**2. CMI Model of the Boost Converter**

Consider the Boost converter schematic of Figure 1. Let us begin by introducing the well-known simplified mathematical model of the Boost converter with a CPL, obtained by an averaging technique in CCM, with infinite switching frequency as well as ideal components [1]:

$$L\frac{di}{dt} = -(1 - \mu\_1)v + E\tag{1}$$

$$\mathcal{C}\frac{dv}{dt} = (1 - u\_1)i - \frac{P}{v} \tag{2}$$

where *L*, *C*, and *P* are the inductance, capacitance, and CPL power demand, respectively; *v* is the averaged output voltage, *i* is the averaged current flowing through the inductor, *R* = *v* <sup>2</sup>/*P* where *P* is the output power, and *u*<sup>1</sup> is the duty cycle of the PWM signal in *Q*. Note that *v* = 0 implies an indeterminate form in Equation (2), characteristic for the Boost converter with a CPL. Since protective circuits can avoid a high output current, it is reasonable to consider the output voltage and current, both greater than zero, for analysis purposes.

**Figure 1.** Basic Boost converter schematic with a variable resistive load whose value is modified to achieve a constant output power (CPL).

On the other hand, using *R* = *v* <sup>2</sup>/*P* in the well-known conduction-mode-inequality for the Boost converter [36], DCM occurs if:

$$k\_{crit}(v) = \frac{2Lfp}{v^2} < u\_1(1 - u\_1)^2\tag{3}$$

where *f* is the operating frequency of the PWM. In principle, the Boost converter cannot be readily designed to operate permanently in CCM or DCM with a CPL. If *P* varies to smaller values (low load), the converter could operate in DCM; conversely, a DCM to CCM change can occur if the CPL level has a high value. Recall that at this point, *v* and *i* both represent averaged values.

In an ideal resistive load scenario, a designer tries to ensure that the range for *u*<sup>1</sup> is as wide as possible to ensure either DCM or CCM (selecting the components to achieve *kcrit*,*<sup>R</sup>* = 2 *f L*/*R* < *u*(*u* − 1) <sup>2</sup> or *<sup>k</sup>crit*,*<sup>R</sup>* <sup>=</sup> <sup>2</sup> *f L*/*<sup>R</sup>* <sup>&</sup>gt; *<sup>u</sup>*(*<sup>u</sup>* <sup>−</sup> <sup>1</sup>) 2 , respectively). Figure 2 illustrates the conduction modes as a function of *u*<sup>1</sup> and *kcrit*,*R*. It is easy to notice that even in such a scenario, it is impossible to ensure a single conduction mode if the load is not a variable to be altered/bounded at will. Furthermore, the components' degradation or aging can alter *kcrit*,*R*, inducing a conduction mode change; hence, the dynamical model of Equations (1) and (2) obtained for CCM operation would not be appropriate for designing a controller. Indeed, many authors have demonstrated the bifurcation phenomena during the CCM to DCM change [20–22].

**Figure 2.** DCM/CCM dependence on *u* and *kcrit*,*R*.

Therefore, to stabilize/regulate the Boost converter with a CPL in any conduction mode, description (1) and (2) is not adequate. In the following, a complementary averaged model for DCM is used to obtain later CMI models for the Boost converter with/without a CPL. As far as the authors know, there are no CMI models such as the one described below.

For a DCM operation of the Boost converter, one can idealize *Q* as two switches *u*¯1, *u*¯<sup>2</sup> as described in Figure 3; *u*¯<sup>2</sup> = 0 stands for the zero current through the inductor (*L*). Reproducing the averaging methodology in [1] to obtain a DCM model, the three operating modes depicted in Figure 4 are possible. Mathematical expressions for each operating mode are presented in Table 2; note that *X* means "do not care" because *u*¯<sup>2</sup> = 0 nullifies the current flow through *L*, and hence *u*¯<sup>1</sup> has no effect on the equations. Note also that *v*ˆ and ˆ*i* are used to differentiate them from the averaged values *v* and *i*.

**Figure 3.** Idealization of the Boost converter operating in DCM through two ideal switches.

**Table 2.** Current flow modes of the inductor.


**Figure 4.** Illustration of the inductor current flow modes: (**a**) Charging, (**b**) discharging, and (**c**) holding.

An elementary analysis allows obtaining an averaged model of the charging, discharging, and holding operating modes as:

$$L\frac{di}{dt} = -(1 - u\_1)v + E(1 + 2u\_1u\_2) \tag{4}$$

$$\mathcal{C}\frac{dv}{dt} = (1 - u\_1)i - \frac{P}{v} \tag{5}$$

where *u*<sup>1</sup> ∈ [0, 1] is the duty cycle (percentage of *T* in charging mode with *T* = 1/ *f* as the PWM period), and *u*<sup>2</sup> is the percentage of *T* in holding mode (zero inductor current); these periods should not be confused with *u*¯1, *u*¯<sup>2</sup> since these last only represent the switches in Figure 3. See Table 2 and Figure 5 for an illustration; *u<sup>d</sup>* is used to describe the period's percentage in discharge mode. Note that *u*<sup>2</sup> = 0 for CCM and *u*<sup>2</sup> > 0 for DCM; hence, there is no ambiguity on using *u*1, *v*, and *i* from the CCM model of Equations (1) and (2), for this model also.

