**2. Floquet-based procedure in PWM switched converters**

We assume that a PWM switched converter can be modelled as a piecewise-linear dynamical system, as it is written in equation (1). **x** is the state (*nx*1)-vector, **A** is the state (*nxn*)-matrix, **B** is the input (*nx*1)-vector, **C** is the output (1*xn*)-vector, *y* is the scalar output and *uPWM* is the control signal.

$$\begin{cases} \dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}u\_{PWM} \\ y = \mathbf{C}\mathbf{x} \end{cases} \tag{1}$$

Replacing the perturbed solution in equation (3) and neglecting the periodic solution, the

**A***T <sup>d</sup>*<sup>∗</sup>

(a) (b)

The relevant roles of *Floquet exponents* on analysis, design and control of PWM switched converters are analyzed in a synchronous buck converter. Its main feature is that the output value *Vo* is lower than the source *E* (step down converter). Figure 1 (*a*) shows a scheme of

The mathematical model for the synchronous buck converter can be written in compact form

For the *ZAD condition*, a piecewise-linear function is defined as equation (7). Figure 1 (*b*)

*<sup>s</sup>*<sup>1</sup> + (*<sup>t</sup>* <sup>−</sup> *kT*)*s*˙1 if *kT* <sup>≤</sup> *<sup>t</sup>* <sup>≤</sup> *kT* <sup>+</sup> *dk*

<sup>2</sup> )*s*˙2 if *kT* <sup>+</sup> *dk*

<sup>2</sup> )*s*˙1 if *kT* + (*<sup>T</sup>* <sup>−</sup> *dk*

� � *x*<sup>1</sup> *x*2 � + � 0 *E L* �

2

<sup>2</sup> <sup>&</sup>lt; *<sup>t</sup>* <sup>&</sup>lt; *kT* + (*<sup>T</sup>* <sup>−</sup> *dk*

<sup>2</sup> ) ≤ *t* ≤ (*k* + 1)*T*

*uPWM* (6)

2 )

(7)

**Figure 1.** (*a*). Scheme of a PWM-controlled Buck converter with ZAD-strategy. (*b*). Piecewise-linear

*T* and **H** is a (*nxn*)-matrix which depends on the time differential related to the

*<sup>s</sup>*2) − Δ*ucle*

<sup>2</sup> − Δ*ulce*

−*μ*(*t*−*t* ∗ *s*1) *δ* (*t* − *t* ∗ *s*1) �

**A***T* � <sup>1</sup><sup>−</sup> *<sup>d</sup>*<sup>∗</sup> 2 �� **H** �

∗ *<sup>s</sup>*<sup>1</sup> = � *τ* + *<sup>d</sup>*<sup>∗</sup> 2 � *T*, *t* ∗ *<sup>s</sup>*<sup>2</sup> =

Floquet Exponents and Bifurcations in Switched Converters 29

**Hp**(0) (4)

**p**(0) (5)

variational equation of the system is obtained by equation (4) where *t*

perturbation of the PWM signal. Matrix **H** depends on the control strategy.

−*μ*(*t*−*t* ∗ *s*2) *δ* (*t* − *t* ∗

The solution of the variational equation is stated in (5) where *z* = *e*−*μT*.

� Δ*urce*

*ze***A***<sup>T</sup>* + *zτ*+<sup>1</sup>

**3. Floquet exponents in a synchronous buck converter**

where *x*<sup>1</sup> = *vC*, *x*<sup>2</sup> = *iL* and *uPWM* belongs to the discrete set {−1, 1}.

*<sup>s</sup>*<sup>2</sup> + (*<sup>t</sup>* <sup>−</sup> *kT* <sup>+</sup> *dk*

*<sup>s</sup>*<sup>3</sup> + (*<sup>t</sup>* <sup>−</sup> *kT* <sup>+</sup> *<sup>T</sup>* <sup>+</sup> *dk*

�

as:

*<sup>τ</sup>* <sup>+</sup> <sup>1</sup> <sup>−</sup> *<sup>d</sup>*<sup>∗</sup> 2 �

**p**˙ = (**A** − *μ***I**)**p** +

**p** (1) =

error dynamic (*spwl*) in a sampling period.

buck converter controlled with ZAD strategy.

shows a scheme of *spwl* in a period sampling.

