Integral Control on Lie Groups

In this paper, we extend the popular integral control technique to systems evolving on Lie groups. More explicitly, we provide an alternative definition of "integral action" for proportional(-derivative)-controlled systems whose configuration evolves on a nonlinear space, where configuration errors cannot be simply added up to compute a definite integral. We then prove that the proposed integral control allows to cancel the drift induced by a constant bias in both first order (velocity) and second order (torque) control inputs for fully actuated systems evolving on abstract Lie groups. We illustrate the approach by 3-dimensional motion control applications.Comment: Resubmitted to Systems and Control Letters, February 201

Advanced Computational-Effective Control and Observation Schemes for Constrained Nonlinear Systems

Constraints are one of the most common challenges that must be faced in control systems design. The sources of constraints in engineering applications are several, ranging from actuator saturations to safety restrictions, from imposed operating conditions to trajectory limitations. Their presence cannot be avoided, and their importance grows even more in high performance or hazardous applications. As a consequence, a common strategy to mitigate their negative effect is to oversize the components. This conservative choice could be largely avoided if the controller was designed taking all limitations into account. Similarly, neglecting the constraints in system estimation often leads to suboptimal solutions, which in turn may negatively affect the control effectiveness. Therefore, with the idea of taking a step further towards reliable and sustainable engineering solutions, based on more conscious use of the plants' dynamics, we decide to address in this thesis two fundamental challenges related to constrained control and observation. In the first part of this work, we consider the control of uncertain nonlinear systems with input and state constraints, for which a general approach remains elusive. In this context, we propose a novel closed-form solution based on Explicit Reference Governors and Barrier Lyapunov Functions. Notably, it is shown that adaptive strategies can be embedded in the constrained controller design, thus handling parametric uncertainties that often hinder the resulting performance of constraint-aware techniques. The second part of the thesis deals with the global observation of dynamical systems subject to topological constraints, such as those evolving on Lie groups or homogeneous spaces. Here, general observability analysis tools are overviewed, and the problem of sensorless control of permanent magnets electrical machines is presented as a case of study. Through simulation and experimental results, we demonstrate that the proposed formalism leads to high control performance and simple implementation in embedded digital controllers

Output regulation for systems with symmetry

The problem of output regulation deals with asymptotic tracking/rejection of a prescribed reference trajectory/disturbance. The main feature of the output regulation is that references/disturbances to be tracked/rejected belong to a family of trajectories generated as solutions of an autonomous system typically referred to as exosystem. Tackling this problem in context of error feedback leads to solutions that embeds a copy of the exosystem properly updated by means of error measurements. The output regulation problem for linear systems has been fully characterized and solved in the mid seven- ties by Davison, Francis and Wonham and then has been generalized to the non-linear context by Isidori and Byrnes. It is worth noting, however, that most of the frameworks considered so far for output regulation deal with systems and exosytems defined on Euclidean real state space and not much efforts have been done to extend the results of output regulation to systems and exosystems whose states live in more general manifolds. The tools available for solutions of the output regulation problem can’t be extented in a straightforward manner to non-linear systems whose states live in more general manifolds due to some restrictive structural assumption. The present thesis focuses on the problem of output regulation for left invariant systems defined on matrix Lie groups. In this framework we extend the idea of internal model-based control to systems defined on matrix Lie-groups taking advantages of the symmetry and invariant structures of the system considered. In particular we propose a general structure of the regulator for left invariant kinematic systems defined on general matrix Lie-group that solves the output regulation problem. Going further we study the output regulation problem for kinematics systems defined on the special orthogonal group and the special Euclidean group