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

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 buck converter controlled with ZAD strategy.

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

$$
\begin{pmatrix}
\dot{\boldsymbol{x}}\_1 \\
\dot{\boldsymbol{x}}\_2
\end{pmatrix} = \begin{pmatrix}
\frac{-1}{\overline{\mathcal{K}}} & \frac{1}{\overline{\mathcal{L}}} \\
\frac{-1}{L} & \frac{-r\_L}{L}
\end{pmatrix} \begin{pmatrix}
\boldsymbol{x}\_1 \\
\boldsymbol{x}\_2
\end{pmatrix} + \begin{pmatrix}
0 \\
\frac{E}{L}
\end{pmatrix} \boldsymbol{\upmu}\_{PWM} \tag{6}
$$

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

For the *ZAD condition*, a piecewise-linear function is defined as equation (7). Figure 1 (*b*) shows a scheme of *spwl* in a period sampling.

$$s\_{pwl}(t) = \begin{cases} s\_1 + (t - kT)\dot{s}\_1 & \text{if } kT \le t \le kT + \frac{d\_k}{2} \\ s\_2 + (t - kT + \frac{d\_k}{2})\dot{s}\_2 & \text{if } kT + \frac{d\_k}{2} < t < kT + (T - \frac{d\_k}{2}) \\ s\_3 + (t - kT + T + \frac{d\_k}{2})\dot{s}\_1 & \text{if } kT + (T - \frac{d\_k}{2}) \le t \le (k + 1)T \end{cases} \tag{7}$$

#### 4 Nonlinearity, Bifurcation and Chaos - Theory and Applications 30 Nonlinearity, Bifurcation and Chaos – Theory and Applications Floquet Exponents and Bifurcations in Switched Converters <sup>5</sup>

where

$$\begin{aligned} \dot{s}\_1 &= (\dot{\mathfrak{x}}\_1 + k\_s \mathfrak{x}\_1) \Big|\_{\mathbf{x} = \mathbf{x}(kT), \mu = 1} & \dot{s}\_2 &= (\dot{\mathfrak{x}}\_1 + k\_s \mathfrak{x}\_1) \Big|\_{\mathbf{x} = \mathbf{x}(kT), \mu = 0} \\ \dot{s}\_1 &= (\mathfrak{x}\_1 - r \mathfrak{e} f + k\_s \dot{\mathfrak{x}}\_1) \Big|\_{\mathbf{x} = \mathbf{x}(kT), \mu = 1} & \dot{s}\_2 = \frac{d}{2} \dot{s}\_1 + s\_1 \ s\_3 = s\_1 + (T - d) \dot{s}\_2 \end{aligned} \tag{8}$$

and *ks* is a positive constant. Therefore, the zero average condition is

$$\int\_{kT}^{(k+1)T} s\_{pvol}(t)dt = 0\tag{9}$$

Therefore, we can find the variational equation based on the dynamical equations of the system. We use a more compact expression applying the following change of variables:

<sup>√</sup>*LC*, thus *<sup>γ</sup>* <sup>=</sup> <sup>1</sup>

� � *x*<sup>1</sup>

We use the parameter values of an experimental prototype reported in [3]. We fix *R* = 20Ω, *C* = 40*μ*F, *L*=2mH, *rL* = 0Ω, *vref* = 32V, *E*=40V and the sampling period *Tc* = 50*μs*..

For simplicity, in the remainder of *Section 3*, we note *dk* as *d* with *d* ∈ [0, *T*], and we present the procedure for one period sampling *t* ∈ [*τT*,(*τ* + 1)*T*] (general notation was used in *Section*

Figure 2 illustrates the mechanism to model the centered pulse (*uPWM*) using unit step

(a) *τ* = 0, *N* = 0 (b) *τ* = 1, *N* = 0

**Figure 3.** The evolution of the real part of the Floquet exponents when *ks* is varied in the ZAD controller.

functions *θ*(*t*). The duty cycle depend on state variables in the instant *t* = (*k* − *τ*)*T*. Equation

1

�

Equation (15) implies that duty cycle can be defined in function of state variables in an initial instant (*t* = 0) because the delay information was included in the unit step functions.

− 2*θ* (*t* − *ts*1)

� 0 1 �

+ 2*θ* (*t* − *ts*2)

� 0 1 �

(15)

*x*2

1 *τT* < *t ts*<sup>1</sup> −1 *ts*<sup>1</sup> < *t ts*<sup>2</sup> 1 *ts*<sup>2</sup> < *t* (*τ* + 1)*T*

� + � 0

*R* � *<sup>L</sup>*

1

�

*<sup>C</sup>* , *β* = *rL*

�*<sup>C</sup>*

Floquet Exponents and Bifurcations in Switched Converters 31

� *τT* + *<sup>d</sup>* 2 �

*<sup>L</sup>* and the sampling

and *ts*<sup>2</sup> =

(14)

*uPWM* (13)

*x*<sup>1</sup> = *vC*/*E*, *x*<sup>2</sup> = <sup>1</sup>

period is *T* = *Tc*/

�

(*<sup>τ</sup>* <sup>+</sup> <sup>1</sup>)*<sup>T</sup>* <sup>−</sup> *<sup>d</sup>*

2 � .

� . *x*1 . *x*2

� = � <sup>−</sup>*<sup>γ</sup>* <sup>1</sup>

−1 −*β*

*E* � *<sup>L</sup>*

<sup>√</sup>*LC* [10].

� . *x*1 . *x*2

*<sup>C</sup> iL* and *t* = *τ*/

� = � <sup>−</sup>*<sup>γ</sup>* <sup>1</sup>

Therefore, the dimensionless parameters are *γ* = 0.35, *β* = 0 and *T* = 0.1767.

⎧ ⎪⎪⎨

⎪⎪⎩

*2*). The signal control *uPWM* is defined as equation (14), where *ts*<sup>1</sup> =

*uPWM* =

(*a*). without delay and without FPIC, (*b*). with one delay and without FPIC

� � *x*<sup>1</sup>

*x*2

(15) shows as two unit step functions can be used to model the centered pulse.

� + � 0

−1 −*β*

Now, finding *dk* means solving equation (9) and redefining the duty cycle as *dzad*

$$d\_{zad} = \frac{2s\_1 + T\dot{s}\_2}{\dot{s}\_2 - \dot{s}\_1} \tag{10}$$

The FPIC control law is given by equation (11), where *N* is the control parameter and *dss* is the duty cycle when the stationary state is reached.

$$d\_k = \frac{d\_{zad} + Nd\_{ss}}{N+1} \tag{11}$$

This result can be expressed as a linear combination of the state variables, where *c*1, *c*<sup>2</sup> and *c*<sup>3</sup> are constants.

$$d\_k = c\_1 \mathbf{x}\_1(k - \tau) + c\_2 \mathbf{x}\_2(k - \tau) + c\_3 \tag{12}$$

$$\begin{array}{c|c} \cdot & 2\theta(t) \\ \cdot & \cdot \\ \hline \\ \cdot & \cdot \\ \cdot & \cdot \\ \cdot & \cdot \end{array} \xrightarrow{T-d/2} \begin{array}{c|c} 2\theta(t) \\ \cdot & \cdot \\ \cdot & \cdot \\ \cdot & \cdot \\ \cdot & \cdot \\ \cdot & \cdot \end{array} \xrightarrow{(\tau+1)T-d/2} \begin{array}{c|c} (\tau+1)T-d/2 \\ \hline \\ \hline \\ \cdot & \cdot \\ (\tau+1)T \\ \hline \\ \tau+d/2 \\ \hline \\ \end{array} \xrightarrow{(\tau+1)T} \begin{array}{c|c} (\tau+1)T-d/2 \\ \hline \\ \hline \\ \tau\tau+d/2 \\ \hline \\ \tau\tau+d/2 \\ \hline \\ \end{array}$$

**Figure 2.** Scheme of centered PWM function depending on delays (*τ*).

Now, we should define the variational equation of the buck converter using the general procedure described in *Section 2*. Basically, we apply to the periodic solution (**x**∗) an appropriate perturbation using exponential functions (*eμ<sup>t</sup>* ). The stability of the digital-PWM power converter can be inferred studying the behavior of the perturbation. If the real part of the exponent *μ* is positive, the perturbation will tend to infinity and the solution will be unstable. If the real part of the exponent *μ* is negative, the perturbation will tend to zero and the solution will be stable.

