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

12 Nonlinearity, Bifurcation and Chaos - Theory and Applications

(a) (b) (c)

**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

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.

<sup>+</sup> 2.79635)*z*<sup>2</sup> <sup>−</sup> 1.90983*<sup>z</sup>* <sup>+</sup> <sup>1</sup> <sup>=</sup> 0 (29)

)*z*<sup>3</sup> + ( 0.344375 *ks*

For *τ* > 1, ZAD strategy is not sufficient to stabilize 1T-periodic orbit and ZAD-FPIC scheme Control Parameters Stability Limit (*τ* = 0), (*ks* = 4.5) *Ncr* <sup>≈</sup> <sup>0</sup> (*τ* = 1), (*ks* = 4.5) *Ncr* <sup>≈</sup> 0.99 (*τ* = 2), (*ks* = 4.5) *Ncr* <sup>≈</sup> 2.32 (*τ* = 3), (*ks* = 4.5) *Ncr* <sup>≈</sup> 3.79 (*τ* = 4), (*ks* = 4.5) *Ncr* <sup>≈</sup> 5.53 (*τ* = 5), (*ks* = 4.5) *Ncr* <sup>≈</sup> 7.55 (*τ* = 6), (*ks* = 4.5) *Ncr* <sup>≈</sup> 9.89 **Table 1.** Critical value of stability (*Ncr*) of buck converter controlled with ZAD-FPIC with several delay

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

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

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.

diagram (*τ* > 1, *ks* > 0 *N* > −1).

*<sup>z</sup>*<sup>4</sup> <sup>−</sup> (1.8867 <sup>+</sup>

increases as the delay number grows.

0.002154 *ks*

numbers. Figure 4 (*b*) shows the evolution of real part of Floquet exponents.

summarizes the behavior of critical value for different delays.

1*e* − 9 *k*2 *s*

> 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 2(*τ* + 1).

> 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 centered PWM control.

$$\mathbf{x}((k+1)T) = e^{\mathbf{A}T}\mathbf{x}(kT) + (e^{\mathbf{A}(T-d\_{k}/2)} + \mathbf{I})\mathbf{A}^{-1}(e^{\mathbf{A}d\_{k}/2} - \mathbf{I})\mathbf{B} - e^{\mathbf{A}d\_{k}/2}\mathbf{A}^{-1}(e^{\mathbf{A}(T-d\_{k})} - \mathbf{I})\mathbf{B} \tag{30}$$

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.

$$\begin{aligned} \mathbf{x}\_1((k+1)T) &= f\_1(\mathbf{x}\_1(kT), \mathbf{x}\_2(kT)) \\ \mathbf{x}\_2((k+1)T) &= f\_2(\mathbf{x}\_1(kT), \mathbf{x}\_2(kT)) \end{aligned} \tag{31}$$

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

$$\mathbf{A}\_{n0} = \begin{bmatrix} \frac{\partial f\_1}{\partial x\_1(kT)} & \frac{\partial f\_1}{\partial x\_2(kT)}\\ \frac{\partial f\_2}{\partial x\_1(kT)} & \frac{\partial f\_2}{\partial x\_2(kT)} \end{bmatrix} \tag{32}$$

#### 14 Nonlinearity, Bifurcation and Chaos - Theory and Applications 40 Nonlinearity, Bifurcation and Chaos – Theory and Applications Floquet Exponents and Bifurcations in Switched Converters <sup>15</sup>

The matrix **A***n*<sup>0</sup> should be evaluated in the fixed point (**A***n*<sup>0</sup> = � *<sup>∂</sup>***<sup>f</sup>** *∂***x***<sup>i</sup>* � *F*.*P*. ). In this case, we define *F*.*P*. as (*ref* , *γref*). Its eigenvalues (or characteristic multipliers) determine stability properties of the fixed point. The 1-periodic orbit is asymptotically stable if all characteristic multipliers have magnitude less than one (|*mi*| < 1); it is unstable if at least one eigenvalue has magnitude greater than one (|*mi*| > 1).

One-delay control law implies that the duty cycle *dk* depends on state variables in the instant (*k* − 1)*T*, i.e., *dk* = *c*1*x*1((*k* − 1)*T*) + *c*2*x*2((*k* − 1)*T*) + *c*3. Two additional state variables can be defined *x*3(*kT*) = *x*1((*k* − 1)*T*) and *x*4(*kT*) = *x*2((*k* − 1)*T*). Therefore, *dk* = *c*1*x*3(*kT*) + *c*2*x*4(*kT*) + *c*3.

