Laura Menini

Trajectory tracking for a particle in elliptical billiards

An infinitely rigid unitary mass (particle) is considered, moving on a planar region delimited by a rigid elliptical barrier (elliptical billiards) under the action of proper control forces. A class of periodic trajectories, involving an infinite sequence of non-smooth impacts between the mass and the barrier at fixed times, is found by using an LMIs based procedure. The jumps in the velocities at the impact times render difficult (if not impossible) to obtain the classical stability and attractivity properties for the dynamic system describing the tracking error behaviour. Hence, the tracking control problem is properly stated using notions similar to the quasi stability concept in V. Lakshmikantham, D.D. Bainov and P.S. Simeonov, Theory of Impulsive Differential Equations, 6, World Scientific, 1989. A controller (whose state is subject to discontinuities) based on the internal model principle is shown to solve the proposed tracking problem, giving rise to control forces that are piecewise continuous function of time, with discontinuities at the desired impact times and at the impact times of the particle with the barrier.

Symmetries and semi-invariants in the analysis of nonlinear systems

Part I: Theory.- Introduction.- Notation and Background.- Analysis of Linear Systems.- Analysis of Nonlinear Systems.- Analysis of Hamiltonian Systems.- Linearization by State Immersion.- Linearization by State Immersion of Hamiltonian Systems.- Extensions Based on the Use of Orbital Symmetries.- Part II: Applications to Control Systems.- Computation of the Flow of Linearizable Systems.- Semi-invariants.- Stability Analysis.- Observer Design.- Exact Sampling of Continuous-time Systems.- Applications to Physically Motivated Systems.

Semi-invariants and their use for stability analysis of planar systems

Semi-invariants and relative characteristic functions extend to nonlinear systems the concept of eigenvector–eigenvalue pair for linear systems, and are, therefore, very useful to depict the behaviour of the system. In this article, semi-invariants are used to construct explicitly Lyapunov functions useful for studying the stability of the origin, for continuous-time systems. Moreover, using well-known tools from differential geometry such as (orbital) symmetries, it is shown how semi-invariants can be found for several classes of systems. Important connections with centre manifold theory are pointed out. By using the proposed general techniques, a new proof of a result claimed to Bendixson (Bendixson, I. (1901), ‘Sur les courbes définies par des équations différentielles’, Acta Mathematics, 24, 1–88) is given.

Brief paper: Linearization through state immersion of nonlinear systems admitting Lie symmetries

In this paper, it is shown that if a nonlinear system admits a Lie symmetry that can be transformed into its Poincare-Dulac normal form by a state diffeomorphism, then, under some technical conditions, such a nonlinear system can be immersed into a linear one. This allows us to compute in closed-form the flow, all algebraic invariant curves (through semi-invariants) of the nonlinear system, and Lyapunov functions to study stability properties.

Velocity observers for non-linear mechanical systems subject to non-smooth impacts ☆

This paper is concerned with the estimation of the velocity variables (when the position variables are the measured outputs) for non-linear mechanical systems subject to non-smooth impacts, both elastic and inelastic (i.e., with coefficient of restitution e=1 and e∈(0,1), respectively). A reduced-order observer is proposed, which guarantees that the corresponding error system, despite the possible presence of an infinite sequence of non-smooth impacts, is locally exponentially stable. An estimate of the basin of attraction is also given.

Computation of the real logarithm for a discrete-time nonlinear system

For a given nonlinear discrete-time dynamical system, the problem is considered of finding, if any, a continuous-time nonlinear dynamical system such that the given system is its exact sampled-data representation. Constructive solutions to the problem are given by geometric tools such as symmetries.

Design of state detectors for nonlinear systems using symmetries and semi-invariants☆

Two techniques are proposed for the design of asymptotic state detectors for nonlinear systems, both in the continuous-time and discrete-time cases. The first technique allows one to design linear state detectors for nonlinear systems belonging to a given class, which is studied and characterized in the paper. The second technique extends the ideas of the first by using semi-invariants to compute invariant sets and study their stability.

A procedure for the computation of semi-invariants

This paper deals with the concept of semi-invariant (Darboux polynomial), which extends to the nonlinear case the concept of eigenfunction. Semi-invariants can be used as elementary bricks for the construction of Lyapunov functions. For planar systems, the construction of semi-invariants is strictly correlated with the existence of an orbital symmetry of the system. This feature is not easily extended when the dimension of the system state is greater than 2. In this paper, a procedure is proposed that does not rely on the existence of orbital symmetries of the system, whence can be easily applied to general systems. The proposed technique is applied to some relevant and physically motivated systems.

Output Regulation of Hybrid Linear Systems with Unpredictable Jumps

In this paper the output regulation problem for a class of hybrid linear systems in the presence of uncertain time domain is considered. Uncertainty is modeled by assuming that the time domain is not known to the controller and by allowing for arbitrarily close, even simultaneous, jumps. Considering the full information setting, the geometric characterization of the relevant regulation manifolds is given. Finally, the theory is illustrated and validated by means of numerical examples.

Exact and approximate feedback linearization without the linear controllability assumption

The feedback linearization problem of nonlinear control systems has been solved in the literature under the assumption that the nonlinear system is linearly controllable. In this paper, the assumption of linear controllability is removed; necessary and sufficient conditions are given through Lie orbital symmetries, thus giving a geometric characterization of the problem in the analytic case. Both the exact and the approximate linearization problems are considered in the analytic case.