This thesis presents several results pertaining two rather distinct research topics within the broader area of the socalled "Switched Systems". The first part of the work features a deep investigation of the structural properties, namely reachability and zerocontrollability, of "Positive Switched Systems", both for the discretetime and the continuoustime 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 zerocontrollability are properly defined within the context of positive switched systems. Then, several results are presented, first dealing with the discretetime case, and subsequently addressing the continuoustime one. More specifically, in Chapter 3, the zerocontrollability of a discretetime 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 discretetime 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 nonswitched) positive systems, but only necessary in our setting, is fully explored and characterized. All the chapters from 5 to 8 tackle with the continuoustime case. In particular, in Chapter 5 the possibility for a continuoustime positive switched system to be zerocontrollable is ruled out. The reachability issue is first addressed in Chapter 6 where, similarly to the discretetime 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 continuoustime 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 nonswitching 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 continuoustime 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.
Topics on switched systems / Santesso, Paolo.  (2008 Jan).
Topics on switched systems
Santesso, Paolo
2008
Abstract
This thesis presents several results pertaining two rather distinct research topics within the broader area of the socalled "Switched Systems". The first part of the work features a deep investigation of the structural properties, namely reachability and zerocontrollability, of "Positive Switched Systems", both for the discretetime and the continuoustime 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 zerocontrollability are properly defined within the context of positive switched systems. Then, several results are presented, first dealing with the discretetime case, and subsequently addressing the continuoustime one. More specifically, in Chapter 3, the zerocontrollability of a discretetime 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 discretetime 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 nonswitched) positive systems, but only necessary in our setting, is fully explored and characterized. All the chapters from 5 to 8 tackle with the continuoustime case. In particular, in Chapter 5 the possibility for a continuoustime positive switched system to be zerocontrollable is ruled out. The reachability issue is first addressed in Chapter 6 where, similarly to the discretetime 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 continuoustime 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 nonswitching 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 continuoustime 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.File  Dimensione  Formato  

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