Therefore, we can find the variational equation based on the dynamical equations of the system. We use a more compact expression applying the following change of variables: *x*<sup>1</sup> = *vC*/*E*, *x*<sup>2</sup> = <sup>1</sup> *E* � *<sup>L</sup> <sup>C</sup> iL* and *t* = *τ*/ <sup>√</sup>*LC*, thus *<sup>γ</sup>* <sup>=</sup> <sup>1</sup> *R* � *<sup>L</sup> <sup>C</sup>* , *β* = *rL* �*<sup>C</sup> <sup>L</sup>* and the sampling period is *T* = *Tc*/ <sup>√</sup>*LC* [10].

4 Nonlinearity, Bifurcation and Chaos - Theory and Applications

*s*<sup>2</sup> = *<sup>d</sup>*

*dzad* <sup>=</sup> <sup>2</sup>*s*<sup>1</sup> <sup>+</sup> *Ts*˙2 *s*˙2 − *s*˙1

The FPIC control law is given by equation (11), where *N* is the control parameter and *dss* is

*dk* <sup>=</sup> *dzad* <sup>+</sup> *Ndss*

This result can be expressed as a linear combination of the state variables, where *c*1, *c*<sup>2</sup> and *c*<sup>3</sup>

Now, we should define the variational equation of the buck converter using the general procedure described in *Section 2*. Basically, we apply to the periodic solution (**x**∗) an

power converter can be inferred studying the behavior of the perturbation. If the real part of the exponent *μ* is positive, the perturbation will tend to infinity and the solution will be unstable. If the real part of the exponent *μ* is negative, the perturbation will tend to zero and

*s*˙2 = (*x*˙1 + *ksx*¨1)

*spwl*(*t*)*dt* = 0 (9)

*<sup>N</sup>* <sup>+</sup> <sup>1</sup> (11)

). The stability of the digital-PWM

*dk* = *c*1*x*1(*k* − *τ*) + *c*2*x*2(*k* − *τ*) + *c*<sup>3</sup> (12)

<sup>2</sup> *s*˙1 + *s*<sup>1</sup> *s*<sup>3</sup> = *s*<sup>1</sup> + (*T* − *d*)*s*˙2

 

*x*=*x*(*kT*),*u*=0

(8)

(10)

where

are constants.

*s*˙1 = (*x*˙1 + *ksx*¨1)

*s*<sup>1</sup> = (*x*<sup>1</sup> − *ref* + *ksx*˙1)

 

the duty cycle when the stationary state is reached.

**Figure 2.** Scheme of centered PWM function depending on delays (*τ*).

appropriate perturbation using exponential functions (*eμ<sup>t</sup>*

the solution will be stable.

*x*=*x*(*kT*),*u*=1

 

*x*=*x*(*kT*),*u*=1

(*k*+1)*T* 

*kT*

Now, finding *dk* means solving equation (9) and redefining the duty cycle as *dzad*

and *ks* is a positive constant. Therefore, the zero average condition is

$$
\begin{pmatrix} \dot{\mathbf{x}}\_1 \\ \dot{\mathbf{x}}\_2 \end{pmatrix} = \begin{pmatrix} -\gamma & 1 \\ -1 & -\beta \end{pmatrix} \begin{pmatrix} \mathbf{x}\_1 \\ \mathbf{x}\_2 \end{pmatrix} + \begin{pmatrix} 0 \\ 1 \end{pmatrix} u\_{\text{PWM}} \tag{13}
$$

We use the parameter values of an experimental prototype reported in [3]. We fix *R* = 20Ω, *C* = 40*μ*F, *L*=2mH, *rL* = 0Ω, *vref* = 32V, *E*=40V and the sampling period *Tc* = 50*μs*.. Therefore, the dimensionless parameters are *γ* = 0.35, *β* = 0 and *T* = 0.1767.

For simplicity, in the remainder of *Section 3*, we note *dk* as *d* with *d* ∈ [0, *T*], and we present the procedure for one period sampling *t* ∈ [*τT*,(*τ* + 1)*T*] (general notation was used in *Section 2*). The signal control *uPWM* is defined as equation (14), where *ts*<sup>1</sup> = � *τT* + *<sup>d</sup>* 2 � and *ts*<sup>2</sup> = � (*<sup>τ</sup>* <sup>+</sup> <sup>1</sup>)*<sup>T</sup>* <sup>−</sup> *<sup>d</sup>* 2 � .

$$u\_{PWM} = \begin{cases} 1 & \text{ $\tau T < t \lesssim t\_{s1}$ } \\ -1 & t\_{s1} < t \lesssim t\_{s2} \\ 1 & t\_{s2} < t \lesssim (\tau + 1)T \end{cases} \tag{14}$$

Figure 2 illustrates the mechanism to model the centered pulse (*uPWM*) using unit step

**Figure 3.** The evolution of the real part of the Floquet exponents when *ks* is varied in the ZAD controller. (*a*). without delay and without FPIC, (*b*). with one delay and without FPIC

functions *θ*(*t*). The duty cycle depend on state variables in the instant *t* = (*k* − *τ*)*T*. Equation (15) shows as two unit step functions can be used to model the centered pulse.

$$
\begin{pmatrix} \dot{\mathbf{x}}\_1 \\ \dot{\mathbf{x}}\_2 \end{pmatrix} = \begin{pmatrix} -\gamma & 1 \\ -1 & -\beta \end{pmatrix} \begin{pmatrix} \mathbf{x}\_1 \\ \mathbf{x}\_2 \end{pmatrix} + \begin{pmatrix} 0 \\ 1 \end{pmatrix} - 2\theta \begin{pmatrix} t - t\_{s1} \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} + 2\theta \begin{pmatrix} t - t\_{s2} \end{pmatrix} \begin{pmatrix} 0 \\ 1 \end{pmatrix} \tag{15}
$$

Equation (15) implies that duty cycle can be defined in function of state variables in an initial instant (*t* = 0) because the delay information was included in the unit step functions.

#### 6 Nonlinearity, Bifurcation and Chaos - Theory and Applications 32 Nonlinearity, Bifurcation and Chaos – Theory and Applications Floquet Exponents and Bifurcations in Switched Converters <sup>7</sup>

Therefore, we can define (*d*/2) such as equation (16) where *c*1, *c*<sup>2</sup> and *c*<sup>3</sup> are given by equations (17), (18) and (19), respectively.

$$\frac{d}{2} = c\_1 \mathbf{x}\_1(0) + c\_2 \mathbf{x}\_2(0) + c\_3 \tag{16}$$

therefore, the step function in function of the perturbed periodic solution is

<sup>1</sup> (0) + *p*1(0)) + *c*2(*x*<sup>∗</sup>

computing a first order Taylor expansion approximation of the unity step function, we obtain

in the second unity step function *θ* (*t* − *ts*2). The dynamic of the perturbed periodic solution

<sup>1</sup> <sup>−</sup> *<sup>γ</sup>eμ<sup>t</sup> <sup>p</sup>*<sup>1</sup> <sup>+</sup> *<sup>x</sup>*<sup>∗</sup>

<sup>2</sup> <sup>−</sup> *<sup>β</sup>eμ<sup>t</sup> <sup>p</sup>*<sup>2</sup> <sup>+</sup> <sup>1</sup> <sup>−</sup> <sup>2</sup>*<sup>θ</sup>*

 *t* − 

(*c*<sup>1</sup> *<sup>p</sup>*1(0) + *<sup>c</sup>*<sup>2</sup> *<sup>p</sup>*2(0))

<sup>2</sup> (0) + *p*2(0)) + *c*<sup>3</sup>

<sup>2</sup> + *<sup>e</sup>μ<sup>t</sup> <sup>p</sup>*<sup>2</sup>

 *t* − *τT* + *<sup>d</sup>*<sup>∗</sup> 2 <sup>+</sup>

(*<sup>τ</sup>* <sup>+</sup> <sup>1</sup>)*<sup>T</sup>* <sup>−</sup> *<sup>d</sup>*<sup>∗</sup>

(*c*<sup>1</sup> *<sup>p</sup>*1(0) + *<sup>c</sup>*<sup>2</sup> *<sup>p</sup>*2(0)) +

