# Robust Stabilization of Linear Switched Systems with Unstable Subsystems
## 1 Introduction  
In recent years, the interest in Switched Linear Systems (SLS) has increased because of its capability to represent complex nonlinear systems in a more tractable math form, and their analysis has spread out as a new branch of stability and control especially for SLS with one or more unstable subsystems, while some of the Lyapunov and other theories can be applied to those with all stable subsystems [1; 2]. State-dependent and time-dependent switching signals are the main approaches to design stabilizing switching laws. In the former, the whole state space is usually divided to facilitate the search for Lyapunov-like functions; unfortunately, the system's states must be measurable or observable. In the latter, time-constrained switching is used, wherein a stable subsystem is activated for enough time to stabilize the entire system [3; 4; 5; 6]. The stability of SLS with one or more unstable subsystems is now well established from the stabilizing switching signal point of view [7; 8; 9; 10; 11; 12; 13]. However, studies on time-dependent switching stabilization with Mode-Dependent Average Dwell Time (MDADT) dedicated to the robust stability of time-varying SLS have remained unfinished. The authors in [14] aimed at the robust stability of a discrete, positive, switched system, with bounded control inputs and with stable and unstable subsystems; the examples presented showed that a stabilizing MDADT signal can be easily designed. In [15] was presented a switching stabilization of SLS composed of both stable and unstable subsystems, easily extendable to all stable or unstable subsystems. Although the study had typographic errors, the authors presented important results on stability and the switching MDADT signal design. In [16], a parameter-dependent MDADTswitching scheme related to a set of parameter-dependent Lyapunov functions was proposed in order to control a class of switching LPVsystems; the proposed approach was applied to satisfy the overall control objectives related to a variable-sweep, wing-morphing aircraft. A full-envelope flight controller using switched linear modeling based on MDADT, with Locally-Overlapped Subsystems (LOSS), was proposed for a full-envelope flight in [17]. A stability proof by using a common Lyapunov function for each LOSS was also reported. It is worth mentioning that each active subsystem can only switch to another adjacent LOSS subsystem, and this particularity relaxed the controller, stability, and control problem formulation. In [18], multiple co-positive Lyapunov functions and an MDADT technique were combined to derive sufficient conditions for the input-output finite time stability problem. Additionally, a controller was derived, and numerical examples were also provided to show the feasibility of the proposed technique. A quasi-time dependent H-\(\infty\) robust controller for a switched system with MDADT was proposed in [19]; a Lyapunov function was used to prove theoretical stability. Numerical results in an SLS and practical results in a power electronic, boost converter were reported. Unfortunately, the switching signal to control both systems had a variable frequency, which in the case of the boost converter produced noise and EMI degradation. The stability analysis of time-varying impulsive positive hybrid systems with time-varying, distributed delays, and all unstable subsystems, by using an MDADT, was reported in [20]. Additionally, a concept of input-output finite time stability was presented. Finally, their numerical simulations showed the feasibility of the proposed method. It is worth mentioning the important contribution in the topic of adaptation and robustness of SLS in [21]. This work was focused on four main areas: adaptive tracking using extended and average dwell times, adaptive asymptotic tracking, robust adaptive tracking, adaptive stabilization with time-varying delays, and robust stability and stabilization with switching delays.