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A differential Lyapunov framework for contraction analysis Forni, Fulvio ; Sepulchre, Rodolphe in IEEE Transactions on Automatic Control (2014), 59(3), 614-628 Lyapunov's second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to ... [more ▼] Lyapunov's second theorem is an essential tool for stability analysis of differential equations. The paper provides an analog theorem for incremental stability analysis by lifting the Lyapunov function to the tangent bundle. The Lyapunov function endows the state-space with a Finsler structure. Incremental stability is inferred from infinitesimal contraction of the Finsler metrics through integration along solutions curves. [less ▲] Detailed reference viewed: 82 (7 ULg)Event-triggered transmission for linear control over communication channels Forni, Fulvio ; ; et al in Automatica (2014), 50(2), 490498 We consider an exponentially stable closed loop interconnection of a continuous linear plant and a continuous linear controller, and we study the problem of interconnecting the plant output to the ... [more ▼] We consider an exponentially stable closed loop interconnection of a continuous linear plant and a continuous linear controller, and we study the problem of interconnecting the plant output to the controller input through a digital channel. We propose a family of "transmission-lazy" sensors whose goal is to transmit the measured plant output information as little as possible while preserving closed-loop stability. In particular, we propose two transmission policies, providing conditions on the transmission parameters. These guarantee global asymptotic stability when the plant state is available or when an estimate of the state is available (provided by a classical continuous linear observer). Moreover, under a specific condition, they guarantee global exponential stability [less ▲] Detailed reference viewed: 23 (1 ULg)Reference mirroring for control with impacts Forni, Fulvio ; ; 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 ▲] Detailed reference viewed: 50 (6 ULg)On differential passivity of physical systems Forni, Fulvio ; Sepulchre, Rodolphe ; in 52nd IEEE Conference on Decision and Control (2013) Differential passivity is a property that allows to check with a pointwise criterion that a system is incrementally passive, a property that is relevant to study interconnected systems in the context of ... [more ▼] Differential passivity is a property that allows to check with a pointwise criterion that a system is incrementally passive, a property that is relevant to study interconnected systems in the context of regulation, synchronization, and estimation. The paper investigates how restrictive is the property, focusing on a class of open gradient systems encountered in the coenergy modeling framework of physical systems, in particular the Brayton-Moser formalism for nonlinear electrical circuits. [less ▲] Detailed reference viewed: 41 (4 ULg)On differentially dissipative dynamical systems Forni, Fulvio ; Sepulchre, Rodolphe in IFAC-PapersOnLine website. (2013) Dissipativity is an essential concept of systems theory. The paper provides an extension of dissipativity, named differential dissipativity, by lifting storage functions and supply rates to the tangent ... [more ▼] Dissipativity is an essential concept of systems theory. The paper provides an extension of dissipativity, named differential dissipativity, by lifting storage functions and supply rates to the tangent bundle. Differential dissipativity is connected to incremental stability in the same way as dissipativity is connected to stability. It leads to a natural formulation of differential passivity when restricting to quadratic supply rates. The paper also shows that the interconnection of differentially passive systems is differentially passive, and provides preliminary examples of differentially passive electrical systems. [less ▲] Detailed reference viewed: 71 (9 ULg)Follow the bouncing ball: global results on tracking and state estimation with impacts Forni, Fulvio ; ; 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 ▲] Detailed reference viewed: 55 (5 ULg)Lyapunov-based sufficient conditions for exponential stability in hybrid systems ; Forni, Fulvio ; in IEEE Transactions on Automatic Control (2013) Lyapunov-based sufficient conditions for exponential stability in hybrid systems are presented. The focus is on converting non-strict Lyapunov conditions, having certain observability properties, into ... [more ▼] Lyapunov-based sufficient conditions for exponential stability in hybrid systems are presented. The focus is on converting non-strict