2 

(*c*<sup>1</sup> *<sup>p</sup>*1(0) + *<sup>c</sup>*<sup>2</sup> *<sup>p</sup>*2(0))

Floquet Exponents and Bifurcations in Switched Converters 33

*dt* . The same considerations are applied

*τT* + *c*1(*x*<sup>∗</sup>

<sup>1</sup> <sup>+</sup> *<sup>μ</sup>eμ<sup>t</sup> <sup>p</sup>*<sup>1</sup> <sup>+</sup> *<sup>e</sup>μ<sup>t</sup> <sup>p</sup>*˙1 <sup>=</sup> <sup>−</sup>*γx*<sup>∗</sup>

<sup>1</sup> <sup>−</sup> *<sup>e</sup>μ<sup>t</sup> <sup>p</sup>*<sup>1</sup> <sup>−</sup> *<sup>β</sup>x*<sup>∗</sup>

(*<sup>τ</sup>* <sup>+</sup> <sup>1</sup>)*<sup>T</sup>* <sup>−</sup> *<sup>d</sup>*<sup>∗</sup>

(*c*<sup>1</sup> *<sup>p</sup>*1(0) + *<sup>c</sup>*<sup>2</sup> *<sup>p</sup>*2(0)) + <sup>2</sup>*<sup>θ</sup>*

2

**Figure 5.** Scheme of equivalent transformations between Floquet exponents (*μi*), Lyapunov exponents

*<sup>μ</sup>eμ<sup>t</sup> <sup>p</sup>*<sup>1</sup> <sup>+</sup> *<sup>e</sup>μ<sup>t</sup> <sup>p</sup>*˙1 <sup>=</sup> <sup>−</sup>*γeμ<sup>t</sup> <sup>p</sup>*<sup>1</sup> <sup>+</sup> *<sup>e</sup>μ<sup>t</sup> <sup>p</sup>*<sup>2</sup>

 *t* − *τT* + *<sup>d</sup>*<sup>∗</sup> 2

, we obtain:

(*c*<sup>1</sup> *<sup>p</sup>*1(0) + *<sup>c</sup>*<sup>2</sup> *<sup>p</sup>*2(0))

2

*p*˙1 = −*μp*<sup>1</sup> − *γp*<sup>1</sup> + *p*<sup>2</sup>

Neglecting the periodic solution **x**∗, we obtain the dynamic of the perturbation.

(*<sup>τ</sup>* <sup>+</sup> <sup>1</sup>)*<sup>T</sup>* <sup>−</sup> *<sup>d</sup>*<sup>∗</sup>

where *δ*(*t*) is the Dirac delta function with *δ*(*t*) = *<sup>d</sup>θ*(*t*)

*x*˙ ∗

<sup>2</sup> <sup>+</sup> *<sup>μ</sup>eμ<sup>t</sup> <sup>p</sup>*<sup>2</sup> <sup>+</sup> *<sup>e</sup>μ<sup>t</sup> <sup>p</sup>*˙2 <sup>=</sup> <sup>−</sup>*x*<sup>∗</sup>

+2*δ t* − 

*θ t* − *τT* + *<sup>d</sup>* 2 <sup>=</sup> *<sup>θ</sup> <sup>t</sup>* <sup>−</sup>

*θ t* − *τT* + *<sup>d</sup>* 2 <sup>=</sup> *<sup>θ</sup> t* − *τT* + *<sup>d</sup>*<sup>∗</sup> 2 <sup>−</sup> *<sup>δ</sup> t* − *τT* + *<sup>d</sup>*<sup>∗</sup> 2

is

*x*˙ ∗

> 2*δ t* − *τT* + *<sup>d</sup>*<sup>∗</sup> 2

(*λi*) and characteristics multipliers (*mi*).

*<sup>μ</sup>eμ<sup>t</sup> <sup>p</sup>*<sup>2</sup> <sup>+</sup> *<sup>e</sup>μ<sup>t</sup> <sup>p</sup>*˙2 <sup>=</sup> <sup>−</sup>*eμ<sup>t</sup> <sup>p</sup>*<sup>1</sup> <sup>−</sup> *<sup>β</sup>eμ<sup>t</sup> <sup>p</sup>*<sup>2</sup> <sup>+</sup> <sup>2</sup>*<sup>δ</sup>*

2*δ t* − 

multiplying both sides of the equation by *e*−*μ<sup>t</sup>*

$$c\_1 = \frac{2 - \gamma(2k\_s + T(1 - \gamma k\_s)) - k\_s T}{-4k\_s(N+1)} \tag{17}$$

$$c\_2 = \frac{2k\_s + T(1 - k\_s(\gamma + \beta))}{-4k\_s(N+1)}\tag{18}$$

$$c\_3 = \frac{2ref + k\_sT}{4k\_s} + \frac{Nd\_{ss}}{2(N+1)}\tag{19}$$

Let the perturbed periodic orbit be

$$x\_1(t) = x\_1^\*(t) + e^{\mu t} p\_1(t), \qquad x\_2(t) = x\_2^\*(t) + e^{\mu t} p\_2(t),$$

where the superstar labels are the periodic solutions and *eμ<sup>t</sup> p*1(*t*), *eμ<sup>t</sup> p*2(*t*) are the perturbations. Then, we replace the perturbed periodic orbit in equation (15).

**Figure 4.** The evolution of the real part of the Floquet exponents for several delays in the Digital-PWM converter based on ZAD or ZAD-FPIC techniques. (*a*). *ks* is varied between [0; 5] with *N* = 0, (*b*). *N* is varied between [0; 30] with *ks* = 4.5.

$$\begin{aligned} \dot{\mathbf{x}}\_1^\* + \mu e^{\mu t} p\_1 + e^{\mu t} \dot{p}\_1 &= -\gamma (\mathbf{x}\_1^\* + e^{\mu t} p\_1) + (\mathbf{x}\_2^\* + e^{\mu t} p\_2) \\ \dot{\mathbf{x}}\_2^\* + \mu e^{\mu t} p\_2 + e^{\mu t} \dot{p}\_2 &= -(\mathbf{x}\_1^\* + e^{\mu t} p\_1) - \beta (\mathbf{x}\_2^\* + e^{\mu t} p\_2) + 1 - 2\theta \left( t - \left( \tau T + \frac{d}{2} \right) \right) + \frac{1}{2} \\ 2\theta \left( t - \left( (\tau + 1) T - \frac{d}{2} \right) \right) \end{aligned}$$

Unit step functions are replaced as follows. The periodic solution of duty cycle is noted:

$$\frac{d^\*}{2} = c\_1 \mathfrak{x}\_1^\*(0) + c\_2 \mathfrak{x}\_2^\*(0) + c\_3$$

then in the first unit step function,

$$\theta\left(t - t\_{s1}^\*\right) = \theta\left(t - \left(\tau T + \frac{d^\*}{2}\right)\right) = \theta\left(t - \left(\tau T + c\_1 x\_1^\*(0) + c\_2 x\_2^\*(0) + c\_3\right)\right)$$

therefore, the step function in function of the perturbed periodic solution is

6 Nonlinearity, Bifurcation and Chaos - Theory and Applications

Therefore, we can define (*d*/2) such as equation (16) where *c*1, *c*<sup>2</sup> and *c*<sup>3</sup> are given by equations

*<sup>c</sup>*<sup>1</sup> <sup>=</sup> <sup>2</sup> <sup>−</sup> *<sup>γ</sup>*(2*ks* <sup>+</sup> *<sup>T</sup>*(<sup>1</sup> <sup>−</sup> *<sup>γ</sup>ks*)) <sup>−</sup> *ksT*

*<sup>c</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup>*ks* <sup>+</sup> *<sup>T</sup>*(<sup>1</sup> <sup>−</sup> *ks*(*<sup>γ</sup>* <sup>+</sup> *<sup>β</sup>*))

+

*p*1(*t*), *x*2(*t*) = *x*<sup>∗</sup>

where the superstar labels are the periodic solutions and *eμ<sup>t</sup> p*1(*t*), *eμ<sup>t</sup> p*2(*t*) are the

(a) ZAD scheme (without FPIC) (b) ZAD-FPIC scheme

<sup>1</sup> <sup>+</sup> *<sup>e</sup>μ<sup>t</sup> <sup>p</sup>*1) <sup>−</sup> *<sup>β</sup>*(*x*<sup>∗</sup>

Unit step functions are replaced as follows. The periodic solution of duty cycle is noted:

*τT* + *c*1*x*<sup>∗</sup>

<sup>1</sup> (0) + *c*2*x*<sup>∗</sup>

<sup>1</sup> + *<sup>e</sup>μ<sup>t</sup> <sup>p</sup>*1)+(*x*<sup>∗</sup>

**Figure 4.** The evolution of the real part of the Floquet exponents for several delays in the Digital-PWM converter based on ZAD or ZAD-FPIC techniques. (*a*). *ks* is varied between [0; 5] with *N* = 0, (*b*). *N* is

<sup>2</sup> + *<sup>e</sup>μ<sup>t</sup> <sup>p</sup>*2)

*Ndss*

*<sup>c</sup>*<sup>3</sup> <sup>=</sup> <sup>2</sup>*ref* <sup>+</sup> *ksT* 4*ks*

perturbations. Then, we replace the perturbed periodic orbit in equation (15).

