systems in cascade %A A. Chaillet %A A. Lor %K Robotics %B Proc. 16th. IFAC World Congress %C Praha, Tcheck Republic %8 July %G eng