arXiv subjects

Pietro Lorenzetti

Publications and source records attributed to Pietro Lorenzetti.

6 recordsLinked to original sources

arXiv2026

The strong stability of the Perron-Frobenius semigroup and almost global attractivity

We discuss some useful properties of the solution map (flow) of a nonlinear dynamical system with a finite-dimensional state space. Then, we introduce the Perron-Frobenius semigroup, and we prove that it is a positive strongly continuous semigroup of contractions. We show that, given a nonlinear system and an invariant set, this set is an almost global attractor if and only if certain Perron-Frobenius semigroups associated to the nonlinear system are strongly stable. Unlike other works on the Perron-Frobenius semigroup from the literature, we do not require the existence of a compact and invariant state-space for the dynamical system, we allow trajectories with finite escape time, and we do not require the attractor to be locally (Lyapunov) stable. Two simple examples are used throughout the paper to illustrate the theory.

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arXiv2025

Projected integral control of impedance passive nonlinear systems

We propose an abstract framework for solving the constrained set-point tracking problem for impedance passive infinite-dimensional nonlinear systems. The class of systems considered is governed by monotone differential inclusions and allows us to exploit the theory of contraction semigroups. To account for possible operational constraints, e.g., bounds on the input, we replace a classical integral controller with a projected integral controller. This guarantees that the integrator state remains in a given closed convex set, where said constraints are satisfied. We showcase our results through three case studies.

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arXiv2023

On singularly perturbed systems that are monotone with respect to a matrix cone of rank $k$

We derive a sufficient condition guaranteeing that a singularly perturbed linear time-varying system is strongly monotone with respect to a matrix cone $C$ of rank $k$. This implies that the singularly perturbed system inherits the asymptotic properties of systems that are strongly monotone with respect to $C$, which include convergence to the set of equilibria when $k=1$, and a Poincaré-Bendixson property when $k=2$. We extend this result to singularly perturbed nonlinear systems with a compact and convex state-space. We demonstrate our theoretical results using a simple numerical example.

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arXiv2022

PI control of stable nonlinear plants using projected dynamical systems

This paper presents a novel anti-windup proportional-integral controller for stable multi-input multi-output nonlinear plants. We use tools from projected dynamical systems theory to force the integrator state to remain in a desired (compact and convex) region, such that the plant input steady-state values satisfy the operational constraints of the problem. Under suitable monotonicity assumptions on the plant steady-state input-output map, we use singular perturbation theory results to prove the existence of a sufficiently small controller gain ensuring closed-loop (local) exponential stability and reference tracking for a feasible set of constant references. We suggest a particular controller design, which embeds (when possible) the right inverse of the plant steady-state input-output map. The relevance of the proposed controller scheme is validated through an application in the power systems domain, namely, the output (active and reactive) power regulation for a grid-connected synchronverter.

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arXiv2021

Saturating PI control of stable nonlinear systems using singular perturbations

This paper presents an anti-windup PI controller, using a saturating integrator, for a single-input single-output stable nonlinear plant, whose steady-state input-output map is increasing. We prove that, under reasonable assumptions, there exists an upper bound on the controller gain such that for any constant reference input, the corresponding equilibrium point of the closed-loop system is exponentially stable, with a "large" region of attraction. When the state of the closed-loop system converges to this equilibrium point, then the tracking error tends to zero. The closed-loop stability analysis employs Lyapunov methods in the framework of singular perturbations theory. Finally, we show that if the plant satisfies the asymptotic gain property, then the closed-loop system is globally asymptotically stable for any sufficiently small controller gain. The effectiveness of the proposed PI controller is proved by showing how it performs as part of the control algorithm of a synchronverter (a special type of DC to AC power converter).

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arXiv2021

The equilibrium points and stability of grid-connected synchronverters

Virtual synchronous machines are inverters with a control algorithm that causes them to behave towards the power grid like synchronous generators. A popular way to realize such inverters are synchronverters. Their control algorithm has evolved over time, but all the different formulations in the literature share the same "basic control algorithm". We investigate the equilibrium points and the stability of a synchronverter described by this basic algorithm, when connected to an infinite bus. We formulate a fifth order model for a grid-connected synchronverter and derive a necessary and sufficient condition for the existence of equilibrium points. We show that the set of equilibrium points with positive field current is a two-dimensional manifold that can be parametrized by the corresponding pair $(P,Q)$, where $P$ is the active power and $Q$ is the reactive power. This parametrization has several surprizing geometric properties, for instance, the prime mover torque, the power angle and the field current can be seen directly as distances or angles in the $(P,Q)$ plane. In addition, the stable equilibrium points correspond to a subset of a certain angular sector in the $(P,Q)$ plane. Thus, we can predict the stable operating range of a synchronverter from its parameters and from the grid voltage and frequency. Our stability result is based on the intrinsic two time scales property of the system, using tools from singular perturbation theory. We illustrate our theoretical results with two numerical examples.

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