<sup>2</sup> <sup>=</sup> *<sup>c</sup>*1*x*1(0) + *<sup>c</sup>*2*x*2(0) + *<sup>c</sup>*<sup>3</sup> (16)

<sup>2</sup> (*t*) + *e μt p*2(*t*),

<sup>2</sup> <sup>+</sup> *<sup>e</sup>μ<sup>t</sup> <sup>p</sup>*2) + <sup>1</sup> <sup>−</sup> <sup>2</sup>*<sup>θ</sup>*

<sup>2</sup> (0) + *c*<sup>3</sup>

 *t* − *τT* + *<sup>d</sup>* 2 +

<sup>−</sup>4*ks*(*<sup>N</sup>* <sup>+</sup> <sup>1</sup>) (17)

<sup>−</sup>4*ks*(*<sup>N</sup>* <sup>+</sup> <sup>1</sup>) (18)

<sup>2</sup>(*<sup>N</sup>* <sup>+</sup> <sup>1</sup>) (19)

*d*

<sup>1</sup> (*t*) + *e μt*

(17), (18) and (19), respectively.

Let the perturbed periodic orbit be

varied between [0; 30] with *ks* = 4.5.

<sup>1</sup> <sup>+</sup> *<sup>μ</sup>eμ<sup>t</sup> <sup>p</sup>*<sup>1</sup> <sup>+</sup> *<sup>e</sup>μ<sup>t</sup> <sup>p</sup>*˙1 <sup>=</sup> <sup>−</sup>*γ*(*x*<sup>∗</sup>

(*<sup>τ</sup>* <sup>+</sup> <sup>1</sup>)*<sup>T</sup>* <sup>−</sup> *<sup>d</sup>*

<sup>1</sup> (0) + *c*2*x*<sup>∗</sup>

then in the first unit step function,

<sup>2</sup> <sup>+</sup> *<sup>μ</sup>eμ<sup>t</sup> <sup>p</sup>*<sup>2</sup> <sup>+</sup> *<sup>e</sup>μ<sup>t</sup> <sup>p</sup>*˙2 <sup>=</sup> <sup>−</sup>(*x*<sup>∗</sup>

2 

<sup>2</sup> (0) + *c*<sup>3</sup>

*x*˙ ∗

*x*˙ ∗

2*θ t* − 

*d*∗ <sup>2</sup> = *c*1*x*<sup>∗</sup>

*θ t* − *t* ∗ *s*1 = *θ t* − *τT* + *<sup>d</sup>*<sup>∗</sup> 2 = *θ <sup>t</sup>* <sup>−</sup>

*x*1(*t*) = *x*<sup>∗</sup>

$$\theta\left(t - \left(\tau T + \frac{d}{2}\right)\right) = \theta\left(t - \left(\tau T + c\_1(\mathbf{x}\_1^\*(0) + p\_1(0)) + c\_2(\mathbf{x}\_2^\*(0) + p\_2(0)) + c\_3\right)\right)$$

computing a first order Taylor expansion approximation of the unity step function, we obtain

$$\theta\left(t - \left(\tau T + \frac{d}{2}\right)\right) = \theta\left(t - \left(\tau T + \frac{d^\*}{2}\right)\right) - \delta\left(t - \left(\tau T + \frac{d^\*}{2}\right)\right) \left(c\_1 p\_1(0) + c\_2 p\_2(0)\right)$$

where *δ*(*t*) is the Dirac delta function with *δ*(*t*) = *<sup>d</sup>θ*(*t*) *dt* . The same considerations are applied in the second unity step function *θ* (*t* − *ts*2). The dynamic of the perturbed periodic solution is

$$\dot{\mathbf{x}}\_1^\* + \mu e^{\mu t} p\_1 + e^{\mu t} \dot{p}\_1 = -\gamma \mathbf{x}\_1^\* - \gamma e^{\mu t} p\_1 + \mathbf{x}\_2^\* + e^{\mu t} p\_2$$

$$\dot{\mathbf{x}}\_2^\* + \mu e^{\mu t} p\_2 + e^{\mu t} \dot{p}\_2 = -\mathbf{x}\_1^\* - e^{\mu t} p\_1 - \beta \mathbf{x}\_2^\* - \beta e^{\mu t} p\_2 + 1 - 2\theta \left( t - \left( \tau T + \frac{d^\*}{2} \right) \right) + \tau$$

$$2\delta \left( t - \left( \tau T + \frac{d^\*}{2} \right) \right) (c\_1 p\_1(0) + c\_2 p\_2(0)) + 2\theta \left( t - \left( (\tau + 1) T - \frac{d^\*}{2} \right) \right)$$

$$+ 2\delta \left( t - \left( (\tau + 1) T - \frac{d^\*}{2} \right) \right) (c\_1 p\_1(0) + c\_2 p\_2(0))$$

**Figure 5.** Scheme of equivalent transformations between Floquet exponents (*μi*), Lyapunov exponents (*λi*) and characteristics multipliers (*mi*).

Neglecting the periodic solution **x**∗, we obtain the dynamic of the perturbation.

$$\begin{aligned} \mu e^{\mu t} p\_1 + e^{\mu t} \dot{p}\_1 &= -\gamma e^{\mu t} p\_1 + e^{\mu t} p\_2 \\ \mu e^{\mu t} p\_2 + e^{\mu t} \dot{p}\_2 &= -e^{\mu t} p\_1 - \beta e^{\mu t} p\_2 + 2\delta \left( t - \left( \tau T + \frac{d^\*}{2} \right) \right) (c\_1 p\_1(0) + c\_2 p\_2(0)) + \\ 2\delta \left( t - \left( (\tau + 1)T - \frac{d^\*}{2} \right) \right) (c\_1 p\_1(0) + c\_2 p\_2(0)) \end{aligned}$$

multiplying both sides of the equation by *e*−*μ<sup>t</sup>* , we obtain:

$$
\dot{p}\_1 = -\mu p\_1 - \gamma p\_1 + p\_2
$$

#### 8 Nonlinearity, Bifurcation and Chaos - Theory and Applications 34 Nonlinearity, Bifurcation and Chaos – Theory and Applications Floquet Exponents and Bifurcations in Switched Converters <sup>9</sup>

**H** =

 0 0 2*c*<sup>1</sup> 2*c*<sup>2</sup>  , **p**˙ =

$$\begin{aligned} \dot{p}\_2 &= -p\_1 - \mu p\_2 - \beta p\_2 + 2e^{-\mu t} \delta \left( t - \left( \tau T + \frac{d^\*}{2} \right) \right) (c\_1 p\_1(0) + c\_2 p\_2(0)) + \dotsb \\ &\quad 2e^{-\mu t} \delta \left( t - \left( (\tau + 1)T - \frac{d^\*}{2} \right) \right) (c\_1 p\_1(0) + c\_2 p\_2(0)) \end{aligned}$$

Simplifying the expressions and writing in matrix notation we obtain the variational equation of the buck converter. Note that the terms *e*−*μ<sup>t</sup>* only have sense in *t* = *ϑ* when these are multiplied by Dirac delta functions *δ*(*t* − *ϑ*).