Lyapunov conditions, having certain observability properties, into strict Lyapunov conditions for exponential stability. Both local and global results are considered. The utility of the results is illustrated through an example. [less ▲] Detailed reference viewed: 149 (4 ULg)Antiwindup regulation of saturated linear systems Forni, Fulvio ; ; Sepulchre, Rodolphe in 51st IEEE Conference on Decision and Control (2012, December) We consider the output regulation problem for linear systems subject to actuators saturation. A state-feedback unconstrained regulator is transformed into an error-feedback bounded regulator by ... [more ▼] We consider the output regulation problem for linear systems subject to actuators saturation. A state-feedback unconstrained regulator is transformed into an error-feedback bounded regulator by introducing an observer which is then decomposed into the sum of "unconstrained" and "antiwindup" dynamics. The unconstrained dynamics are regulated towards a predefined reference trajectory, by a control signal u which may violate the saturation bounds during transients. The antiwindup dynamics transiently store the mismatch between unconstrained and constrained dynamics. The antiwindup design also applies to a predefined error-feedback dynamic regulator for the unconstrained system, as in standard antiwindup setting. As a particular case of distinct interest, the design provides a new global asymptotic stabilizer for saturated null-controllable systems. [less ▲] Detailed reference viewed: 89 (21 ULg)Model recovery anti-windup for continuous-time rate and magnitude saturated linear plants Forni, Fulvio ; ; 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 ▲] Detailed reference viewed: 73 (40 ULg)Global tracking and state estimation with non- smooth impacts for a mass confined to an n-dimensional half-space Forni, Fulvio ; ; in IFAC Conference on Analysis and Design of Hybrid Systems (ADHS 12) (2012) We formulate tracking and state-estimation problems of a translating mass moving in an n-dimensional space and impacting on a hyperplane. Due to the discontinuous trajectories arising from the nonsmooth ... [more ▼] We formulate tracking and state-estimation problems of a translating mass moving in an n-dimensional space and impacting on a hyperplane. Due to the discontinuous trajectories arising from the nonsmooth impacts, we use hybrid systems stability analysis tools to establish that 1) a tracking control algorithm and 2) an observer algorithm guarantee global exponential stability. Then, based on a separation principle, we combine the two constructions above to design a dynamic output feedback controller ensuring asymptotic tracking. [less ▲] Detailed reference viewed: 56 (19 ULg)Tracking Control in Billiards Using Mirrors without Smoke, Part I: Lyapunov-Based Local Tracking in Polyhedral Regions Forni, Fulvio ; ; 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 ▲] Detailed reference viewed: 14 (5 ULg)Tracking Control in Billiards Using Mirrors without Smoke, Part II: Additional Lyapunov-Based Local and Global Results Forni, Fulvio ; in 50th IEEE Conference on Decision and Control (cdc 2011) (2011) Two control results are described: 1) local tracking control for convex billiards with piecewise locally Lipschitz boundary, and 2) global tracking control for special polyhedral billiards, including ... [more ▼] Two control results are described: 1) local tracking control for convex billiards with piecewise locally Lipschitz boundary, and 2) global tracking control for special polyhedral billiards, including rectangles and equilateral triangles. The controllers are based on Lyapunov functions and a mirroring concept introduced in a companion paper. The local results require the impacts to satisfy an average dwell-time condition with parameters that depend on the Lipschitz constant of the function that characterizes the boundary. For piecewise constant boundary, and for the global results, the average dwell-time parameters are arbitrary. Tools from stability analysis for hybrid systems are used to establish the results. [less ▲] Detailed reference viewed: 11 (2 ULg)Reset passivation of nonlinear controllers via a suitable time-regular reset map Forni, Fulvio ; ; 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 ▲] Detailed reference viewed: 43 (12 ULg)Lazy sensors for the scheduling of measurement samples transmission in linear closed loops over networks Forni, Fulvio ; ; et al in 49th IEEE Conference on Decision and Control (cdc 2010) (2010) We consider an exponentially stable close-loop interconnection of a continuous linear plant and a continuous linear controller, and we study the problem