2.4.1. The Piecewise Linear State-Space Switched Model Close to the Maximum Power Point

The periodic equation to be considered for our purposes will be expressed in the state-space representation (or matrix representation). This form lends itself well to the calculation of the solution of a differential equation system. The dynamics will be expressed as a function of a vector of variables, which we will call the state vector. Assuming that our subsystem is linear and time-invariant, the evolution of each subsystem is defined by:

$$\begin{aligned} \dot{\mathfrak{x}} &= A\_1 \mathfrak{x} + B\_1 W \text{ if } \mathfrak{u} = 1, \\ \dot{\mathfrak{x}} &= A\_0 \mathfrak{x} + B\_0 W \text{ if } \mathfrak{u} = 0. \end{aligned} \tag{12}$$

where *x* = (*x*1*x*2*x*3*x*4) *<sup>T</sup>*, *A*1, *B*1, *A*0, *B*<sup>0</sup> are the state and input matrices corresponding to the different switch states and the external input parameters vector are given by:

$$A\_1 = \begin{pmatrix} -\frac{1}{\mathcal{R}\_{pN}\mathcal{C}\_1} & -\frac{1}{\mathcal{C}\_1} & 0 & 0\\ \frac{1}{L} & -\frac{1}{L}r & 0 & 0\\ 0 & 0 & -\frac{1}{\mathcal{R}\mathcal{C}\_2} & 0\\ 0 & w\_ck\_p & 0 & -w\_c \end{pmatrix} \tag{13}$$

$$A\_0 = \begin{pmatrix} 1 & 0 & 0 \\ -\frac{1}{\mathcal{R}\_{pN}\mathbb{C}\_1} & -\frac{1}{\mathcal{C}\_1} & 0 & 0 \\ \frac{1}{L} & -\frac{1}{L}r & -\frac{1}{L} & 0 \\ 0 & \frac{1}{\mathcal{C}\_2} & -\frac{1}{\mathcal{R}\mathcal{C}\_2} & 0 \\ 0 & w\_c k\_p & 0 & -w\_c \end{pmatrix} \tag{14}$$

$$B\_1 = \begin{pmatrix} \frac{2}{C\_1} & 0\\ 0 & 0\\ 0 & \frac{1}{\overline{\mathcal{R}C\_2}}\\ 0 & 0 \end{pmatrix} \tag{15}$$

$$B\_0 = \begin{pmatrix} \frac{2}{\overline{\mathbb{C}\_1}} & 0 \\ 0 & 0 \\ 0 & \frac{1}{\overline{\mathbb{R} \mathbb{C}\_2}} \\ 0 & 0 \end{pmatrix}' \tag{16}$$

$$\mathcal{W} = \begin{pmatrix} i\_{mpp} \\ E \end{pmatrix}. \tag{17}$$

The switching condition in Figure 2 is given by:

$$h(\mathbf{x}, t) = R\_s \left( i\_{ref} - \mathbf{K} \mathbf{x} \right) - \frac{V\_M}{T} t\_\prime \tag{18}$$

with *VM <sup>T</sup> t* = *Vramp*, the vector K is given by:

$$K = \begin{pmatrix} 0 \\ 0 \\ 0 \\ -1 \end{pmatrix}^T. \tag{19}$$