$$\dot{\mathbf{p}} = \begin{pmatrix} -\gamma - \mu & 1 \\ -1 & -\beta - \mu \end{pmatrix} \mathbf{p} + e^{-\mu t\_{s1}^\*} \delta \begin{pmatrix} t - t\_{s1}^\* \end{pmatrix} \mathbf{H} \mathbf{p} \begin{pmatrix} 0 \end{pmatrix} + e^{-\mu t\_{s2}^\*} \delta \begin{pmatrix} t - t\_{s2}^\* \end{pmatrix} \mathbf{H} \mathbf{p} \begin{pmatrix} 0 \end{pmatrix} \tag{20}$$

 *p*<sup>1</sup> *p*2  , **<sup>a</sup>***e*(0) = *<sup>p</sup>*1(0)

*p*2(0)

The particular selection of **M**<sup>1</sup> and **M**<sup>2</sup> allows a easier solution of state-transition matrix *e***M***<sup>t</sup>*

The first exponential matrix *e***M**1*<sup>t</sup>* is computed using the identity matrix, while the second

<sup>2</sup> <sup>+</sup> *<sup>β</sup>* <sup>2</sup> +*μ*)*t* **I**

*<sup>α</sup>*<sup>2</sup> *sen*(*α*2*t*)

Floquet Exponents and Bifurcations in Switched Converters 35

*<sup>α</sup>*<sup>2</sup> *sen*(*α*2*t*) + cos(*α*2*t*)

2

<sup>2</sup> )**Hp**

<sup>2</sup> )**H p** (0)

<sup>2</sup> ) and *<sup>t</sup>* = (*<sup>T</sup>* <sup>−</sup> *<sup>d</sup>*

2 ).

<sup>2</sup> **p** (0)

<sup>2</sup> *e***M**<sup>2</sup> *<sup>d</sup>*

*e***M**1*<sup>t</sup>* = *e*−( *<sup>γ</sup>*

*<sup>α</sup>*<sup>2</sup> *sen*(*α*2*t*) + cos(*α*2*t*) <sup>1</sup>

*<sup>α</sup>*<sup>2</sup> *sen*(*α*2*t*) *<sup>α</sup>*<sup>1</sup>

(a) (b)

**Figure 7.** Locus of Floquet exponents and characteristic multipliers when *N* is varied between [0; 30] for

The piecewise-smooth ordinary differential equation can be solved in each interval with

<sup>−</sup> <sup>=</sup> *<sup>e</sup>***M**<sup>1</sup> *<sup>d</sup>*

<sup>+</sup> integrating about the discontinuity as follows.

<sup>−</sup>*μ*(*τT*<sup>+</sup> *<sup>d</sup>*

<sup>−</sup>*μ*(*τT*<sup>+</sup> *<sup>d</sup>*

**p** *<sup>d</sup>* 2 

> *<sup>d</sup>* 2 <sup>−</sup> <sup>+</sup> *<sup>e</sup>*

exponential matrix *e***M**2*<sup>t</sup>* is computed using sine and cosine functions.

− 1

 <sup>1</sup> <sup>−</sup> *<sup>α</sup>*<sup>2</sup> 1.

*e***M**2*<sup>t</sup>* =

where *α*<sup>1</sup> =

 *γ* <sup>2</sup> <sup>−</sup> *<sup>β</sup>* 2 

then we compute **p**

 <sup>−</sup>*α*<sup>1</sup>

and *α*<sup>2</sup> =

*τ* = 1. (*a*). Floquet locus, (*b*). Characteristic multipliers locus.

 *<sup>d</sup>* 2 

> **p** *<sup>d</sup>* 2 <sup>+</sup> <sup>=</sup> *e***M**<sup>1</sup> *<sup>d</sup>* <sup>2</sup> *e***M**<sup>2</sup> *<sup>d</sup>* <sup>2</sup> + *e*

special attention in the discontinuities due to Dirac delta functions.

1). Initially, we compute the solution between *t* = 0 and *t* = *<sup>d</sup>*

**p** *<sup>d</sup>* 2 <sup>+</sup> <sup>=</sup> **<sup>p</sup>**

2). Now, we compute the solution in the interval between *t* = ( *<sup>d</sup>*

.

Therefore, equation (20) is the variational equation where *t* ∗ *<sup>s</sup>*<sup>1</sup> <sup>=</sup> *<sup>τ</sup><sup>T</sup>* <sup>+</sup> *<sup>d</sup>*<sup>∗</sup> <sup>2</sup> , *t* ∗ *<sup>s</sup>*<sup>2</sup> <sup>=</sup> (*<sup>τ</sup>* <sup>+</sup> <sup>1</sup>) *<sup>T</sup>* <sup>−</sup> *<sup>d</sup>*<sup>∗</sup> 2 and,

> *p*˙1 *p*˙2 , **p** =

**Figure 6.** The evolution of the real part of the Floquet and Lyapunov exponents when *N* is varied. (*a*). without delay, (*b*). with one delay.

Note that equation (20) can be written in a compact form as:

$$\dot{\mathbf{p}} = \mathbf{M}\mathbf{p} + e^{-\mu t} \left[ \delta \left( t - \left( \tau T + \frac{d}{2} \right) \right) + \delta \left( t - \left( \tau + 1 \right) T + \frac{d}{2} \right) \right] \mathbf{H} \mathbf{p}(0) \tag{21}$$

with **M** and **H** defined according to the equation (20), and again, for simplicity we note *d* = *d*∗. For solving this piecewise-smooth ordinary differential equation, we write *z* = *e*−*μT*, **M** = **M**<sup>1</sup> + **M**<sup>2</sup> with *e***M***<sup>t</sup>* = *e***M**1*<sup>t</sup> e***M**2*<sup>t</sup>* .

$$\mathbf{M}\_1 = \begin{pmatrix} -\frac{\gamma}{2} - \frac{\mathcal{G}}{2} - \mu & 0\\ 0 & -\frac{\gamma}{2} - \frac{\mathcal{G}}{2} - \mu \end{pmatrix} \tag{22}$$

$$\mathbf{M}\_2 = \begin{pmatrix} -\frac{\gamma}{2} + \frac{\beta}{2} & 1\\ -1 & \frac{\gamma}{2} - \frac{\beta}{2} \end{pmatrix} \tag{23}$$

The particular selection of **M**<sup>1</sup> and **M**<sup>2</sup> allows a easier solution of state-transition matrix *e***M***<sup>t</sup>* . The first exponential matrix *e***M**1*<sup>t</sup>* is computed using the identity matrix, while the second exponential matrix *e***M**2*<sup>t</sup>* is computed using sine and cosine functions.

$$e^{\mathbf{M}\_1 t} = e^{-(\frac{\gamma}{\xi} + \frac{\xi}{\xi} + \mu)t} \mathbf{I}\_1$$

$$e^{\mathbf{M}\_2 t} = \begin{pmatrix} -\frac{\mathfrak{a}\_1}{\mathfrak{a}\_2} \operatorname{sen}(\mathfrak{a}\_2 t) + \cos(\mathfrak{a}\_2 t) & \frac{1}{\mathfrak{a}\_2} \operatorname{sen}(\mathfrak{a}\_2 t) \\\ -\frac{1}{\mathfrak{a}\_2} \operatorname{sen}(\mathfrak{a}\_2 t) & \frac{\mathfrak{a}\_1}{\mathfrak{a}\_2} \operatorname{sen}(\mathfrak{a}\_2 t) + \cos(\mathfrak{a}\_2 t) \end{pmatrix}.$$

where *α*<sup>1</sup> = *γ* <sup>2</sup> <sup>−</sup> *<sup>β</sup>* 2 and *α*<sup>2</sup> = <sup>1</sup> <sup>−</sup> *<sup>α</sup>*<sup>2</sup> 1.

