3 may be carried out, to calculate the nominal values of the quantities V_boost, V_buckboost, I_L, D₁, D₂and D₃, relating to the so-called working point of the switched inductive electrical network 3; in other words, the value of the input voltage V_gand the values of the first and second output voltages V_boost, V_buckboostmay be imposed, so as to obtain the nominal values of the current I_Land the durations D₁, D₂and D₃, both in case of continuous-current mode and in case of discontinuous-current mode, in which case the duration D₄is determined as previously described. Furthermore, as regards the state of the switched inductive electrical network 3, it is governed by a set of three non-linear equations, which may be obtained in a per se known manner by implementing the “state-space averaging” technique, i.e., by averaging the equations of state referring to each phase over a switching period. For example, referring for simplicity to the continuous-current mode, for each of the three phases a corresponding set of three equations of state apply, and these three sets of three equations of state are averaged over the switching period; similar considerations apply to the case of discontinuous current, wherein however four sets of three equations of state are present, since the fourth phase is also present.

In this manner, the three equations of state comprise:

$V_{boost}^{'} = {dV}_{b oost} / dt = f_{1} (V_{boost}, V_{b u c kboost}, I_{L}, D_{1}, D_{2})$

$V_{b u c kboost}^{'} = {dV}_{b u c k b oost} / dt = f_{2} (V_{boost}, V_{b u c kboost}, I_{L}, D_{1}, D_{2})$

$I_{L}^{'} = {dI}_{L} / dt = f_{3} (V_{boost}, V_{b u c kboost}, I_{L}, D_{1}, D_{2})$

where f₁, f₂and f₃represent functions that express the dependencies of the time derivatives of the first and second output voltages V_boost, V_buckboostand the current I_Lwith respect to the set of five quantities formed by the first and second output voltages V_boost, V_buckboost, by the current I_Land the durations D₁and D₂. The functions f₁, f₂and f₃vary depending on whether the continuous-current mode or the discontinuous-current mode is considered, however the following description does not vary depending on the current mode considered.

By linearizing, by a first-order Taylor approximation, the three above equations around the aforementioned working point of the switched inductive electrical network 3, the following is obtained:

$x^{'} = A \cdot x + B \cdot u$

$y = C \cdot x + D \cdot u$

where x′ expresses the time derivative of x, with x=[V_boost,V_buckboost,sI_L]^T(the superscript T indicates the transpose). Furthermore y=[V_boost,V_buckboost]^Tand u=[u₁(t),u₂(t)]^T, where u₁(t), u₂(t) represent a first and a second control signal, respectively, which are voltage analog signals that control the durations D₁and D₂, respectively, such that the durations D₁and D₂are directly proportional, respectively, to the values of the first and second control signals u₁(t), u₂(t), as explained in more detail below. Furthermore, A is a matrix of real numbers having dimensions 3×3, while B is a matrix of real numbers having dimensions 3×2. Additionally, there are the following matrices:

$C = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}]$

$D = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$

In practice, the matrices A, B, C and D define a time-invariant linear dynamic model of the switched inductive electrical network 3 of the switching converter 1. As previously mentioned, the values of the elements of the matrices A and B vary depending on whether the continuous-current mode or the discontinuous-current mode is considered, however the following description still applies. Furthermore, hereinafter reference is made to the augmented state xaug to indicate the signal vector composed of the first and second output voltages V_boost, V_buckboost, the signal sI_L(t) and the first and second additional signals int₁(t), int₂(t); for this reason, there apply the relationships:

${xaug}^{'} = A^{*} \cdot xaug + B^{*} \cdot u$

$y = C^{*} \cdot xaug + D^{*} \cdot u$

where the matrices A*, B*, C* and D* are obtained in a per se known manner from the matrices A, B, C and D. In particular, the matrix A* has dimensions 5×5 and is of the type:

$[\begin{matrix} A 11 & A 12 & A 13 & 0 & 0 \\ A 21 & A 22 & A 23 & 0 & 0 \\ A 31 & A 32 & A 33 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}]$

where Aij indicates the element on the i-th row and on the j-th column of the matrix A, while the matrix B* has dimensions 5×2 and is of the type:

$[\begin{matrix} B 11 & B 12 \\ B 21 & B 22 \\ B 31 & B 32 \\ 0 & 0 \\ 0 & 0 \end{matrix}]$

where Bij indicates the element on the i-th row and on the j-th column of the matrix B.

All this having been said, the gain stage 28 is formed, for example, by a corresponding configurable analog electrical network, which implements a so-called Linear-Quadratic Regulator (LQR).

In particular, the gain stage 28 multiplies the augmented state xaug by a state feedback matrix −K_LQR, which has dimensions 2×5, such that the outcome of the multiplication is equal to the first and second control signals u₁(t), u₂(t). In this regard, in FIG. 3 the symbols u and u′ are used to indicate the set of the first and second control signals u₁(t), u₂(t) and the set of the first and second intermediate signals u₁′(t), u₂′(t), respectively.

In practice, the gain stage 28 implements the operation u=−K_LQR·xaug. Furthermore, the state feedback matrix −K_LQRmay be set in a per se known manner, on the basis of a cost function J which depends, in addition to xaug and u, on a first and a second weight matrix Q, R, which have dimensions equal to 5×5 and 2×2, respectively, and may be chosen during the design step; in particular, the augmented state xaug represents an indicator of the performances of the switching converter 1, understood as speed or bandwidth of the control loop, while the vector u represents an indicator of the effort required to the switching converter 1, understood as the use of the inputs (in this case, the first and second control signals u₁(t), u₂(t), i.e. the durations D₁, D₂).

By choosing the values of the elements of the matrices Q and R, the speed of the control loop may be optimized, without causing saturation of the durations D₁, D₂, i.e. preventing them from extending for more than one switching period. In fact, the cost function J may be expressed as:

J
=

∫
0

+
∞

(

xaug
_

T

·
Q
·

xaug
_

+

u
_

T

·
R