## 1 Introduction  
Even if the control of power converters is a topic which soon interested the control community [1; 2; 3], recent years have seen a renewed interest due to the increasing importance of renewable energies, which are an alternative to the fossil ones, leading to a compelling need of processing power energy provided by multiple available sources with an adequate efficiency level [4; 5; 6]. In that context, power converters play a central role and, motivated by the significant recent progresses done in the domain of materials, electronic devices or components [7; 8], as well as in the one of control of switched or more generally hybrid systems [9; 10], several new perspectives exist to easily address, at least from a theoretical point of view, problems whose solutions are difficult to obtain by the use of standard techniques more or less well suited. Indeed, power converters are switched systems that can be handled in several ways, but the discontinuous nature of their operation is a source of complexity which is not easy to capture by the standard mathematical tools. A general model describing a large class of power converters is a bilinear differential model whose state, composed of currents and voltages, belongs to a finite dimensional vector space and the control variables associated with the switches devices belong to a finite discrete set [1]. In many practical cases, for example for DC-DC converters, the objective is to maintain the system around an equilibrium state. In such a situation, a possible approach consists in deducing an averaged model which, although bilinear, exhibits constrained continuous control variables allowing the use of all the methods developed for nonlinear systems (feedback linearization, sliding mode, flatness, and passivity (see, e.g., [11] and references therein)). To go a step further in the simplification of the control design problem, it is possible to linearize the averaged bilinear model around a desired operation state and invoke the powerful robust linear control design techniques (see, e.g., [12; 13; 14]). Among the main limitations of these approaches, we can point out a difficulty to quantify the precision of the resulting approximations and the fact that the deduced control laws are, in general, only valid locally. The control design problem is more complicated for the DC-AC converters where the objective is to enforce the output of the converter to follow a given reference, for example a sinusoidal signal whose frequency and amplitude are a priori prescribed. In that case, the associated control design problem can be formulated as a tracking problem [2] for which a solution with qualified properties, e.g. stability and performances, is not so easy to determine. Recently, some efforts have been made to propose efficient control design techniques for switched systems, a subclass of hybrid systems whose control is a switching rule that selects, at each instant of time, a mode within a set of possible modes, each of them representing a dynamical subsystem [9; 10]. These techniques, well adapted for taking into account the explicit discontinuous nature of the dynamics of power converters, have been successfully applied to the control of DC-DC converters. Among the numerous works, we can cite [15; 16; 17; 18], where the problem is formulated in terms of a switched system whose modes are described by affine differential models. It is possible to show that the obtained switching rules can be interpreted, in general, as sliding mode control laws with sliding surfaces implicitly determined in terms of state space variables and whose stability properties are proven without invoking the equivalent control paradigm [19], the application of which, in that context, may not be so easy. To constrain the resulting switching frequency to adequate values regarding semiconductor limitations, some of these results have been extended to take into account these specific and important implementation constraints [18; 20]. In [18], a sampling time is explicitly introduced.