8 Nonlinearity, Bifurcation and Chaos - Theory and Applications

2 

Simplifying the expressions and writing in matrix notation we obtain the variational equation of the buck converter. Note that the terms *e*−*μ<sup>t</sup>* only have sense in *t* = *ϑ* when these are

(a) *τ* = 0, *ks* = 0.125 (b) *τ* = 1, *ks* = 0.125

**Figure 6.** The evolution of the real part of the Floquet and Lyapunov exponents when *N* is varied. (*a*).

*d* 2 ) + *δ* 

with **M** and **H** defined according to the equation (20), and again, for simplicity we note *d* = *d*∗. For solving this piecewise-smooth ordinary differential equation, we write *z* = *e*−*μT*,

> <sup>2</sup> − *μ* 0 <sup>0</sup> <sup>−</sup>*<sup>γ</sup>*

<sup>2</sup> <sup>−</sup> *<sup>β</sup>* <sup>2</sup> − *μ*

<sup>2</sup> <sup>−</sup> *<sup>β</sup>* 2 (*c*<sup>1</sup> *p*1(0) + *c*<sup>2</sup> *p*2(0)) +

<sup>2</sup> , *t* ∗

 *p*1(0) *p*2(0) *<sup>s</sup>*2) **Hp** (0) (20)

*<sup>s</sup>*<sup>2</sup> <sup>=</sup> (*<sup>τ</sup>* <sup>+</sup> <sup>1</sup>) *<sup>T</sup>* <sup>−</sup> *<sup>d</sup>*<sup>∗</sup>

2

(*c*<sup>1</sup> *p*1(0) + *c*<sup>2</sup> *p*2(0))

−*μt* ∗ *<sup>s</sup>*<sup>2</sup> *δ* (*t* − *t* ∗

*<sup>s</sup>*<sup>1</sup> <sup>=</sup> *<sup>τ</sup><sup>T</sup>* <sup>+</sup> *<sup>d</sup>*<sup>∗</sup>

*<sup>s</sup>*1) **Hp** (0) + *e*

 *p*<sup>1</sup> *p*2  ∗

, **a***e*(0) =

*t* − (*τ* + 1) *T* +

*d* 2 

**Hp**(0) (21)

(22)

(23)

*δ t* − *τT* + *<sup>d</sup>*<sup>∗</sup> 2 

(*<sup>τ</sup>* <sup>+</sup> <sup>1</sup>)*<sup>T</sup>* <sup>−</sup> *<sup>d</sup>*<sup>∗</sup>

 *p*˙1 *p*˙2 , **p** =

*<sup>p</sup>*˙2 <sup>=</sup> <sup>−</sup>*p*<sup>1</sup> <sup>−</sup> *<sup>μ</sup>p*<sup>2</sup> <sup>−</sup> *<sup>β</sup>p*<sup>2</sup> <sup>+</sup> <sup>2</sup>*e*−*μ<sup>t</sup>*

2*e*−*μ<sup>t</sup> δ t* − 

multiplied by Dirac delta functions *δ*(*t* − *ϑ*).

−1 −*β* − *μ*

 0 0 2*c*<sup>1</sup> 2*c*<sup>2</sup>  **p** + *e* −*μt* ∗ *<sup>s</sup>*<sup>1</sup> *δ* (*t* − *t* ∗

 , **p**˙ =

Note that equation (20) can be written in a compact form as:

*t* − (*τT* +

*e***M**2*<sup>t</sup>* .

> −*γ* <sup>2</sup> <sup>−</sup> *<sup>β</sup>*

**M**<sup>2</sup> =

 −*γ* <sup>2</sup> <sup>+</sup> *<sup>β</sup>* <sup>2</sup> 1 <sup>−</sup><sup>1</sup> *<sup>γ</sup>*

**M**<sup>1</sup> =

−*μt δ* 

Therefore, equation (20) is the variational equation where *t*

<sup>−</sup>*<sup>γ</sup>* <sup>−</sup> *<sup>μ</sup>* <sup>1</sup>

**H** =

without delay, (*b*). with one delay.

**M** = **M**<sup>1</sup> + **M**<sup>2</sup> with *e***M***<sup>t</sup>* = *e***M**1*<sup>t</sup>*

**p**˙ = **Mp** + *e*

**p**˙ =

and,

**Figure 7.** Locus of Floquet exponents and characteristic multipliers when *N* is varied between [0; 30] for *τ* = 1. (*a*). Floquet locus, (*b*). Characteristic multipliers locus.

The piecewise-smooth ordinary differential equation can be solved in each interval with special attention in the discontinuities due to Dirac delta functions.

1). Initially, we compute the solution between *t* = 0 and *t* = *<sup>d</sup>* 2

$$\mathbf{p}\begin{pmatrix} \frac{d}{2} \end{pmatrix}\_{-} = e^{\mathbf{M}\_1 \frac{d}{2}} e^{\mathbf{M}\_2 \frac{d}{2}} \mathbf{p}\begin{pmatrix} 0 \end{pmatrix}$$

then we compute **p** *<sup>d</sup>* 2 <sup>+</sup> integrating about the discontinuity as follows.

$$\begin{aligned} \mathbf{p}\left(\frac{d}{2}\right)\_+ &= \mathbf{p}\left(\frac{d}{2}\right)\_- + e^{-\mu\left(\tau T + \frac{d}{2}\right)} \mathbf{H} \mathbf{p} \\\\ \mathbf{p}\left(\frac{d}{2}\right)\_+ &= \left(e^{\mathbf{M}\_1\frac{d}{2}}e^{\mathbf{M}\_2\frac{d}{2}} + e^{-\mu\left(\tau T + \frac{d}{2}\right)} \mathbf{H}\right) \mathbf{p}\left(0\right) \end{aligned}$$

2). Now, we compute the solution in the interval between *t* = ( *<sup>d</sup>* <sup>2</sup> ) and *<sup>t</sup>* = (*<sup>T</sup>* <sup>−</sup> *<sup>d</sup>* 2 ).

10 Nonlinearity, Bifurcation and Chaos - Theory and Applications 36 Nonlinearity, Bifurcation and Chaos – Theory and Applications Floquet Exponents and Bifurcations in Switched Converters <sup>11</sup>

$$\begin{aligned} \mathbf{p}\left(T - \frac{d}{2}\right)\_- &= e^{\mathbf{M}\_1(T-d)} e^{\mathbf{M}\_2(T-d)} \mathbf{p}\left(\frac{d}{2}\right)\_+ \\\\ \mathbf{p}\left(T - \frac{d}{2}\right)\_- &= e^{\mathbf{M}\_1(T-d)} e^{\mathbf{M}\_2(T-d)} \left(e^{\mathbf{M}\_1\frac{d}{2}} e^{\mathbf{M}\_2\frac{d}{2}} + e^{-\mu\left(\tau T + \frac{d}{2}\right)} \mathbf{H}\right) \mathbf{p}\left(0\right) \end{aligned}$$

*e***M**1(*T*−*d*)*e***M**2(*T*−*d*)*e*

Equation (24) is the solution of the variational system where *z* = *e*−*μT*.

*e* <sup>−</sup>*α*3(*T*<sup>−</sup> *<sup>d</sup>* 2 )*e*

The matrix **<sup>Q</sup>**<sup>1</sup> is given by equation (25), where *<sup>α</sup>*<sup>3</sup> = *<sup>γ</sup>*

**<sup>M</sup>**2*<sup>T</sup>* + *z*(*τ*+1)

exponent is negative or not. If det(**Q**<sup>1</sup> − **I**) = 0 implies *μ* negative.

is displayed in figure 3(*a*). The T-periodic solution is stable for *ks* > 3.24.

bifurcation zones (*τ* > 1, *ks* > 0 *N* > −1).

polynomial in *z* of degree 2(*τ* + 1).

<sup>−</sup> 0.00215387 *ks*

( 1*e* − 10 *k*2 *s*

**Q**<sup>1</sup> = *ze*−*α*3*Te* <sup>−</sup>*μ*((*τ*+1)*T*<sup>−</sup> *<sup>d</sup>*

(a) (b) (c)

**Figure 9.** Evolution of characteristic multipliers in several bifurcation zones. (*a*). flip bifurcation zone (*τ* = 0, *ks* > 0 *N* > −1). (*b*). fold bifurcation zones (*τ* > 0, *ks* > 0 *N* < −1). (*c*). Neimark-Sacker

<sup>2</sup> )**H p** (0)

Floquet Exponents and Bifurcations in Switched Converters 37

**p** (*T*) = **Q**1**p***<sup>e</sup>* (0) (24)

*e* <sup>−</sup>*α*3( *<sup>d</sup>* 2 )*e* **M**2( *<sup>d</sup>* <sup>2</sup> )**H** 

(**Q**<sup>1</sup> − *I*) **p** (0) = **0** (26)

− 0.0535075)*z* + 1 = 0 (28)

(25)

<sup>2</sup> <sup>+</sup> *<sup>β</sup>* 2 .