⎧ ⎪⎨

⎪⎩

*spwl*(*t*) =

� *x*˙1 *x*˙2 � = � −1 *RC* 1 *C* −1 *<sup>L</sup>* <sup>−</sup>*rL L*

� Δ*urce*

�

The PWM signal is a function of *dk* and the duty cycle is a function of the delayed state variables (*dk* = *f*(**x**(*k* − *τ*))). We consider the centered scheme given by equation (2).

$$u\_{PWM}(t) = \begin{cases} u\_l & \text{if } \ k \le t \le k + d\_k/2\\ u\_c & \text{if } \ k + d\_k/2 < t < k + 1 - d\_k/2\\ u\_r & \text{if } \ k + 1 - d\_k/2 \le t \le k + 1 \end{cases} \tag{2}$$

First, we define the PWM Switched System including the discontinuity by using a unit step function (*θ*), as in equation (3) where *<sup>t</sup>* <sup>∈</sup> {0,(*<sup>τ</sup>* <sup>+</sup> <sup>1</sup>) *<sup>T</sup>*}, *ts*<sup>1</sup> <sup>=</sup> � *τ* + *d*<sup>0</sup> � 2 � *<sup>T</sup>*, *ts*<sup>2</sup> <sup>=</sup> � *τ* + 1 − *d*<sup>0</sup> � 2 � *T*, Δ*ucl* = *uc* − *ul* and Δ*urc* = *ur* − *uc*.

$$\dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \boldsymbol{\mu}\_l \mathbf{B} + \Delta \boldsymbol{\mu}\_{cl} \theta \left(t - t\_{\rm s1}\right) \mathbf{B} + \Delta \boldsymbol{\mu}\_{rc} \theta \left(t - t\_{\rm s2}\right) \mathbf{B} \tag{3}$$

According to Floquet theory we define, for the perturbed solution,

$$\mathbf{x} = \mathbf{x}^\* + e^{\mu T} \mathbf{p}(t)\_{\prime\prime}$$

where

*μ* ∈ *C* is the so-called Floquet exponent and **p**(*t* + *T*) = **p**(*t*) is an associated T-periodic function.

Replacing the perturbed solution in equation (3) and neglecting the periodic solution, the variational equation of the system is obtained by equation (4) where *t* ∗ *<sup>s</sup>*<sup>1</sup> = � *τ* + *<sup>d</sup>*<sup>∗</sup> 2 � *T*, *t* ∗ *<sup>s</sup>*<sup>2</sup> = � *<sup>τ</sup>* <sup>+</sup> <sup>1</sup> <sup>−</sup> *<sup>d</sup>*<sup>∗</sup> 2 � *T* and **H** is a (*nxn*)-matrix which depends on the time differential related to the perturbation of the PWM signal. Matrix **H** depends on the control strategy.

$$\dot{\mathbf{p}} = (\mathbf{A} - \mu \mathbf{I})\mathbf{p} + \left(\Delta u\_{\ell \mathcal{I}} e^{-\mu \left(t - t\_{s2}^\*\right)} \delta \left(t - t\_{s2}^\*\right) - \Delta u\_{\ell \mathcal{I}} e^{-\mu \left(t - t\_{s1}^\*\right)} \delta \left(t - t\_{s1}^\*\right)\right) \mathbf{H} \mathbf{p}(0) \tag{4}$$

The solution of the variational equation is stated in (5) where *z* = *e*−*μT*.

