Corrado Possieri

On observer design for a class of continuous-time affine switched or switching systems

The main goal of this paper is to design a state observer for a class of affine switched or switching dynamical systems, without requiring the knowledge of the switching signal. To reach such a goal some tools, taken from Algebraic Geometry, are used to express the switching signal as a function of the output and its time derivatives. Then, similar tools are used to design an observer to estimate both the switching signal and the state of the system. A physically motivated example of application is reported.

Switching Signal Estimator Design for a Class of Elementary Systems

The goal of this technical note is to design a switching signal estimator for a class of elementary continuous-time switching or switched systems. First, the elementary system is recast into a polynomial form and, secondly, some tools borrowed from Algebraic Geometry are used to express the switching signal as a function of the time derivatives of the output and of the input.

Boolean network representation of a continuous-time system and finite-horizon optimal control: application to the single-gene regulatory system for the lac operon

ABSTRACT In this paper, the single-gene regulatory system for the lac operon is considered and a method to build a Boolean network able to represent its dynamics is given. A procedure, based on techniques borrowed from algebraic geometry, to compute an optimal control policy minimising a cost function is given for such a Boolean network. A tool to apply such an optimal control policy to the continuous-time system is reported. The proposed control method is tested in two simulative examples.

On the computation of the continuous-time reference trajectory for mechanical juggling systems

A particular class of jugglers, i.e., linear mechanical systems, subject to impacts, which would not be controllable, if the impacts were absent, is considered. The main objective is to give a fully algorithmic procedure for the computation of the continuous-time reference trajectory for this class of systems, that, combined with an existing control law, constitutes the solution of the global dead-beat regulation of the “uncontrollable” subsystem, just by using impacts.

On polynomial vector fields having a given affine variety as attractive and invariant set: application to robotics

The main goal of this paper is to compute a class of polynomial vector fields, whose associated dynamical system has a given affine variety as attractive and invariant set, a given point in such an affine variety as invariant and attractive and another given affine variety as invariant set, solving the application of this technique in the robotic area. This objective is reached by using some tools taken from algebraic geometry. Practical examples of how these vector fields can be computed are reported. Moreover, by using these techniques, two feedback control laws, respectively, for a unicycle-like mobile robot and for a car-like mobile robot, which make them move, within the workspace, approaching to a selected algebraic curve, are given.

On polynomial feedback Nash equilibria for two-player scalar differential games

In this paper, two-player scalar differential games are thoroughly studied, in the presence of polynomial dynamics and focusing on the notion of solution provided by polynomial feedback Nash equilibria. It is well-known that such strategies are related to the solution of coupled partial differential equations, namely the so-called Hamilton-Jacobi-Isaacs equations. Herein, we firstly prove a somewhat negative result, stating that, for a generic choice of the parameters, two-player scalar polynomial differential games do not admit polynomial Nash equilibria. Then, we focus on the class of Linear-Quadratic (LQ) games and we propose an algorithm that, by borrowing techniques from algebraic geometry, allows to recast the problem of computing all stabilizing Nash feedback strategies into that of finding the zero of a single polynomial function in a scalar variable. This permits a comprehensive characterization-in terms of number and values-of the set of solutions to the associated game.

An algebraic geometry approach for the computation of all linear feedback Nash equilibria in LQ differential games

In this paper, Linear-Quadratic (LQ) differential games are studied, focusing on the notion of solution provided by linear feedback Nash equilibria. It is well-known that such strategies are related to the solution of coupled algebraic Riccati equations, associated to each player. Herein, we propose an algorithm that, by borrowing techniques from algebraic geometry, allows to recast the problem of computing all stabilizing Nash strategies into that of finding the zeros of a single polynomial function in a scalar variable, regardless of the number of players and the dimension of the state variable. Moreover, we show that, in the case of a scalar two-player differential game, the proposed approach permits a comprehensive characterization - in terms of number and values - of the set of solutions to the associated game.

Application of algebraic geometry techniques in permanent-magnet DC motor fault detection and identification

Abstract This paper presents a fault detection method, which is able to identify the occurrence of a fault and, possibly, its nature, in permanent-magnet DC motors. Techniques arising from Algebraic Geometry are used to compute the motor dynamical model parameters in function of the output and its time derivatives, up to a finite order. Similar tools are used to compute auxiliary expressions of the state of the motor, independent of the estimated parameter. Simulative and experimental examples show the effectiveness of the proposed technique.

On f-invariant and attractive affine varieties for continuous-time polynomial systems: The case of robot motion planning

The main objective of this paper is to describe a class of polynomial vector fields f, whose associated dynamic system has one or more affine varieties as f-invariant and attractive sets. This result can be used for robot motion planning, thus computing robot paths, avoiding collisions with obstacles and reaching a target point.

Asymptotic stability in probability for Stochastic Boolean Networks

Abstract In this paper, a new class of Boolean networks, called Stochastic Boolean Networks, is presented. These systems combine some features of the classical deterministic Boolean networks (the state variables admit two operation levels, either 0 or 1) and of Probabilistic Boolean Networks (at each time instant the transition map is selected through a random process), enriching the set of admissible dynamical behaviors, thanks to the set-valued nature of the transition map. Necessary and sufficient Lyapunov conditions are given to guarantee global asymptotic stability (resp., global asymptotic stability in probability) of a given set for a deterministic Boolean network with set–valued transition map (resp., for a Stochastic Boolean Network). A constructive procedure to compute a Lyapunov function (resp., stochastic Lyapunov function) relative to a given set for a deterministic Boolean network with set–valued transition map (resp., Stochastic Boolean Network) is reported.