<sup>2</sup> )**H** + *z*(*τ*+1)

**<sup>M</sup>**2(*T*<sup>−</sup> *<sup>d</sup>*

The existence of T-periodic solution, i.e. **p** (*T*) = **p** (0), depends on equation (26) is satisfied.

The stability of the periodic solution depends on whether the real part of each Floquet

We fix the parameter values in *γ* = 0.35; *β* = 0; *T* = 0.1767; *ks* = 4.5; *N* = 0 and *ref* = 0.8. Equation (27) shows equation (26) in function of the number of delays (*τ*). The equation is a

Assuming real-time behavior, i.e. *τ* = 0, the determinant is the second order polynomial of equation (28). The evolution of the real part of the Floquet exponents as parameter *ks* varies

<sup>−</sup> 0.9466771)*z*<sup>2</sup> + ( 0.3443753

0.1*<sup>e</sup>* <sup>−</sup> <sup>20</sup>*z*2(*τ*+1) <sup>−</sup> 1.867*z*(*τ*+2) <sup>+</sup> 1.911*z*(*τ*+1) <sup>+</sup> 0.93*z*<sup>2</sup> <sup>−</sup> 1.899*<sup>z</sup>* <sup>+</sup> <sup>1</sup> <sup>=</sup> 0 (27)

*ks*

in the discontinuity,

$$\begin{aligned} \mathbf{p}\left(T-\frac{4}{2}\right)\_+ &= \mathbf{p}\left(T-\frac{4}{2}\right)\_- + e^{-\mathbf{p}\left((\tau+1)T-\frac{4}{2}\right)}\mathbf{H}\mathbf{p}\left(0\right) \\\\ \mathbf{p}\left(T-\frac{4}{2}\right)\_+ &= \left(e^{\mathbf{M}\_1\left(T-d\right)}e^{\mathbf{M}\_1\left(T-d\right)}e^{\mathbf{M}\_1\frac{4}{2}}e^{\mathbf{M}\_1\frac{4}{2}} + e^{\mathbf{M}\_1\left(T-d\right)}e^{\mathbf{M}\_2\left(T-d\right)}e^{-\mathbf{p}\left((\tau+1)T-\frac{4}{2}\right)}\mathbf{H}\right)\mathbf{p}\left(0\right) \\\\ &\stackrel{\text{def}}{=} \\\\ \mathbf{r}\frac{\mathbf{r}}{\tau} &= \mathbf{\frac{1}{2}\mathbf{Q}} \\\\ \mathbf{r}\frac{\mathbf{r}}{\tau} &= \mathbf{0} \\\\ \mathbf{\frac{r}{\tau}} &= \mathbf{0} \\\\ \mathbf{\frac{\mathbf{p}}{\tau}} &= \mathbf{0} \\\\ \mathbf{\frac{\mathbf{p}}{\tau}} &= \mathbf{0} \\\\ \mathbf{\frac{\mathbf{p}}{\tau}} &= \mathbf{0} \\\\ \mathbf{\frac{\mathbf{p}}{\tau}} &= \mathbf{0} \\\\ \mathbf{\frac{\mathbf{p}}{\tau}} &= \mathbf{0} \\\\ \mathbf{\frac{\mathbf{p}}{\tau}} &= \mathbf{0} \end{aligned} \qquad \text{(a)} \qquad \begin{\mathbf{\frac{\mathbf{p}}{\tau}}} \mathbf{\frac{\mathbf{p}}{\tau}} \\\\ \mathbf{\frac{\mathbf{p}}{\tau}} &= \mathbf{0} \\\\ \mathbf{\frac{\mathbf{p}}{\tau}} &= \mathbf{0} \end{aligned} \qquad \begin{\mathbf{\frac{\mathbf{p}}}}$$

**Figure 8.** 3D parameter space of Digital-PWM switched converter and bifurcation zones. (*a*). (*ks*, *N*, *τ*) space where the lines *ks* = 0 and *N* = −1 separate each plane in four sub-spaces. (*b*). flip bifurcation zone. (*c*). fold bifurcation zones. (*d*). Neimark-Sacker bifurcation zones.

3). Finally, we compute the solution in the last interval between *<sup>t</sup>* = (*<sup>T</sup>* <sup>−</sup> *<sup>d</sup>* <sup>2</sup> ) and *T*.

$$\begin{aligned} \mathbf{p}\left(T\right) &= e^{\mathbf{M}\_1 \frac{d}{2}} e^{\mathbf{M}\_2 \frac{d}{2}} \mathbf{p}\left(T - \frac{d}{2}\right)\_+ \\\\ \mathbf{p}\left(T - \frac{d}{2}\right)\_+ &= \left(e^{\mathbf{M}\_1 \left(T - d\right)} e^{\mathbf{M}\_2 \left(T - d\right)} e^{\mathbf{M}\_1 \frac{d}{2}} e^{\mathbf{M}\_2 \frac{d}{2}} + \right) \end{aligned}$$

#### 36 Nonlinearity, Bifurcation and Chaos – Theory and Applications Floquet Exponents and Bifurcations in Switched Converters <sup>11</sup> Floquet Exponents and Bifurcations in Switched Converters 37

(a) (b) (c)

<sup>−</sup>*μ*((*τ*+1)*T*<sup>−</sup> *<sup>d</sup>*

<sup>2</sup> )**H p** (0)

**Figure 9.** Evolution of characteristic multipliers in several bifurcation zones. (*a*). flip bifurcation zone (*τ* = 0, *ks* > 0 *N* > −1). (*b*). fold bifurcation zones (*τ* > 0, *ks* > 0 *N* < −1). (*c*). Neimark-Sacker bifurcation zones (*τ* > 1, *ks* > 0 *N* > −1).

Equation (24) is the solution of the variational system where *z* = *e*−*μT*.

*e***M**1(*T*−*d*)*e***M**2(*T*−*d*)*e*

$$\mathbf{p}\left(T\right) = \mathbf{Q}\_1 \mathbf{p}\_\ell\left(0\right) \tag{24}$$

The matrix **<sup>Q</sup>**<sup>1</sup> is given by equation (25), where *<sup>α</sup>*<sup>3</sup> = *<sup>γ</sup>* <sup>2</sup> <sup>+</sup> *<sup>β</sup>* 2 .

10 Nonlinearity, Bifurcation and Chaos - Theory and Applications

<sup>−</sup> <sup>=</sup> *<sup>e</sup>***M**1(*T*−*d*)*e***M**2(*T*−*d*)**<sup>p</sup>**

 *e***M**<sup>1</sup> *<sup>d</sup>* <sup>2</sup> *e***M**<sup>2</sup> *<sup>d</sup>* <sup>2</sup> + *e*

<sup>2</sup> *e***M**<sup>2</sup> *<sup>d</sup>*

(a) (b)

(c) (d)

zone. (*c*). fold bifurcation zones. (*d*). Neimark-Sacker bifurcation zones.

**p** *<sup>T</sup>* <sup>−</sup> *<sup>d</sup>* 2 <sup>+</sup> <sup>=</sup> 

3). Finally, we compute the solution in the last interval between *<sup>t</sup>* = (*<sup>T</sup>* <sup>−</sup> *<sup>d</sup>*

**p** (*T*) = *e***M**<sup>1</sup> *<sup>d</sup>*

**Figure 8.** 3D parameter space of Digital-PWM switched converter and bifurcation zones. (*a*). (*ks*, *N*, *τ*) space where the lines *ks* = 0 and *N* = −1 separate each plane in four sub-spaces. (*b*). flip bifurcation

> <sup>2</sup> *e***M**<sup>2</sup> *<sup>d</sup>* <sup>2</sup> **p** *<sup>T</sup>* <sup>−</sup> *<sup>d</sup>* 2 +

*e***M**1(*T*−*d*)*e***M**2(*T*−*d*)*e***M**<sup>1</sup> *<sup>d</sup>*

<sup>2</sup> *e***M**<sup>2</sup> *<sup>d</sup>* <sup>2</sup> +

 *<sup>d</sup>* 2 +

<sup>−</sup>*μ*((*τ*+1)*T*<sup>−</sup> *<sup>d</sup>*

<sup>2</sup> + *e***M**1(*T*−*d*)*e***M**2(*T*−*d*)*e*

<sup>−</sup>*μ*(*τT*<sup>+</sup> *<sup>d</sup>*

<sup>2</sup> )**Hp** (0)

<sup>2</sup> )**H p** (0)

<sup>−</sup>*μ*((*τ*+1)*T*<sup>−</sup> *<sup>d</sup>*

<sup>2</sup> )**H p** (0)

<sup>2</sup> ) and *T*.

