## 1 Introduction  
Hybrid systems play an important role in practical applications, such as intelligent transportation systems and manufacturing systems. In fact, hybrid systems are formed by the discrete event dynamical subsystem and the continuous variable dynamical subsystem that interact and intermix with each other [1; 2]. Generally, the former can be modeled as a Petri net [3; 4], or a finite state machine (FSM) [5; 6]. Some basic research results on FSMs have been obtained [6; 7; 8], concerning its structure, stability, observability, and liveliness. If hybrid systems consist of a family of dynamical subsystems and a switching signal which determines the switching manner between the subsystems, they are called switched systems [9; 10; 11]. A system is positive if for any nonnegative initial condition its state variables and outputs naturally take non-negative values for all nonnegative times. Generally, positive systems have many special and interesting properties [12; 13; 14; 15]. Moreover, positive switched systems switch between several positive subsystems [16]. The importance of positive switched systems has received much attention due to their broad applications in communication systems [17], formation flying [18], systems theory [19], and so on. Actually, the study of positive switched systems is more challenging than that of general switched systems because the features of positive systems and the features of switched systems have to be combined to obtain elegant results [20]. It should be pointed out that many previous results on positive switched systems focus mainly on stability analysis and controller synthesis [21; 22; 23; 24; 25; 26; 27; 28; 29], such as exponential stability [21; 22; 23; 24; 25], asymptotic stability [26], finite time stability (FTS) [27; 28], and input-output finite time stability (IO-FTS) [30]. If the state of a system does not exceed a prescribed threshold during a fixed finite-time interval, then the system is said to have FTS [31]. This is different from Lyapunov asymptotic stability. Obviously, a finite-time stable system may not have Lyapunov stability, and a Lyapunov stable system may not be finite-time stable. The study of FTS is useful for dealing with the behavior of a system within a finite time interval. Much work on FTS has been done [27; 28; 31; 32; 33]. Furthermore, input-to-state FTS [34], robust-FTS [35] and IO-FTS [29; 30; 36; 37; 38] have also been investigated. In particular, in [27; 28], the FTS is investigated with respect to positive switched linear systems by designing average dwell time (ADT) as the switching strategy. The ADT approach is proposed and used to investigate the stability and stabilization problems for time-dependent hybrid systems [39; 40]. Recently, a new switching strategy called the mode-dependent average dwell time (MDADT) technique was proposed [41]. It allows each mode in the underlying systems to have its own ADT, therefore, it is more applicable in practice than the ADT technique. Based on the MDADT technique, the periodic switching law was designed for periodic switching systems in order to achieve optimal switching control [42]. This technique is also used to deal with the stability and stabilization problems of positive switched systems [23]. The concept of IO-FTS was proposed for a linear time-varying system in [37]. Roughly speaking, a system is said to have IO-FTS if, given a class of norm bounded input signals over a specified time interval \([t_{0},T]\), the outputs of the system do not exceed an assigned threshold during \([t_{0},T]\). The author of [37] points out that IO-FTS is dependent on IO stability, because it involves signals defined over a finite-time interval, does not require the inputs and outputs to belong to the same class, and quantitative bounds on both inputs and outputs must be specified. Based on [37], some research results were derived.