In this case, Poincaré map (30) can be written as equation (33).

$$\begin{array}{ll} \mathbf{x}\_1((k+1)T) = f\_1(\mathbf{x}\_1(kT), \mathbf{x}\_2(kT), \mathbf{x}\_3(kT), \mathbf{x}\_4(kT))\\ \mathbf{x}\_2((k+1)T) = f\_2(\mathbf{x}\_1(kT), \mathbf{x}\_2(kT), \mathbf{x}\_3(kT), \mathbf{x}\_4(kT))\\ \mathbf{x}\_3((k+1)T) = & \mathbf{x}\_1(kT)\\ \mathbf{x}\_4((k+1)T) = & \mathbf{x}\_2(kT) \end{array} \tag{33}$$

example, we can compute the Floquet exponents (*μi*) for any delay and later we can apply the relations *mi* = *eμiT* and *λ<sup>i</sup>* = *μiT* to find characteristic multipliers and Lyapunov exponents,

Floquet Exponents and Bifurcations in Switched Converters 41

Figure 6 shows the evolution of Floquet and Lyapunov exponents when the duty cycle is computed without delay and with one delay. The critic values are the equals using any

Figure 7 shows the evolution of Floquet exponents and characteristic multipliers in the complex plane of each representation. In both cases, the parameter *N* is varied in the range [0; 30] with *ks* = 4.5 and *τ* = 1. Imaginary axis is the stability limit of the Floquet exponents

In this section, we analyze types of bifurcations in the buck converter controlled with Delayed ZAD-FPIC scheme using the procedure based on Floquet exponents described in previous sections. We transform Floquet exponents in characteristic multipliers using the equivalences

If at least one characteristic multiplier is outside of the unit circle then the system has an unstable fixed point and nonlinear phenomena as quasi-periodicity and chaos could be present. In the boundary, the smooth bifurcations (flip, fold and Neimark-Sacker) are present. The presence of the three smooth bifurcations in the same converter is not common and this

Control parameters *ks* and *N* can be varied in **R** with the exception of *ks* = 0 and *N* + 1 = 0 (because the control law is not defined there). Parameter *τ* can be varied in **Z**. The 3D-parameter space (*ks*, *N*, *τ*) is discontinuous due to the discrete delays (*τ* = 0, 1, 2, 3, ) and the undefined planes (*ks* = 0 and *N* + 1 = 0). Figure 8 (*a*) shows a representation of the

The two-dimensional plane (*ks*, *N*) can be divided into four regions: region I: *ks* > 0 and *N* > −1; region II: *ks* < 0 and *N* > −1; region III: *ks* < 0 and *N* < −1 ; and region IV: *ks* > 0 and *N* < −1. Fold zones, flip zones and Neimark-Sacker zones can be identified in the control space. The fold bifurcation is an alarm for duty cycle saturation in *d* = 0% or *d* = 100%; the flip bifurcation signals a doubling period and the Neimark-Sacker bifurcation

Computer simulations are given for the purpose of illustration and verification. Next, we

This bifurcation is associated with the appearance of a negative real characteristic multiplier in the unit cycle boundary (*mi* = −1). Figure 9 (*a*) shows the evolution of characteristic

locus, while unity circle is the stability limit of the characteristic multipliers locus.

**5. Bifurcations in Buck converter with delayed ZAD-FPIC**

fact has not been reported widely in Digital-PWM switched converters [14].

present the three bifurcations types in the 3D-parameter space.

multipliers when *N* is varied in a positive range for several *ks* values.

**5.1. Flip bifurcations in** (*ks*, *N*, *τ*) **space**

method. Therefore, both methods give the same information.

respectively.

shown in figure 5.

control parameter space.

is related to 2D-torus birth.

Jacobian matrix of the system with *τ* = 1 can be computed with equation (34).

$$\mathbf{A}\_{\eta1} = \begin{bmatrix} \frac{\partial f\_1}{\partial \mathbf{x}\_1(kT)} & \frac{\partial f\_1}{\partial \mathbf{x}\_2(kT)} & \frac{\partial f\_1}{\partial \mathbf{x}\_3(kT)} & \frac{\partial f\_1}{\partial \mathbf{x}\_4(kT)}\\ \frac{\partial f\_2}{\partial \mathbf{x}\_1(kT)} & \frac{\partial f\_2}{\partial \mathbf{x}\_2(kT)} & \frac{\partial f\_2}{\partial \mathbf{x}\_3(kT)} & \frac{\partial f\_2}{\partial \mathbf{x}\_4(kT)}\\ 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \end{bmatrix} \tag{34}$$

The matrix **A***n*<sup>1</sup> should be evaluated in the fixed point (**A***n*<sup>1</sup> = � *<sup>∂</sup>***<sup>f</sup>** *∂***x***<sup>i</sup>* � *F*.*P*. ). In this case, we define *F*.*P*. as (*ref* , *γref* ,*ref* , *γref*).

Four characteristic multipliers are computed. The 1-periodic orbit is asymptotically stable if the four characteristic multipliers have magnitude less than one.

Characteristic multipliers for delayed PWM control law with *τ* > 1 can be computed following the same procedure. However, The order of Jacobian matrix increases as the delay number grows.

Now, we compute Lyapunov exponents using a numeric routine. This algorithm is based on the definition of Lyapunov exponents. Equation (35) synthesizes this procedure. Poincaré map is used to compute the values of state variables. Jacobian matrix should be known to compute the eigenvalues *qi* in each iteration *k*.

$$\lambda\_i = \lim\_{M \to \infty} \left\{ \frac{1}{M} \sum\_{k=0}^{M} \log \left| q\_i \left( A\_{\eta j} \left( \mathfrak{x}(k) \right) \right) \right| \right\} \tag{35}$$