**p** *<sup>T</sup>* <sup>−</sup> *<sup>d</sup>* 2 

<sup>−</sup> <sup>=</sup> *<sup>e</sup>***M**1(*T*−*d*)*e***M**2(*T*−*d*)

*e***M**1(*T*−*d*)*e***M**2(*T*−*d*)*e***M**<sup>1</sup> *<sup>d</sup>*

**p** *<sup>T</sup>* <sup>−</sup> *<sup>d</sup>* 2 

in the discontinuity,

**p** *<sup>T</sup>* <sup>−</sup> *<sup>d</sup>* 2 <sup>+</sup> <sup>=</sup>  **p** *<sup>T</sup>* <sup>−</sup> *<sup>d</sup>* 2 <sup>+</sup> <sup>=</sup> **<sup>p</sup>** *<sup>T</sup>* <sup>−</sup> *<sup>d</sup>* 2 <sup>−</sup> <sup>+</sup> *<sup>e</sup>*

$$\mathbf{Q}\_1 = \left( z e^{-a\_3 T} e^{\mathbf{M}\_2 T} + z^{(\tau+1)} e^{-a\_3 \left(T - \frac{d}{2}\right)} e^{\mathbf{M}\_2 \left(T - \frac{d}{2}\right)} \mathbf{H} + z^{(\tau+1)} e^{-a\_3 \left(\frac{d}{2}\right)} e^{\mathbf{M}\_2 \left(\frac{d}{2}\right)} \mathbf{H} \right) \tag{25}$$

The existence of T-periodic solution, i.e. **p** (*T*) = **p** (0), depends on equation (26) is satisfied.

$$(\mathbf{Q}\_1 - I)\,\mathbf{p}\,(0) = \mathbf{0} \tag{26}$$

The stability of the periodic solution depends on whether the real part of each Floquet exponent is negative or not. If det(**Q**<sup>1</sup> − **I**) = 0 implies *μ* negative.

We fix the parameter values in *γ* = 0.35; *β* = 0; *T* = 0.1767; *ks* = 4.5; *N* = 0 and *ref* = 0.8. Equation (27) shows equation (26) in function of the number of delays (*τ*). The equation is a polynomial in *z* of degree 2(*τ* + 1).

$$0.1e-20z^{2(\tau+1)} - 1.867z^{(\tau+2)} + 1.911z^{(\tau+1)} + 0.93z^2 - 1.899z + 1 = 0\tag{27}$$

Assuming real-time behavior, i.e. *τ* = 0, the determinant is the second order polynomial of equation (28). The evolution of the real part of the Floquet exponents as parameter *ks* varies is displayed in figure 3(*a*). The T-periodic solution is stable for *ks* > 3.24.

$$(\frac{1e-10}{k\_s^2} - \frac{0.00215387}{k\_s} - 0.9466771)z^2 + (\frac{0.3443753}{k\_s} - 0.0535075)z + 1 = 0\tag{28}$$

Control Parameters Stability Limit (*τ* = 0, *N* = 0) *ks* <sup>≈</sup> 3.24 (*τ* = 1, *N* = 2) *ks* <sup>≈</sup> 0.46 (*τ* = 2, *N* = 3) *ks* <sup>≈</sup> 1.19 (*τ* = 3, *N* = 4) *ks* <sup>≈</sup> 2.99 (*τ* = 4, *N* = 6) *ks* <sup>≈</sup> 2.72 (*τ* = 5,*N* = 8) *ks* <sup>≈</sup> 3.25 (*τ* = 6,*N* = 10) *ks* <sup>≈</sup> 4.21

**Table 2.** Critical value of stability (*kscr*) of buck converter controlled with ZAD-FPIC with several delay

In previous section, we show that the procedure based on variational equation can be used to compute Floquet exponents for any number of delays (*τ*). In this section, 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

The evaluation of the jacobian matrix is necessary to compute characteristic multipliers and Lyapunov exponents in PWM switched converters. The dimension of the jacobian matrix depends on the delay number considered in the control law. The order of Jacobian matrix is

Poincaré map of the PWM switched converter can be used to determine the stability of 1-periodic orbit. Equation (30) presents the Poincaré map of synchronous buck converter with

Real-time control law implies that the duty cycle *dk* depends on state variables in the instant *kT*, i.e., *dk* = *c*1*x*1(*kT*) + *c*2*x*2(*kT*) + *c*3. Therefore, Poincaré map (30) can be written as follows.

> *∂ f*<sup>1</sup> *∂x*1(*kT*)

> *∂x*1(*kT*)

= = **<sup>A</sup>***dk*/2 <sup>−</sup>**I**)**B**<sup>−</sup> *<sup>e</sup>*

*f*1(*x*1(*kT*), *x*2(*kT*))

*∂ f*<sup>1</sup> *<sup>∂</sup>x*2(*kT*) *<sup>∂</sup> <sup>f</sup>*<sup>2</sup> ⎤

*∂ f*<sup>2</sup> *∂x*2(*kT*) **<sup>A</sup>***dk*/2**A**−1(*e*

Floquet Exponents and Bifurcations in Switched Converters 39

*<sup>f</sup>*2(*x*1(*kT*), *<sup>x</sup>*2(*kT*)) (31)

⎦ (32)

**<sup>A</sup>**(*T*−*dk* ) <sup>−</sup>**I**)**<sup>B</sup>** (30)

**<sup>A</sup>**(*T*−*dk*/2) +**I**)**A**−1(*e*

Jacobian matrix of the system with *τ* = 0 can be computed with equation (32).

⎡ ⎣

**A***n*<sup>0</sup> =

*x*1((*k* + 1)*T*) *x*2((*k* + 1)*T*)

**4. Stability of fixed points in delayed PWM switched converters**

**4.1. Stability of 1-periodic orbit using Jacobian matrix**

numbers.(*τ* = 0 to *τ* = 6)

disadvantage [13].

centered PWM control.

**<sup>A</sup>***T***x**(*kT*)+(*e*

**x**((*k* +1)*T*) = *e*

2(*τ* + 1).

**Figure 10.** Examples of bifurcation diagrams in each zone. (*a*). flip bifurcation diagram (*τ* = 0, *ks* > 0 *N* > −1). (*b*). fold bifurcation diagram (*τ* > 0, *ks* > 0 *N* < −1). (*c*). Neimark-Sacker bifurcation diagram (*τ* > 1, *ks* > 0 *N* > −1).

Assuming one-delay period, i.e. *τ* = 1, the determinant is the forth order polynomial of equation (29). The evolution of the real part of the Floquet exponents as parameter *ks* varies is displayed in figure 3(*b*). In this case, one Floquet exponent has positive real part for any *ks*. Therefore, ZAD strategy should be combined with FPIC (*N* �= 0) to reach stable solutions.

$$\frac{1\varepsilon - 9}{k\_s^2} z^4 - (1.8867 + \frac{0.002154}{k\_s}) z^3 + (\frac{0.344375}{k\_s} + 2.79635) z^2 - 1.90983z + 1 = 0\tag{29}$$

For *τ* > 1, ZAD strategy is not sufficient to stabilize 1T-periodic orbit and ZAD-FPIC scheme


**Table 1.** Critical value of stability (*Ncr*) of buck converter controlled with ZAD-FPIC with several delay numbers. Figure 4 (*b*) shows the evolution of real part of Floquet exponents.

is necessary. Figure 4(*a*) shows the evolution of Floquet exponents when *ks* is varied for several delay numbers and *N* = 0. The number of Floquet exponents with positive real part increases as the delay number grows.

Figure 4 (*b*) shows the results of ZAD-FPIC scheme when *ks* = 4.5 and *N* is varied between [0; 30]. The critical value of stability (*Ncr*) increases as the delay number grows. Table 1 summarizes the behavior of critical value for different delays.

The behavior of the critical value is similar when *N* is fixed and *ks* is varied for several delay numbers. The value *kscr* increases as the delay number grows. Table 2 shows this condition.


**Table 2.** Critical value of stability (*kscr*) of buck converter controlled with ZAD-FPIC with several delay numbers.(*τ* = 0 to *τ* = 6)
