**3. Proposed Backstepping Control**

The principle of the backstepping controller is to summarize a control law in an iterative way. Some components of the state representation would be examined as "virtual controls" and intermediate control laws will be prepared [20]. It holds the conception of stability in the sense of Lyapunov, in order to ensure that a certain Lyapunov function, is positive, and that its derivative is always negative. The method allows the system to be divided into a set of nested subsystems of decreasing order. At each step, the order of the system is increased and the treatment of the non-stable part of the previous step is carried out, until the appearance of the control law which is the last step. This consists in guaranteeing, at all times, the overall stability of the system [21–25]. We will apply this control technique to control the whole hybrid filter. The equations of the system, in the stationary reference frame is given by:

$$\begin{split} \frac{d\dot{i}\_{f\alpha}}{dt} &= -\frac{\mathbb{R}}{\mathbb{L}} \dot{i}\_{f\alpha} - \frac{V\_c}{L} + \frac{1}{\mathbb{L}} \boldsymbol{\upsilon}\_{f\alpha}^\* - \frac{1}{\mathbb{L}} \boldsymbol{\upsilon}\_{c\text{ch}\alpha} \\ \frac{d\dot{i}\_{f\beta}}{dt} &= -\frac{\mathbb{R}}{\mathbb{L}} \dot{i}\_{f\beta} - \frac{V\_c}{L} + \frac{1}{\mathbb{L}} \boldsymbol{\upsilon}\_{f\beta}^\* - \frac{1}{\mathbb{L}} \boldsymbol{\upsilon}\_{c\text{ch}\beta} \\ \frac{d\boldsymbol{\upsilon}\_{dc}}{dt} &= \frac{1}{\mathbb{C}\_{dc}} \dot{i}\_{dc} = -\frac{\mathbb{R}\_{dc}}{\mathbb{C}\_{dc} \boldsymbol{\upsilon}\_{dc}} \end{split} \tag{13}$$

*R* and *L* is the internal resistance of the inductance and the inductance of the passive filter, respectively, and *Vc* is the voltage across the capacitance *C* of the passive filter.

The system can be divided into subsystems, the first two equations of the system (13) are used for current regulation *if*<sup>α</sup>, *if*β where voltages *<sup>v</sup>*\**f*<sup>α</sup>, *<sup>v</sup>*\**f*β are considered as control variables.
