A discrete-time positive switched system (DPSS) consists of a family of positive state-space models and a switching law, specifying when and how the switching among the various models takes place. These systems have some interesting practical applications: they have been adopted for describing networks employing TCP and other congestion control applications, for modeling consensus and synchronization problems, and, quite recently, for describing the viral mutation dynamics under drug treatment.
As for the broader classes of hybrid and switched systems, stability and stabilizability properties have been the two major issues to attract the researchers' attention. Clearly, all results so far obtained for general discrete-time switched systems hold true for DPSS's.
In particular, the asymptotic stability of a DPSS switching into a finite set of matrices, say A, is equivalent to the fact that the joint spectral radius of A, is smaller than 1.
Even if a number of algorithms was proposed to evaluate the joint spectral radius of a set of matrices in quite general conditions, research efforts about stability and henceforth about stabilizability have also taken alternative directions and focused on different approaches.The most popular approach to the investigation of stability and stabilizability is undoubtedly the one based on common Lyapunov functions or multiple Lyapunov functions.
In this talk we concentrate our attention on discrete-time positive switched systems, and we investigate in detail stability and stabilizability properties for them. We propose several sufficient conditions for testing stability, based on the existence of special classes of common Lyapunov functions, and we mutually relate them, thus proving that if a linear copositive common Lyapunov function can be found, then a quadratic positive definite common Lyapunov function can be found, too, and this latter, in turn, ensures the existence of a quadratic copositive common Lyapunov function.
Stabilizability is also introduced and characterized. It is shown that if a DPSS is stabilizable, it can be stabilized by means of a periodic switching sequence, which asymptotically drives to zero every positive initial state. Conditions for the existence of state-dependent stabilizing switching laws, based on the values of a copositive (linear/quadratic) Lyapunov function, are investigated and related to each other. Interestingly enough, the mutual relationship between the various conditions for the existence of these special Lyapunov functions are very close to the analogous ones obtained for the stability characterization.
Maria Elena Valcher
Maria Elena Valcher received the Master Degree (cum laude) in Electronic Engineering in July 1991 from the University of Padova (Italy) and the Ph.D. Degree in Systems Engineering in October 1995 from the same university. From January 2005 she is Full Professor of Control Theory at the University of Padova.
She is author/co-author of more than 50 papers appeared on international journals, 70 conference papers and 14 book chapters. Her research interests include multidimensional systems theory, polynomial matrix theory, behavior theory, convolutional coding, fault detection and observer design, delay-differential systems, switched systems and positive systems.
She has been involved in the Organizing Committees and in the Program Committees of several conferences. She is currently the Program Chair of the CDC 2012, Maui, Hawaii, 2012.
She was in the Editorial Board of the IEEE Transactions on Automatic Control (1999-2002) of Systems and Control Letters (2204-2010) and she is currently in the Editorial Boards of
- Automatica (2006-today),
- Multidimensional Systems and Signal Processing (2004-today).
She was Appointed Member of the CSS BoG (2003), Elected Member of the CSS BoG (2004-2006; 2010-2012), Vice President Member Activities (2006-2007) and Vice President Conference Activities (2008-2010) of the IEEE Control System Society.
Location
Distinguished Lecture Program
Talk Title: Stability and stabilizability of discrete-time switched systems with positivity constraints
Talk Title: A behavioral approach to dead-beat control
As first suggested in the pioneering papers by Jan C. Willems, the behavioral approach provides the natural framework where control problems can be addressed in the utmost generality and without any a priori assumption regarding input/output partition and feedback connection. Indeed, in this setting, the control target is that of restricting the behavior trajectories to a subset of "good ones" and this goal is achieved by interconnection, namely by constraining either all or a subset of the system variables to obey an additional family of
laws, which represent the controller laws.
Starting from these inspiring contributions, there has been quite a number of papers on the control within the behavioral framework. In these papers two fundamental control set-ups have been explored: the full interconnection case and the partial interconnection case.
In the former, it is assumed that all variables are the target of the control problem (for instance, the stabilization problem) and at the same time they are all available for interconnection. In the latter, the system variables are partitioned in two (or possibly more) groups, by distinguishing between to be controlled variables, which are the object of the control specifications, and control variables, which are the means through which the control target is achieved. Indeed, the controller achieves the desired result by restricting the behavior of the control variables which are the only ones available for interconnection.
In this talk we explore the dead-beat control (DBC) problem for discrete-time behaviors defined on Z+. In investigating this problem, we take a perspective that differs both from the full interconnection and from the partial interconnection set-ups, in that we assume that only the to be controlled variable must be "driven to zero in a finite number of steps", but the control laws restrict the evolutions of all the variable. So, there is full interconnection for control purposes, but the control target is just the to be controlled variable.
The investigation of the DBC problem under this perspective requires to introduce new concepts of controllability and of zero-controllability. Consistently, it will turn out that the possibility of designing DBC's with good properties (admissibility or regularity) is just equivalent to the zero-controllability property. Special attention will be devoted to the class of minimal DBC's, by this meaning DBC's with the least number of rows, for which a parametrization will be provided. Such parametrization is quite complex, as it resorts both to polynomial and to rational parameters, under the constraint that the obtained result is polynomial. Open problems will be illustrated.