Reference mirroring for control with impacts

in Daafouz, Jamal; Tarbouriech, Sophie; Sigalotti, Mario (Eds.) Hybrid Systems with Constraints (2013)

We illustrate a peculiar concept of mirrored reference to address reference tracking control systems for plants subject to impacts. The idea is based on the constructions of artificial references ... [more ▼]

We illustrate a peculiar concept of mirrored reference to address reference tracking control systems for plants subject to impacts. The idea is based on the constructions of artificial references corresponding to suitable reflections of the actual reference performed with respect to the impacting surfaces. We illustrate the application of this idea to reference tracking and state observation for a number of examples of control systems subject to impacts: a one-dimensional hammer hitting a surface, a two-dimensional Newton’s cradle, a two-dimensional mass moving in a planar equilateral triangle and a mass moving in an n-dimensional polyhedral region. For each one of these examples, we provide controllers inducing global exponential stability of the error dynamics. Despite the apparently different nature among the examples, we show that the controller/observer design follows the same paradigm based on mirrored references and we highlight the similarities of the different proofs by showing that they are all based on a specific result about Lyapunov conditions for global exponential stability of hybrid dynamical systems. [less ▲]

Follow the bouncing ball: global results on tracking and state estimation with impacts

in IEEE Transactions on Automatic Control (2013)

In this paper we formulate tracking and state- estimation problems of a translating mass in a polyhedral billiard as a stabilization problem for a suitable set. Due to the discontinuous trajectories ... [more ▼]

In this paper we formulate tracking and state- estimation problems of a translating mass in a polyhedral billiard as a stabilization problem for a suitable set. Due to the discontinuous trajectories arising from the impacts, we use hybrid systems stability analysis tools to establish the results. Using a novel concept of mirrored images of the target mass we prove that 1) a tracking control algorithm, and 2) an observer algorithm guarantee global exponential stability results for specific classes of polyhedral billiards, including rectangles. Moreover, we combine these two algorithms within dynamic controllers that guarantee global output feedback tracking. The results are illustrated via simulations. [less ▲]

Model recovery anti-windup for continuous-time rate and magnitude saturated linear plants

in Automatica (2012), 48(8), 15021513

In this paper two approaches are given for anti-windup design for nonlinear control systems with linear plants subject to limitations both in the magnitude and the rate of variation of the control input ... [more ▼]

In this paper two approaches are given for anti-windup design for nonlinear control systems with linear plants subject to limitations both in the magnitude and the rate of variation of the control input. Both approaches are based on the so-called Model Recovery Anti-windup (MRAW) framework. The first approach is built by treating the rate+magnitude saturation as a single dynamic nonlinearity, while in the second one, the dynamic compensator dynamics is extended with extra states to treat the two saturations separately. Both approaches lead to global stability with exponentially stable plants and local stability in all other cases. For both approaches, stability and performance guarantees are proven, numerical recipes are given and the relative merits are comparatively highlighted on a simulation example. [less ▲]

Tracking Control in Billiards Using Mirrors without Smoke, Part I: Lyapunov-Based Local Tracking in Polyhedral Regions

in 50th IEEE Conference on Decision and Control (cdc 2011) (2011)

In this paper we formulate the tracking problem of a translating mass in a polyhedral billiard as a stabilization problem for a suitable set. Due to the discontinuous dynamics arising from nonsmooth ... [more ▼]

In this paper we formulate the tracking problem of a translating mass in a polyhedral billiard as a stabilization problem for a suitable set. Due to the discontinuous dynamics arising from nonsmooth impacts, the tracking problem is formulated within a hybrid systems framework and a Lyapunov function is given, which decreases during flow (continuous motion) and remains constant across jumps (impacts of the masses). To guarantee non-increase of the Lyapunov function at jumps, we introduce a novel concept of mirrored images of the target mass and prove that, with this concept, local tracking is achieved. Several simulations illustrate the effectiveness of the proposed approach as compared to alternative solutions. In a companion paper [6] we address global results and generalize the local approach to curved billiards. [less ▲]

Reset passivation of nonlinear controllers via a suitable time-regular reset map

in Automatica (2011), 47(9), 2099-2106

For a class of square continuous-time nonlinear controllers we design a suitable resetting rule inspired by the resetting rule for Clegg integrators and First Order Reset Elements (FORE). With this rule ... [more ▼]

