2.4.2. Stability Analysis Using Floquet Theory

The differential equations describing the dynamics of switching converters are timeperiodic with the switching period T determining the periodicity of solutions at the fast switching scale. Floquet theory has been widely used in the analysis of the stability of dynamical systems [30] in general and in switching converters in particular [11,31,32]. For DC-DC converters, the stability dynamics at the fast switching cycle can be accurately predicted by analyzing the stability of the fixed points of the Poincare map of the system using its Jacobian matrix or using Floquet theory combined with the Filippov method which leads to the same results as the Poincare map [33]. The main tool for studying the stability of periodic orbits using Floquet theory is the principal fundamental matrix or the monodromy matrix M. This matrix plays a key role in the accurate stability analysis of switching systems [34–36]. The dynamics in the vicinity of a quasi-static periodic orbit can be expressed in the monodromy matrix as follows:

$$
\mathfrak{X}(t+T) = M\mathfrak{X}(t) \quad \forall t,\tag{20}
$$

where the overhat stands for small signal variations. Its eigenvalues are called the characteristic multipliers or Floquet multipliers and it can be seen that they determine the amount of contraction or expansion near a periodic orbit and hence they determine the stability of these periodic orbits. It can be obtained by computing the state transition matrices before and after each switch and the saltation matrix that describes the behaviors of the solution 7 switching [24] which are described in the following.

Let *X*(*DT*) = (*I* − Φ) −1 Ψ be the steady-state value of X at time instant *DT*, where Φ = Φ1Φ0, Φ<sup>1</sup> = *eA*1*DT*, Φ<sup>0</sup> = *eA*0(1−*D*)*T*,

$$\Psi = e^{A\_1 D T} A\_0^{-1} \left( e^{A\_0 (1 - D) T} - I \right) B\_0 W + A\_1^{-1} \left( e^{A\_1 D T} - I \right) B\_1 W. \tag{21}$$

The monodromy matrix use [11] can be expressed as follows *P* = Φ<sup>0</sup> *S* Φ<sup>1</sup>

$$P = \Phi\_0 \, \mathcal{S} \, \Phi\_{1\prime} \tag{22}$$

where S is saltation matrix and is given by:

$$S = I + \frac{\left(A\_0 X(DT) - A\_1 X(DT)\right) K^T}{K^T \left(A\_1 X(DT) + B\_1 W\right) + m\_a}.\tag{23}$$

Equations (22) and (23) will allow us to determine the Floquet multipliers. According to Floquet's theory, if only one multiplier has a value equal to −1, then the dynamics of the system presents period doubling bifurcation. In general, the occurrence of the critical point of this bifurcation has two regions: a region of subharmonic oscillation and a region of periodic oscillation of period-1. The second region described above is the stability region in power systems [24].
