Total stability and integral action for discrete-time nonlinear systems

Robustness guarantees are important properties to be looked for during control design. They ensure stability of closed-loop systems in face of uncertainties, unmodeled effects and bounded disturbances. While the theory on robust stability is well established in the continuous-time nonlinear framework, the same cannot be stated for its discrete-time counterpart. In this paper, we propose the discrete-time parallel of total stability results for continuous-time nonlinear system. This enables the analysis of robustness properties via simple model difference in the discrete-time context. First, we study how existence of equilibria for a nominal model transfers to sufficiently similar ones. Then, we provide results on th

Generalized Lyapunov conditions for k-contraction: analysis and feedback design

Recently, the concept of k-contraction has been introduced as a promising generalization of contraction for dynamical systems. However, the study of k-contraction properties has faced significant challenges due to the reliance on complex mathematical objects called matrix compounds. As a result, related control design methodologies have yet to appear in the literature. In this paper, we overcome existing limitations and propose new sufficient conditions for k-contraction which do not rely on matrix compounds. Our design-oriented conditions stem from a strong geometrical interpretation and establish a connection between kcontraction and p-dominance. Notably, these conditions are also necessary in the linear time-invariant framework. Leveraging on these findings, we propose a feedback design methodology for both the linear and the nonlinear scenarios

Continuous locomotion over smooth surfaces with aerial manipulators

This document addresses the analysis and the synthesizing of a controller aimed at allowing a UAV (Unmanned Aerial Vehicle) to exert a force on the surrounding environment in a sustained fashion. The vehicle is assumed to be equipped with a 1 DoF manipulator endowed with a tool. The goal is to enable the system to push onto arbitrary oriented smooth surfaces, while dealing with constraints on the applied force and on the manipulator workspace.\\ At first the problem is analyzed and the main challenges evidenced. Then the proposed approach is presented and the system model computed by means of Screw Theory, highlighting salient features. Since the controller relies on linearization, particular attention is posed in the equilibrium setpoints analysis and choice, which are designed to satisfy the force and stability requirements while allowing the system to perform some movements, in order to tackle the bounded workspace challenge.\\ Finally a linear quadratic regulator is adopted, switching between different equilibrium points with a gain scheduling technique. In the last section the simulation results obtained via ROS and Gazebo are shown and commented, evaluating the control algorithm and approach performances with respect to the addressed task and the implementation. \\Finally some interesting possible future developments and improvements for the project are suggested

Deep learning-based output tracking via regulation and contraction theory

In this paper, we deal with output tracking control problems for input-affine nonlinear systems. We propose a deep learning-based solution whose foundations lay in control theory. We design a two-step state-feedback controller: a contraction-based feedback stabilizer and a feedforward action. The first component guarantees convergence to the steady-state trajectory on which the tracking error is zero. The second one is inherited from output regulation theory and provides forward invariantness of such a trajectory along the solutions of the system. To alleviate the need for heavy analytical computations or online optimization, we rely on deep neural networks and link their approximation error to the tracking one. Mimicking the analytical control structure, we split the learning task into two separate modules. For the stabilizer module, we propose a switching objective function balancing feasibility of the solution and performance improvement.We test our solution in a challenging environment to validate the proposed design

LMI conditions for k-contraction analysis: a step towards design

International audienceRecently, k-contraction has been proposed as a generalization of contraction properties for nonlinear timevariant systems. Existing tools for k-contraction analysis exploit complex mathematical tools known as matrix compounds. This prevented the development of related design methodologies. In this paper, we link k-contraction properties to partial stability analysis tools. This leads to new, design-oriented sufficient conditions for k-contraction analysis which do not involve matrix compounds. We also show that such sufficient conditions are necessary for the linear time-invariant framework. Finally, we compare our results to existing methods and highlight their advantages

Generalized Lyapunov conditions for k-contraction: analysis and feedback design

Recently, the concept of k-contraction has been introduced as a promising generalization of contraction for dynamical systems. However, the study of k-contraction properties has faced significant challenges due to the reliance on complex mathematical objects called matrix compounds. As a result, related control design methodologies have yet to appear in the literature. In this paper, we overcome existing limitations and propose new sufficient conditions for k-contraction which do not rely on matrix compounds. Our design-oriented conditions stem from a strong geometrical interpretation and establish a connection between kcontraction and p-dominance. Notably, these conditions are also necessary in the linear time-invariant framework. Leveraging on these findings, we propose a feedback design methodology for both the linear and the nonlinear scenarios

Generalized Lyapunov conditions for k-contraction: analysis and feedback design

Recently, the concept of k-contraction has been introduced as a promising generalization of contraction for dynamical systems. However, the study of k-contraction properties has faced significant challenges due to the reliance on complex mathematical objects called matrix compounds. As a result, related control design methodologies have yet to appear in the literature. In this paper, we overcome existing limitations and propose new sufficient conditions for k-contraction which do not rely on matrix compounds. Our design-oriented conditions stem from a strong geometrical interpretation and establish a connection between kcontraction and p-dominance. Notably, these conditions are also necessary in the linear time-invariant framework. Leveraging on these findings, we propose a feedback design methodology for both the linear and the nonlinear scenarios

Incremental stabilization and multi-agent synchronization of discrete-time nonlinear systems

In this paper, we propose a novel distributed statefeedback design for robust synchronization of networks of identical discrete-time nonlinear agents under generic time-invariant communication graphs. We focus on the class of almost differentiable (possibly time-varying) dynamics that are linear in the input. By generalizing results on synchronization of linear agents, we build strong links between the solution to the synchronization problem in the linear and nonlinear framework. This is also enabled by the introduction of new results on design of incrementally stabilizing controllers based on contraction analysis. Finally, we propose numerically tractable sufficient conditions for the synchronization of networks of non-smooth Lur'e systems

LMI conditions for k-contraction analysis: a step towards design

International audienceRecently, k-contraction has been proposed as a generalization of contraction properties for nonlinear timevariant systems. Existing tools for k-contraction analysis exploit complex mathematical tools known as matrix compounds. This prevented the development of related design methodologies. In this paper, we link k-contraction properties to partial stability analysis tools. This leads to new, design-oriented sufficient conditions for k-contraction analysis which do not involve matrix compounds. We also show that such sufficient conditions are necessary for the linear time-invariant framework. Finally, we compare our results to existing methods and highlight their advantages

Reinforcement Learning Policies With Local LQR Guarantees For Nonlinear Discrete-Time Systems

International audienceOptimal control of nonlinear systems is a difficult problem which has been addressed by both the Control Theory (CT) and Reinforcement Learning (RL) communities. Frequently, the former relies on the linearization of the system thus obtaining only local guarantees. The latter relies on data to build model-free controllers, focused solely on performances. In this paper we propose a methodology to combine the advantages of both approaches, casting the formulation of an optimal local Linear Quadratic Regulator (LQR) into a Deep RL problem. Our solution builds on the linear framework to derive a learnt nonlinear controller showing local stability properties and global performances