2 Nonlinearity, Bifurcation and Chaos - Theory and Applications

Stability, bifurcations and transient response of switched converters with *Delayed PWM controllers* can be studied more efficiently using an analysis of disturbances based on *Floquet theory*. We show that this procedure can be generalized to compute Floquet exponents for any number of delays (*τ*) in the control law (*dk*, so-called *duty cycle*). We compare this approach with other methods which determine stability in switched converters. One of them is the computation of characteristic multipliers based on the jacobian matrix. Another one is the computation of Lyapunov exponents using a numeric routine. Each method gives equivalent information. However Floquet approach is the most appropriated when delays appear since this method does not require the evaluation of the jacobian matrix (its dimension increases when the number of delays is higher). The other two methods have this disadvantage [13]. The chapter is organized as follows. In *Section 2* we present the general procedure to compute Floquet exponents in PWM switched converters. The particular case of a buck converter controlled with digital-PWM controller based on ZAD, FPIC and DELAY schemes, is presented in *Section 3*. The stability of fixed points in delayed PWM switched converters is discussed in *Section 4*, while fold, flip and Neimark-Sacker bifurcations are presented in

*Section 5*. Finally, the conclusions and future work are presented in *Section 6*.

We assume that a PWM switched converter can be modelled as a piecewise-linear dynamical system, as it is written in equation (1). **x** is the state (*nx*1)-vector, **A** is the state (*nxn*)-matrix, **B** is the input (*nx*1)-vector, **C** is the output (1*xn*)-vector, *y* is the scalar output and *uPWM* is

� **x**˙ = **Ax** + **B***uPWM*

The PWM signal is a function of *dk* and the duty cycle is a function of the delayed state

*ul* if *k* ≤ *t* ≤ *k* + *dk*

First, we define the PWM Switched System including the discontinuity by using a unit

*<sup>T</sup>*, *ts*<sup>2</sup> <sup>=</sup> �

**x** = **x**<sup>∗</sup> + *eμT***p**(*t*),

*μ* ∈ *C* is the so-called Floquet exponent and **p**(*t* + *T*) = **p**(*t*) is an associated T-periodic

�

�

variables (*dk* = *f*(**x**(*k* − *τ*))). We consider the centered scheme given by equation (2).

*uc* if *k* + *dk*

*ur* if *k* + 1 − *dk*

*<sup>y</sup>* <sup>=</sup> **Cx** (1)

� 2

> *τ* + *d*<sup>0</sup> � 2 �

(2)

� 2

**x**˙ = **Ax** + *ul***B** + Δ*uclθ* (*t* − *ts*1) **B** + Δ*urcθ* (*t* − *ts*2) **B** (3)

2 < *t* < *k* + 1 − *dk*

2 ≤ *t* ≤ *k* + 1

**2. Floquet-based procedure in PWM switched converters**

⎧ ⎨ ⎩

step function (*θ*), as in equation (3) where *<sup>t</sup>* <sup>∈</sup> {0,(*<sup>τ</sup>* <sup>+</sup> <sup>1</sup>) *<sup>T</sup>*}, *ts*<sup>1</sup> <sup>=</sup> �

*T*, Δ*ucl* = *uc* − *ul* and Δ*urc* = *ur* − *uc*.

According to Floquet theory we define, for the perturbed solution,

*uPWM* (*t*) =

the control signal.

*τ* + 1 − *d*<sup>0</sup>

where

function.

� 2 �

$$\mathbf{p}\left(1\right) = \left(z\mathbf{e}^{\mathbf{A}\mathbf{T}} + z^{\tau+1}\left(\Delta u\_{l\tau}\mathbf{e}^{\mathbf{A}\mathbf{T}\frac{\mathbf{e}^{\mathbf{s}}}{2}} - \Delta u\_{l\mathbf{c}}\mathbf{e}^{\mathbf{A}\mathbf{T}\left(1 - \frac{\mathbf{e}^{\mathbf{s}}}{2}\right)}\right)\mathbf{H}\right)\mathbf{p}(0) \tag{5}$$

**Figure 1.** (*a*). Scheme of a PWM-controlled Buck converter with ZAD-strategy. (*b*). Piecewise-linear error dynamic (*spwl*) in a sampling period.
