Santesso, Paolo (2008) *Topics on switched systems.* [Ph.D. thesis]

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## Abstract (english)

This thesis presents several results pertaining two rather distinct research topics within the broader area of the so-called "Switched Systems".

The first part of the work features a deep investigation of the structural properties, namely reachability and zero-controllability, of "Positive Switched Systems", both for the discrete-time and the continuous-time case.

All the notation relative to this contribution is defined in Chapter 1. Together with considerations on the motivational aspect, in Chapter 2, the familiar concepts of reachability and zero-controllability are properly defined within the context of positive switched systems. Then, several results are presented, first dealing with the discrete-time case, and subsequently addressing the continuous-time one. More specifically, in Chapter 3, the zero-controllability of a discrete-time positive switched system is proved to be equivalent to the mortality property of the set of system matrices; some sufficient conditions for this property to hold are then provided, together with an algorithm designed to find the correct switching sequence, if any, which is needed to drive any positive state

vector to the zero vector.

In Chapter 4 the reachability issue for discrete-time positive switched systems is addressed. First, the problem is restated into a geometric form, then the property of monomial reachability, known to be equivalent to the reachability for standard (meaning non-switched) positive systems, but only necessary in our setting, is fully explored and characterized.

All the chapters from 5 to 8 tackle with the continuous-time case.

In particular, in Chapter 5 the possibility for a continuous-time positive switched system to be zero-controllable is ruled out.

The reachability issue is first addressed in Chapter 6 where, similarly to the discrete-time case, we investigate the monomial as well as the pattern reachability property, which represent two necessary conditions for the general reachability of the system. Then, in Chapter 7, a useful sufficient condition for the reachability is provided; a geometric equivalent description of a reachable system is also introduced. Finally, further contributions to the problem of finding conditions ensuring the reachability of a continuous-time positive switched system are presented in Chapter 8, where the useful concept of asymptotic exponential cone of a Metzler matrix (an ordered set of Metzler matrices) is first defined and then fully characterized.

Results pertaining to a different stream of research 1 are included in the chapters 9 and 10.

More specifically, in Chapter 9 the case when a traditional Linear Time Invariant plant is controlled by a switching multicontroller whose transfer function may commute among different ones, each of them stabilizing the system, is considered. In particular, we focus our interest on the design of

the function, called Reset Map, ruling the update of the multicontroller state vector at every switching time. It turns out that a proper choice of it may deeply improve the controlled system transient behaviour.

The application of the same principles is then suggested in Chapter 10 in the context of non-switching reset controllers. The result presented within this chapter represents a substantial enhancement with respect to the traditional

approach which is known in the literature under the name of Reset Control Strategy.

The Appendix, besides a series of technical results which are preliminary to those presented in this thesis, features an extensive contribution to the study of the exponential of a Metzler matrix. This topic has been initially addressed as a mathematical mean for solving certain specific problems within

the setting of positive switched systems. Indeed, the analysis of reachability property for continuous-time positive switched systems requires a deep knowledge of the behaviour of these exponential matrices. For this reason, we decided to include the results in the Appendix. However, we believe that they deserve some interest by themselves, as their significance and extension

exceed by far what we needed for their initial application.

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