# Geometric Methods and Computational Tools for Switching Structured Linear Systems
## 1 Introduction  
Switching structured linear systems are dynamical systems that can be described by a family of structured linear systems with the same dimension, the _modes_, and by a time-dependent signal, the _switching signal_, that determines the active mode at each time instant. Each mode, as a structured linear system, is such that the direct relations between its variables are described either by fixed zeros or by real parameters whose value (possibly including zero) is unknown. In particular, the switching from one structured linear system of the family to another enables the direct relations between the variables to change from non-existent (fixed zeros) to existent (unknown real parameters) and vice versa. Therefore, the interconnections between the variables of switching structured linear systems are not only described by unknown real values wherever they exist, but they also exhibit a topology subject to change. The main motivation for introducing and studying this class of dynamical systems is that they are a powerful means to model multiagent systems whose network features an _evolving topology_, as is explained through the literature review that follows. Structured linear systems were introduced in the mid-seventies by [PERSON] [1]. In the following two decades, they were extensively studied and the literature of those years mainly emphasizes their role in coping with model uncertainties. Since the state-space representation of a structured linear system consists of linear maps whose entries are either fixed zeros or unknown real parameters, a single structured linear system is, actually,representative of a family of classic linear time-invariant systems. For this reason, many authors resorted to structured linear systems to investigate systems' properties featuring some robustness to parameter variations (see, e.g., [2; 3; 4]) or, as was more precisely pointed out by [PERSON] and coauthors in their survey paper [5], properties that hold true for almost any value of the parameters, briefly referred to as _generic properties_. More recently, structured linear systems have received renewed attention, as is well-documented by another survey paper [6]. However, in the latest years, the focus of interest for structured linear systems has turned to their use in the study of _complex systems_ and _networked systems_ (see, e.g., [7; 8; 9; 10; 11]). Indeed, a consolidated approach to the study of networked systems leverages the network topology (i.e., the map of the interconnections between the agents) to investigate the network properties, whereas the behavior of each agent is modeled by quite simple dynamics--in the most elementary case, the agents are described by single integrators [12]. The widespread use of structured linear systems in the investigation of the properties of networked systems has led to various _generalizations_ of the original paradigm, such as switching structured linear systems [13; 14], structured switching descriptor linear systems [15], and structured linear systems with pattern matrices [16; 17; 18]. In particular, switching structured linear systems were introduced in [13], while in [14] they were considered in connection with electrical networks and complex systems, like the glucose metabolism in the human body, as a means to model networked systems where the topology of the respective interconnections is subject to change. Since, as mentioned earlier, switching structured linear systems consist of a family of modes, which are structured linear systems, and of an exogenous signal, which rules the switching from one mode to another, the system properties worth investigating are those that are true not only for almost any value of the parameters, but also for any admissible switching sequence.