Using algebraic geometry for control probems: from observer design to game theory

The development of inexpensive and fast computers, coupled with the discovery of ecient algorithms for dealing with polynomial equations, gave rise to some exciting new applications of algebraic geometry and commutative algebra. One of the main goals of this work is to show how some tools borrowed from these two elds can be eciently employed to solve relevant control problem, as, for instance, nonlinear observer design, fault detection and isolation, motion planning of mobile robots, optimal control and game theory. In order to do this, a brief introduction to some algebraic objects and techniques is given in Chapter 1, where concepts that nd useful applications in control theory are highlighted. To keep the exposition reasonably short, the classical results in algebraic geometry are given without proofs, although complete references are given. Such tools and methodologies are then applied to a wide variety of topics concerning control theory, as detailed in the following. In Chapter 2, a procedure that, coupled with classical control techniques, allows to easily design nonlinear observers is proposed, with a particular attention on switching and switched nonlinear systems and on disturbance identication. In Chapter 3, a model{based tool is given to design fault detectors. In Chapter 4, it is shown how algebraic geometry can be eciently employed to design nonlinear systems having an ane variety as attractive and invariant set, with a focus on how these techniques can be employed to plan the motion of mobile robots. In Chapter 5, a method to build a Boolean network able to represent the dynamics of a continuous{time system is given. A procedure to compute a nite{horizon optimal policy for a Boolean network is given and a tool to apply the optimal control to the continuous{time system is reported. i In Chapter 6, a novel approach for robust model predictive control (MPC) for constrained linear discrete{time systems with bounded time{varying structural uncertainties is presented. In Chapter 7, a novel algorithm to compute the set of all the solutions to a polynomial system of equalities is given. Such an algorithm is then employed in three relevant control applications: the static output feedback stabilization of a linear parametrically varying plant, the dead{beat regulation of mechanical juggling systems, and the solution to Linear Quadratic dierential games. In Chapter 8, by exploiting some tools borrowed from algebraic geometry, interesting game{theoretical results are given. An attempt has been made to keep the exposition as self{contained as possible, by reporting proofs of the new results that, sometimes, provide a deeper understanding of the topic and of its relationship with other results presented in the literature.