For a class of square continuous-time nonlinear controllers we design a suitable resetting rule inspired by the resetting rule for Clegg integrators and First Order Reset Elements (FORE). With this rule, we prove that the arising hybrid system with temporal regularization is passive in the conventional continuous-time sense with a small shortage of input passivity decreasing with the temporal regularization constant. Based on the passivity property, we then investigate the finite gain stability of the interconnection between this passive controller and a passive nonlinear plant. [less ▲]

Results on Passification of Nonlinear Controllers Via a Suitable Time-Regular Reset Map

in 8th IFAC Symposium on nonlinear control systems (2010)

For a class of square continuous time nonlinear controllers we design a suitable resetting rule inspired by the resetting rule for Clegg integrators and First Order Reset Elements (FORE). With this rule ... [more ▼]

For a class of square continuous time nonlinear controllers we design a suitable resetting rule inspired by the resetting rule for Clegg integrators and First Order Reset Elements (FORE). With this rule, we prove that the arising hybrid system with temporal regularization is passive in the conventional continuous time sense with a small shortage of input passivity decreasing with the temporal regularization constant. Based on the passivity property, we then investigate the finite gain stability of the interconnection between this passive controller and a passive nonlinear plant. [less ▲]

Stability For A Class of Homogeneous Hybrid Systems By Annular Lyapunov Analysis

in 49th IEEE Conference on Decision and Control (cdc 2010) (2010)

For a class of homogeneous hybrid systems we present a set of annular Lyapunov-like conditions for inferring global pre-asymptotic stability of systems. Then, we prove that such conditions are mild ... [more ▼]

For a class of homogeneous hybrid systems we present a set of annular Lyapunov-like conditions for inferring global pre-asymptotic stability of systems. Then, we prove that such conditions are mild, namely, that each globally pre-asymptotically stable system must satisfy them. Based on these results, we design a sum of squares algorithm that constructs a suitable Lyapunov-like function to fulfill such annular conditions. Finally, based on recent results on homogeneous approximations of hybrid systems, we point out that such conditions can also be used to deduce local pre-asymptotic stability for a wider class of hybrid systems. [less ▲]

Gain-scheduled, model-based anti-windup for LPV systems

in Automatica (2010), 46(1), 222-225

The aim of this paper is to show that a recently proposed technique for anti-windup control of exponentially unstable plants can be easily extended to solve the corresponding robust anti-windup problem ... [more ▼]

The aim of this paper is to show that a recently proposed technique for anti-windup control of exponentially unstable plants can be easily extended to solve the corresponding robust anti-windup problem for linear parameter varying systems, for which the time varying parameters are measured online. The proposed technique is minimally conservative with respect to the size of the resulting operating region (which coincides, up to an arbitrarily small quantity, with the largest set on which asymptotic stability can be guaranteed for the considered plant with the given saturation level and uncertainty characteristics), and is not limited to plants having only small uncertainties or being open-loop stable. [less ▲]

Model Recovery Anti-Windup for plants with rate and magnitude saturation

in European Control Conference (2009) (2009)

In this paper two alternative approaches are provided to design an anti-windup compensator for control systems subject to limitations both in the magnitude and the rate of variation of the plant input ... [more ▼]

In this paper two alternative approaches are provided to design an anti-windup compensator for control systems subject to limitations both in the magnitude and the rate of variation of the plant input. For both approaches, constructive numerical recipes are given and the relative merits are highlighted. [less ▲]

Model based, gain-scheduled anti-windup control for LPV systems

in 46th IEEE Conference on Decision and Control (cdc 2007) (2007)

In this paper we show that a recently proposed technique for anti-windup control of exponentially unstable plants can be easily extended to solve the corresponding robust anti-windup problem for linear ... [more ▼]

In this paper we show that a recently proposed technique for anti-windup control of exponentially unstable plants can be easily extended to solve the corresponding robust anti-windup problem for linear parameter varying systems, for which the time varying parameters are measured online. For this class of plants, it is shown that the proposed technique is minimally conservative with respect to the size of the resulting operating region: in particular, such a region is (up to an arbitrarily small quantity) exactly the largest set on which asymptotic stability can be guaranteed for the considered plant, for the given saturation level and uncertainty characteristics. [less ▲]