# A Unified Approach for the Control of Power Electronics Converters. Part II: Tracking
## 1 Introduction
More closely related to the control of power converters, in [33], two closed-loop boost power converters are proposed for a DC-AC power conversion problem controlled using a sliding mode strategy. In [34], a cascade control scheme also based on sliding mode control is developed for the same boost inverter. One of the main difficulties is the non-minimum-phase property of such a converter, motivating the design of an indirect control law whose objective is to impose a tracking reference signal to the current inductors, the problem being then translated into the determination of an appropriate current reference signal. Several methods have been invoked for such a determination. In [35], a linearized internal dynamics is use, while in [36; 37], the reference signal is obtained, respectively, by means of a uniform sequence of [PERSON]'s approximations or from periodic solutions of [PERSON]'s ordinary differential equation of the second kind. Several papers combining the previous approaches using sliding mode control or other nonlinear control techniques exist [38; 39; 40; 41]. Recently, in [42], a solution for the half bridge inverter was proposed, and in [43], the case of the full bridge is considered using bipolar or unipolar strategies. The specificity of the aforementioned works is to formulate the control problems as tracking control problems directly for the switched models, seeking to extend the results of [21]. The specificity of the half-bridge and full-bridge inverters is that the dynamical matrix does not depend on the discrete control variable, drastically simplifying the resolution of the problem. For inverters whose dynamical matrix depends on the control variable, such as the boost inverter introduced in [33], the problem is more complicated and the method used in [42] cannot be directly extended. One of the main technical problems is to solve the set of algebraic matrix equations, which can be seen as an extension of the one in [21]. The main objective of this paper is to extend the approach proposed in [19], developed for stabilization problems, to the case of problems that can be formulated as tracking ones. As in [19], the idea is to derive formulations in a general setting and propose solutions that are justified and qualified from a theoretical point of view using efficient theoretical and numerical tools. In this sense, a crucial step consists of manipulating the original bilinear converter model in an appropriate way, leading to a model that is closely related to the switched model considered in the literature--for example, in [15] or [18]--but being better suited for solving control problems that cannot be easily solved when considering the original switched model. The approaches proposed in this paper could be easily extended in the context of the hybrid formulation paradigm proposed in [44] with the associated time and space regularization techniques. This extension is not considered and can be addressed in a future work. The paper is organized as follows. The next section recalls the main models considered in [19] and used in this work. In Section 3, the tracking problem is precisely stated and, in particular, the notion of admissible reference is defined. Section 4 develops a solution to the problem stated in Section 3, whose derivation is based on a central assumption discussed in detail. Section 5 considers the tracking problem in the presence of perturbations. Two cases are considered, the case in which the perturbations are not available for measurement and then not usable by the control law, and the case in which the perturbations are measurable using some low-cost sensors. In Section 6, the proposed methods are illustrated for two converters: the classical full-bridge inverter feeding nonlinear loads, and the boost inverter introduced in [33]. For this last case, the more involved determination of the state reference is discussed and obtained by two methods corresponding to the two means of controlling such a converter.