# A Unified Approach for the Control of Power Electronics Converters. Part II: Tracking
## 1 Introduction  
For many years, the control of power electronics converters has been a challenge for the control community [1; 2; 3]. Despite the fact that there has been renewed interest motivated by the increasing importance of renewable energies [4; 5; 6], the control problems associated with energy processing have been the main focus of the efforts of researchers because they possess several specific features that introduce significant difficulties into the underlying control problems of interest. Among the more stringent ones, we can highlight the switching nature combined with nonlinearities, present in almost all problems concerning energy conversion. Another motivation is related to the significant recent progress achieved in the domain of materials, electronic devices or components [7; 8], but also in the control of switched or, more generally, hybrid systems [9; 10], which have led to several new perspectives to deal with problems whose solutions cannot be easily obtained by the standard techniques. 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 control variables associated with the switched devices belong to a finite discrete set [1]. From this original model, it is possible to derive approximated models, such as 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, passivity (see [11] and references therein)). It is also possible to linearize the averaged bilinear model around an equilibrium state and invoke the powerful robust linear control design techniques [12; 13; 14]. 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 and concern principally stabilization problems. Still in the context of stabilization problems, in [19], in which the original bilinear converter model is manipulated in such a way that allows it to exhibit a constant dynamicalmatrix (i.e., independent of the control variable), the authors show how to solve problems that cannot be easily solved when considering the original switched model. When the objective is to enforce the output of the converter to follow a given periodic reference, the problem is more involved because it is formulated as a tracking problem [2] for which a solution with qualified properties such as stability and performance can be very difficult to obtain. Due to its importance, tracking control with disturbance rejection has been extensively studied in the control literature. If the reference signals are generated by an appropriate autonomous exosystem, the problem is known as the output regulation problem [20]. For linear systems, it has been solved in [21] from the solution of a set of algebraic matrix equations. Its extension to nonlinear systems involves the solution of a set of nonlinear differential equations, the Francis-Byrnes-Isidori equations, which allow the determination of an attractive invariant steady state. A few general works have been presented in the context of switched or hybrid control design methods. We can cite [22; 23; 24]. More closely related to the problem addressed in this paper, we can also cite [25], where the problem of nonlinear sliding mode output regulation is addressed. Using the concepts connected to the zero output tracking sub-manifold, a solution is proposed for nonlinear affine control systems with time-varying disturbance including both minimum and non-minimum phase systems extending some previous results proposed in [26; 27; 28; 29] for minimum-phase systems, in [30; 31] for linear systems, or in [32] for non-minimum-phase non-linear systems with a unitary relative degree.