**Figure 5.** Exemplification of the inductor current behavior in DCM concerning the triggering on *Q*. PWM stands for the voltage signal in the gate pin.

From inductor and diode current waveforms in steady-state, it is well known that the load current is [36]:

$$\frac{v}{R} = \frac{P}{v} = \frac{Eu\_1 u\_d}{2fL}.\tag{6}$$

Furthermore, a time balance over the inductor current can be represented as (see Figure 5):

$$T = \mu\_1 T + \mu\_d T + \mu\_2 T \tag{7}$$

such that:

$$
\mu\_2 = 1 - \mu\_1 - \mu\_d = 1 - \mu\_1 - \frac{2PLf}{Evu\_1}.\tag{8}
$$

Then, it is easy to see that:

$$
\mu\_2 = 1 - \mu\_1 - \frac{2PLf}{Evu\_1} \tag{9}
$$

is a *C* 1 -diffeomorphism (for the inverse, it is enough to take only the positive root to have a bijection; this inverse is differentiable) for Equations (4) and (5) with *v* > 0 and 1 ≥ *u*<sup>1</sup> > 0 (*u*<sup>1</sup> = 0 is a trivial case), allowing to eliminate the dependency on *u*<sup>2</sup> [37]:

$$L\frac{di}{dt} = -(1 - u\_1)v - \frac{4PLf}{v} + (1 + 2u\_1 - 2u\_1^2)E\tag{10}$$

$$\mathcal{C}\frac{dv}{dt} = (1 - u\_1)i - \frac{P}{v}.\tag{11}$$

The following section provides validations showing that the result of this transformation is quantitatively favorable in terms of the mean squared error (MSE).

By comparison of Equations (10) and (11) with Equations (1) and (2) and from inequality (3), one can design/approximate (recall that the CCM and DCM models were obtained using an infinite frequency consideration, while the change between conduction modes (3) is obtained from a steady-state analysis. Since a common Lyapunov function is used to design a controller and demonstrate stability even with arbitrary switching, it is not relevant to obtain a precise design of the switching signal; these following switching signals are presented for completeness purposes) a switching signal as *γ*(*u*1, *v*) ∈ {0, 1}:

$$\gamma(u\_1, v) \triangleq 0.5 \left( 1 + \text{sign} \left( \left( u\_1 (1 - u\_1)^2 - \frac{2LfP}{v^2} \right) \right) \right) \tag{12}$$

where:

$$\text{sign}(\mathbf{x}) = \begin{cases} -1, & \mathbf{x} < \mathbf{0} \\ \mathbf{0}, & \mathbf{x} = \mathbf{0} \\ 1, & \mathbf{x} > \mathbf{0} \end{cases}. \tag{13}$$

That is, *γ*(*u*1, *v*) = 1 for DCM, and *γ*(*u*1, *v*) = 0 for CCM; the switched model is then:

$$L\frac{di}{dt} = -(1 - u\_1)v - \frac{4PLf}{v}\gamma + (1 + 2\gamma u\_1 - 2\gamma u\_1^2)E\tag{14}$$

$$\mathcal{C}\frac{dv}{dt} = (1 - u\_1)\mathbf{i} - \frac{P}{v}\mathbf{.}\tag{15}$$

Alternatively, one can approximate the switching signal with a *C* <sup>∞</sup> function <sup>0</sup> <sup>≤</sup> *<sup>ρ</sup>* <sup>≤</sup> <sup>1</sup> to obtain a continuous (non-switched) CMI model of the Boost converter:

$$\rho(u\_1, v) = 0.5 \left( 1 + \tanh\left( a \left( u\_1 (1 - u\_1)^2 - \frac{2LfP}{v^2} \right) \right) \right) \tag{16}$$

with *a* 1, and the switched model is:

$$L\frac{di}{dt} = -(1 - u\_1)v - \frac{4PLf}{v}\rho + (1 + 2\rho u\_1 - 2\rho u\_1^2)E\tag{17}$$

$$\mathcal{C}\frac{dv}{dt} = (1 - u\_1)i - \frac{P}{v}.\tag{18}$$

Note that for *γ* = 0 and *ρ* = 0, the operating point congruently coincides with that of the classic CCM model.

For the non-CPL case (resistive load), the switched/CMI model can be obtained by a similar procedure:

$$L\frac{di}{dt} = -\left(1 - u\_1 + \frac{4Lf}{R}\sigma\right)v + (1 + 2\sigma u\_1 - 2\sigma u\_1^2)E\tag{19}$$

$$\mathcal{C}\frac{dv}{dt} = (1 - u\_1)i - \frac{v}{R} \tag{20}$$

where:

$$\sigma(u\_1) \stackrel{\Delta}{=} 0.5 \left( 1 + \text{sign} \left( \left( u\_1 (1 - u\_1)^2 - \frac{2Lf}{R} \right) \right) \right),\tag{21}$$

or alternatively:

$$L\frac{di}{dt} = -\left(1 - u\_1 + \frac{4Lf}{R}\varphi\right)v + (1 + 2\varphi u\_1 - 2\varphi u\_1^2)E\tag{22}$$

$$\mathcal{C}\frac{dv}{dt} = (1 - u\_1)i - \frac{v}{R} \tag{23}$$

with,

$$\varphi(u\_1) = 0.5 \left( 1 + \tanh\left( a \left( u\_1 (1 - u\_1)^2 - \frac{2Lf}{R} \right) \right) \right). \tag{24}$$
