Carlo Savorgnan

Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions

Counterexamples are given which show that a linear switched system (with controlled switching) that can be stabilized by means of a suitable switching law does not necessarily admit a convex Lyapunov function. Both continuous and discrete-time cases are considered. This fact contributes in focusing the difficulties encountered so far in the theory of stabilization of switched systems. In particular the result is in contrast with the case of uncontrolled switching in which it is known that if a system is stable under arbitrary switching then admits a polyhedral norm as a Lyapunov function

Brief paper: Stability results for linear parameter varying and switching systems

In this paper, stabilizability problems for discrete and continuous time linear systems with switching or polytopic uncertainties are dealt with. Both state feedback and full-information controllers are considered, possibly with an integral action. Several cases will be analyzed, depending on the control, the uncertainty and the presence of uncertainties in the input matrix. A complete overview of the stabilizability implications among these different cases will be given.

MAXIS-G: a software package for computing polyhedral invariant sets for constrained LPV systems

In this paper, a software package for computing the maximal invariant set of constrained linear parameter varying (LPV) systems is presented. The theoretical details at the basis of the proposed algorithm are first briefly illustrated along with the extensions which allow to deal with systems affected by disturbances and different kinds of uncertainty. Then, the algorithm is presented together with some comments on its actual implementation.

A Mixed Convex/Nonconvex Distributed Localization Approach for the Deployment of Indoor Positioning Services

In this paper the indoor wireless localization problem is addressed both from the theoretical and application standpoints. The main result of the paper is on the theoretical side: the topological definition of regular and irregular nodes is introduced, and formal results are presented to support regularity as a desirable network property for the attainment of precise node localization. In force of this definition, a mixed convex/non-convex optimization approach has been derived for the solution of the positioning problem. The two procedures, suitably combined, allow the achievement of better convergence towards the best positioning of a multitude of blind wireless nodes. A completely decentralized, partially asynchronous algorithm is presented, which proceeds locally on each node based on the sole knowledge of the distances measured from, and of the estimated positions of the connected nodes only. Its repeated asynchronous application on each nodes guarantees the convergence of the algorithm to the positioning of the whole network, even in presence of a limited number of peripheral reference points. Indeed, no global information is required for the proper functioning of the algorithm. Simulations of relevant case studies have been performed to qualify the proposed scheme in realistic conditions, and the results are presented.

Dynamic augmentation and complexity reduction of set-based constrained control

Computing polytopic controlled invariant sets which are maximal inside a prescribed region often yields sets which have a really complex representation. Since the associated control has a complexity which drastically increases with that of the region, this turns out to be a major problem in the implementation of the theory of controlled invariant regions. In this paper, we consider the problem of reducing the complexity of these regions and/or the complexity of the associated compensator. We propose two heuristic techniques based on spectral properties of some relevant matrices and on vertex-elimination methods. The paper presents several preliminary results which are interesting on their own such as dynamic augmentation and the properties of complex (as opposed to real) polytopic invariant regions.

Complex polytopic control Lyapunov functions

In this paper, the class of real Lyapunov functions induced by complex polytopes is investigated. Such polytopes are in some sense the extension of real polytopes to the complex space, but have some peculiar aspects which might be relevant for the solution of constrained and/or robust control problems. In detail, analysis and synthesis results (which rely on real polytopes) for a class of linear uncertain systems are extended to the complex case. A relevant contribution of this new extensions is that of showing that the stabilizability (stability) conditions imply that the projection (projection and intersection) on the real space of such complex polytopes are level sets of control Lyapunov functions, say they can be used to derive a stabilizing control law

Stability results for continuous and discrete time linear parameter varying systems

Abstract In this paper, stabilizability problems for discrete and continuous time linear systems with switching or polytopic uncertainties are dealt with. Both state feedback and full-information controllers are considered, possibly with an integral action. Several cases will be analyzed, depending on the control, the uncertainty and the presence of uncertainties in the input matrix. A complete overview of the stabilizability implications among these different cases will be given.

Mixed Convex/Non-Convex Distributed Localization Algorithm for the Deployment of Indoor Positioning Services

In this paper a fully decentralized algorithm for indoor wireless localization is presented. The algorithm combines the use of convex and non-convex optimization procedures, nested to achieve better converge towards the best positioning of a multitude of blind wireless nodes. The algorithm proceeds locally on each node, based on the sole knowledge of the measured distances from, and estimated positions of, the connected nodes only. Repeated asynchronous application on each node of the local procedure guarantees hierarchical convergence of the algorithm to the positioning of the whole network, even in presence of a limited number of peripheral anchors. No global information is required for the proper functioning of the algorithm. From the theoretical standpoint, the paper introduces the geometrical definition of regular and irregular nodes, which is shown to be a useful concept, given a network topology, for the characterization of the convergence properties of the localization problem.

Polyhedral lyapunov functions computation for robust and gain scheduled design

Abstract In this paper, the problem of efficiently computing polyhedral Lyapunov functions for poly topic time-varying linear discrete-time systems, is considered. Two strictly related problems are considered: the robust case, when the designer is unaware of the value of the uncertain parameter entering the system, and the gain scheduled case, when such value is available and exploitable for synthesis design. The relation between the two is evidenced, together with an efficient algorithm for the determination of the respective Lyapunov functions. Finally, it is shown how such algorithm can be used for the determination of proper domains of attraction when constraints on the state and/or input are present.

Set based control synthesis for state and velocity constrained systems

In the present contribution, the control synthesis for state and input constrained systems for linear systems via set-valued techniques will be presented. The results will be developed for both uncertain and gain scheduled constrained control problems, where the difference between the two is the knowledge of the uncertain parameter entering the system at each time instant in the gain scheduled case. It will be shown how slew-rate constrained control problems can be recast into standard constrained control problems by means of an extended system. To present such technique, an overview of some of the existing results with some numerical algorithms to derive the stability conditions and the control will be given.