of interconnecting the plant output to the ... [more ▼] We consider an exponentially stable close-loop interconnection of a continuous linear plant and a continuous linear controller, and we study the problem of interconnecting the plant output to the controller input through a network. We propose a family of "lazy sensors" whose goal is to transmit the measured plant output information as little as possible while preserving closed-loop stability. In particular, we propose three transmission policies, providing conditions on the transmission parameters. These guarantee global asymptotic stability when the plant state is available, and global practical stability when the plant state is not available. Simulation results confirm the effectiveness of the proposed strategies. [less ▲] Detailed reference viewed: 20 (8 ULg)A Family of Global Stabilizers for Quasi-Optimal Control of Planar Linear Saturated Systems Forni, Fulvio ; ; in IEEE Transactions on Automatic Control (2010), 55(5), 1175-1180 We propose a family of nonlinear state feedback global stabilizers for all planar linear systems which are globally stabilizable by bounded inputs (namely, all non exponentially unstable linear systems ... [more ▼] We propose a family of nonlinear state feedback global stabilizers for all planar linear systems which are globally stabilizable by bounded inputs (namely, all non exponentially unstable linear systems). This family is parametrized by a nonlinear function whose selection can yield quasi time-optimal responses, where the "quasi" is required to achieve local exponential stability of the closed loop. The arising trajectories are quasi-time-optimal for arbitrarily large initial conditions; so, we expect the very simple proposed nonlinear control law to be very useful for embedded control applications with strong computational constraints. [less ▲] Detailed reference viewed: 32 (11 ULg)Results on Passification of Nonlinear Controllers Via a Suitable Time-Regular Reset Map Forni, Fulvio ; ; 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 ▲] Detailed reference viewed: 16 (5 ULg)An almost Anti-Windup scheme for plants with magnitude, rate and curvature saturation Forni, Fulvio ; ; in American Control Conference (ACC 2010) (2010) We address the anti-windup augmentation problem for plants with saturations on the magnitude, rate and curvature of the control input. To this aim, given an unconstrained closed-loop, we generate a ... [more ▼] We address the anti-windup augmentation problem for plants with saturations on the magnitude, rate and curvature of the control input. To this aim, given an unconstrained closed-loop, we generate a slightly modified strictly proper controller for which the derivatives of the control signal are available and we solve the anti-windup problem for this modified control scheme (namely, an almost anti-windup for the original closed-loop). Based on this “almost” approach, we revisit an existing Model Recovery anti-windup solution for rate and magnitude saturated plants and then we extend the results to the case of rate, magnitude and curvature saturation, by providing a Model Recovery solution to this additional problem. An example illustrates the peculiarities and the effectiveness of the proposed solutions. [less ▲] Detailed reference viewed: 16 (0 ULg)Stability For A Class of Homogeneous Hybrid Systems By Annular Lyapunov Analysis Forni, Fulvio ; 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 ▲] Detailed reference viewed: 12 (3 ULg)Instability and Overshoots of Solutions For A Class of Homogeneous Hybrid Systems By Lyapunov-like Analysis Forni, Fulvio ; in 49th IEEE Conference on Decision and Control (cdc 2010) (2010) For a class of homogeneous hybrid systems we present a generalization to the hybrid systems framework of Chetaev's theorem and we propose a set of Lyapunov-like conditions for studying instability of the ... [more ▼] For a class of homogeneous hybrid systems we present a generalization to the hybrid systems framework of Chetaev's theorem and we propose a set of Lyapunov-like conditions for studying instability of the point x e = 0 and overshoots of solutions (namely when the norm of the solution vector x at some time instant exceeds the norm of the initial condition of x). Based on these results, we design a sum of squares algorithm that constructs a suitable Lyapunov-like function to fulfill such conditions. [less ▲] Detailed reference viewed: 14 (7 ULg)Gain-scheduled, model-based anti-windup for LPV systems Forni, Fulvio ; 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 ▲] Detailed reference viewed: 53 (6 ULg) |
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